diff --git "a/OE_MM_maths_en_COMP.tsv" "b/OE_MM_maths_en_COMP.tsv"
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+++ "b/OE_MM_maths_en_COMP.tsv"
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+id	question	solution	final_answer	context	image	modality	difficulty	is_multiple_answer	unit	answer_type	error	question_type	subfield	subject	language
+0	"Turbo the snail sits on a point on a circle with circumference 1. Given an infinite sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \ldots$. Turbo successively crawls distances $c_{1}, c_{2}, c_{3}, \ldots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
+
+For example, if the sequence $c_{1}, c_{2}, c_{3}, \ldots$ is $0.4,0.6,0.3, \ldots$, then Turbo may start crawling as follows:
+<image_1>
+
+Determine the largest constant $C>0$ with the following property: for every sequence of positive real numbers $c_{1}, c_{2}, c_{3}, \ldots$ with $c_{i}<C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across."	"['The largest possible $C$ is $C=\\frac{1}{2}$.\n\nFor $0<C \\leqslant \\frac{1}{2}$, Turbo can simply choose an arbitrary point $P$ (different from its starting point) to avoid. When Turbo is at an arbitrary point $A$ different from $P$, the two arcs $A P$ have total length 1; therefore, the larger of the two the arcs (or either arc in case $A$ is diametrically opposite to $P$ ) must have length $\\geqslant \\frac{1}{2}$. By always choosing this larger arc (or either arc in case $A$ is diametrically opposite to $P$ ), Turbo will manage to avoid the point $P$ forever.\n\nFor $C>\\frac{1}{2}$, we write $C=\\frac{1}{2}+a$ with $a>0$, and we choose the sequence\n\n$$\n\\frac{1}{2}, \\quad \\frac{1+a}{2}, \\quad \\frac{1}{2}, \\quad \\frac{1+a}{2}, \\quad \\frac{1}{2}, \\ldots\n$$\n\nIn other words, $c_{i}=\\frac{1}{2}$ if $i$ is odd and $c_{i}=\\frac{1+a}{2}<C$ when $i$ is even. We claim Turbo must eventually visit all points on the circle. This is clear when it crawls in the same direction two times in a row; after all, we have $c_{i}+c_{i+1}>1$ for all $i$. Therefore, we are left with the case that Turbo alternates crawling clockwise and crawling counterclockwise. If it, without loss of generality, starts by going clockwise, then it will always crawl a distance $\\frac{1}{2}$ clockwise followed by a distance $\\frac{1+a}{2}$ counterclockwise. The net effect is that it crawls a distance $\\frac{a}{2}$ counterclockwise. Because $\\frac{a}{2}$ is positive, there exists a positive integer $N$ such that $\\frac{a}{2} \\cdot N>1$. After $2 N$ crawls, Turbo will have crawled a distance $\\frac{a}{2}$ counterclockwise $N$ times, therefore having covered a total distance of $\\frac{a}{2} \\cdot N>1$, meaning that it must have crawled over all points on the circle.\n\nNote: Every sequence of the form $c_{i}=x$ if $i$ is odd, and $c_{i}=y$ if $i$ is even, where $0<x, y<C$, such that $x+y \\geqslant 1$, and $x \\neq y$ satisfies the conditions with the same argument. There might be even more possible examples.'
+ ""To show that $C\\le \\frac12$\n\nWe consider the following related problem:\n\nWe assume instead that the snail Chet is moving left and right on the real line. Find the size $M$ of the smallest (closed) interval, that we cannot force Chet out of, using a sequence of real numbers $d_{i}$ with $0<d_{i}<1$ for all $i$.\n\nThen $C=1 / M$. Indeed if for every sequence $c_{1}, c_{2}, \\ldots$, with $c_{i}<C$ there exists a point that Turbo can avoid, then the circle can be cut open at the avoided point and mapped to an interval of size $M$ such that Chet can stay inside this interval for any sequence of the from $c_{1} / C, c_{2} / C, \\ldots$, see Figure 5 . However, all sequences $d_{1}, d_{2}, \\ldots$ with $d_{i}<1$ can be written in this form. Similarly if for every sequence $d_{1}, d_{2}, \\ldots$, there exists an interval of length smaller or equal $M$ that we cannot force Chet out of, this projects to a subset of the circle, that we cannot force Turbo out of using any sequence of the form $d_{1} / M, d_{2} / M, \\ldots$. These are again exactly all the sequences with elements in $[0, C)$.\n\n<img_3381>\n\nFigure 5: Chet and Turbo equivalence\n\nClaim: $M \\geqslant 2$.\n\nProof. Suppose not, so $M<2$. Say $M=2-2 \\varepsilon$ for some $\\varepsilon>0$ and let $[-1+\\varepsilon, 1-\\varepsilon]$ be a minimal interval, that Chet cannot be forced out of. Then we can force Chet arbitrarily close to $\\pm(1-\\varepsilon)$. In partiular, we can force Chet out of $\\left[-1+\\frac{4}{3} \\varepsilon, 1-\\frac{4}{3} \\varepsilon\\right]$ by minimality of $M$. This means that there exists a sequence $d_{1}, d_{2}, \\ldots$ for which Chet has to leave $\\left[-1+\\frac{4}{3} \\varepsilon, 1-\\frac{4}{3} \\varepsilon\\right]$, which means he ends up either in the interval $\\left[-1+\\varepsilon,-1+\\frac{4}{3} \\varepsilon\\right)$ or in the interval $\\left(1-\\frac{4}{3} \\varepsilon, 1-\\varepsilon\\right]$.\n\nNow consider the sequence,\n\n$$\nd_{1}, 1-\\frac{7}{6} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{7}{6} \\varepsilon, d_{2}, 1-\\frac{7}{6} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{7}{6} \\varepsilon, d_{3}, \\ldots\n$$\n\nobtained by adding the sequence $1-\\frac{7}{6} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{7}{6} \\varepsilon$ in between every two steps. We claim that this sequence forces Chet to leave the larger interval $[-1+\\varepsilon, 1-\\varepsilon]$. Indeed no two consecutive elements in the sequence $1-\\frac{7}{6} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{2}{3} \\varepsilon, 1-\\frac{7}{6} \\varepsilon$ can have the same sign, because the sum of any two consecutive terms is larger than $2-2 \\varepsilon$ and Chet would leave the interval $[-1+\\varepsilon, 1-\\varepsilon]$. It follows that the $\\left(1-\\frac{7}{6} \\varepsilon\\right)$ 's and the $\\left(1-\\frac{2}{3} \\varepsilon\\right)$ 's cancel out, so the position after $d_{k}$ is the same as before $d_{k+1}$. Hence, the positions after each $d_{k}$ remain the same as in the original sequence. Thus, Chet is also forced to the boundary in the new sequence.\n\nIf Chet is outside the interval $\\left[-1+\\frac{4}{3} \\varepsilon, 1-\\frac{4}{3} \\varepsilon\\right]$, then Chet has to move $1-\\frac{7}{6} \\varepsilon$ towards 0 , and ends in $\\left[-\\frac{1}{6} \\varepsilon, \\frac{1}{6} \\varepsilon\\right]$. Chet then has to move by $1-\\frac{2}{3} \\varepsilon$, which means that he has to leave the interval $[-1+\\varepsilon, 1-\\varepsilon]$. Indeed the absolute value of the final position is at least $1-\\frac{5}{6} \\varepsilon$. This contradicts the assumption, that we cannot force Chet out of $[-1+\\varepsilon, 1-\\varepsilon]$. Hence $M \\geqslant 2$ as needed.""]"	['$\\frac{1}{2}$']		['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+1	"In the diagram, $\angle A B F=41^{\circ}, \angle C B F=59^{\circ}, D E$ is parallel to $B F$, and $E F=25$. If $A E=E C$, determine the length of $A E$, to 2 decimal places.
+
+<image_1>"	['Let the length of $A E=E C$ be $x$.\n\nThen $A F=x-25$.\n\nIn, $\\triangle B C F, \\frac{x+25}{B F}=\\tan \\left(59^{\\circ}\\right)$.\n\nIn $\\triangle A B F, \\frac{x-25}{B F}=\\tan \\left(41^{\\circ}\\right)$.\n\nSolving for $B F$ in these two equations and equating,\n\n$$\nB F=\\frac{x+25}{\\tan 59^{\\circ}}=\\frac{x-25}{\\tan 41^{\\circ}}\n$$\n\nso $\\quad\\left(\\tan 41^{\\circ}\\right)(x+25)=\\left(\\tan 59^{\\circ}\\right)(x-25)$\n\n$$\n\\begin{aligned}\n25\\left(\\tan 59^{\\circ}+\\tan 41^{\\circ}\\right) & =x\\left(\\tan 59^{\\circ}-\\tan 41^{\\circ}\\right) \\\\\nx & =\\frac{25\\left(\\tan 59^{\\circ}+\\tan 41^{\\circ}\\right)}{\\tan 59^{\\circ}-\\tan 41^{\\circ}} \\\\\nx & \\doteq 79.67 .\n\\end{aligned}\n$$\n\n<img_3420>\n\nTherefore the length of $A E$ is 79.67 .']	['79.67']		['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iigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKrX9lb6jZTWV3Ak9tOhjljcZDqeCKs0UAeeaZ4A1vwp5kHhTxKIdMZi66fqNr9oSInk7XDKwGe38zk1LqPgzxV4it3tda8ZtFZSgrLbaXZLAXU9QZGZjj26V31FAFXTLCHStLtdOtgwgtYUhiDHJ2qAoz+Aq1RRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAH/9k=']	Multimodal	Competition	False		Numerical	1e-1	Open-ended	Geometry	Math	English
+2	"In triangle $A B C, A B=B C=25$ and $A C=30$. The circle with diameter $B C$ intersects $A B$ at $X$ and $A C$ at $Y$. Determine the length of $X Y$.
+
+<image_1>"	"['Join $B Y$. Since $B C$ is a diameter, then $\\angle B Y C=90^{\\circ}$. Since $A B=B C, \\triangle A B C$ is isosceles and $B Y$ is an altitude in $\\triangle A B C$, then $A Y=Y C=15$.\n\nLet $\\angle B A C=\\theta$.\n\nSince $\\triangle A B C$ is isosceles, $\\angle B C A=\\theta$.\n\nSince $B C Y X$ is cyclic, $\\angle B X Y=180-\\theta$ and so $\\angle A X Y=\\theta$.\n\n<img_3475>\n\nThus $\\triangle A X Y$ is isosceles and so $X Y=A Y=15$.\n\nTherefore $X Y=15$.'
+ 'Join $B Y . \\angle B Y C=90^{\\circ}$ since it is inscribed in a semicircle.\n\n$\\triangle B A C$ is isosceles, so altitude $B Y$ bisects the base.\n\nTherefore $B Y=\\sqrt{25^{2}-15^{2}}=20$.\n\nJoin $C X . \\angle C X B=90^{\\circ}$ since it is also inscribed in a semicircle.\n\n<img_3739>\n\nThe area of $\\triangle A B C$ is\n\n$$\n\\begin{aligned}\n\\frac{1}{2}(A C)(B Y) & =\\frac{1}{2}(A B)(C X) \\\\\n\\frac{1}{2}(30)(20) & =\\frac{1}{2}(25)(C X) \\\\\nC X & =\\frac{600}{25}=24 .\n\\end{aligned}\n$$\n\nFrom $\\triangle A B Y$ we conclude that $\\cos \\angle A B Y=\\frac{B Y}{A B}=\\frac{20}{25}=\\frac{4}{5}$.\n\nIn $\\Delta B X Y$, applying the Law of Cosines we get $(X Y)^{2}=(B X)^{2}+(B Y)^{2}-2(B X)(B Y) \\cos \\angle X B Y$.\n\nNow (by Pythagoras $\\triangle B X C$ ),\n\n\n\n$$\n\\begin{aligned}\nB X^{2} & =B C^{2}-C X^{2} \\\\\n& =25^{2}-24^{2} \\\\\n& =49 \\\\\nB X & =7 .\n\\end{aligned}\n$$\n\nTherefore $X Y^{2}=7^{2}+20^{2}-2(7)(20) \\frac{4}{5}$\n\n$$\n=49+400-224\n$$\n\n$$\n=225 \\text {. }\n$$\n\nTherefore $X Y=15$.']"	['15']		['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ooAKKKKACiiigAooooA//9k=']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+3	"Points $P$ and $Q$ are located inside the square $A B C D$ such that $D P$ is parallel to $Q B$ and $D P=Q B=P Q$. Determine the minimum possible value of $\angle A D P$.
+
+<image_1>"	"['Placing the information on the coordinate axes, the diagram is indicated to the right.\n\nWe note that $P$ has coordinates $(a, b)$.\n\nBy symmetry (or congruency) we can label lengths $a$ and $b$ as shown. Thus $Q$ has coordinates $(2-a, 2-b)$.\n\nSince $P D=P Q, a^{2}+b^{2}=(2-2 a)^{2}+(2-2 b)^{2}$\n\n$$\n\\begin{aligned}\n& 3 a^{2}+3 b^{2}-8 a-8 b+8=0 \\\\\n& \\left(a-\\frac{4}{3}\\right)^{2}+\\left(b-\\frac{4}{3}\\right)^{2}=\\frac{8}{9}\n\\end{aligned}\n$$\n\n<img_3483>\n\n$P$ is on a circle with centre $O\\left(\\frac{4}{3}, \\frac{4}{3}\\right)$ with $r=\\frac{2}{3} \\sqrt{2}$.\n\nThe minimum angle for $\\theta$ occurs when $D P$ is tangent to the circle.\n\nSo we have the diagram noted to the right.\n\nSince $O D$ makes an angle of $45^{\\circ}$ with the $x$-axis then $\\angle P D O=45-\\theta$ and $O D=\\frac{4}{3} \\sqrt{2}$.\n\nTherefore $\\sin (45-\\theta)=\\frac{\\frac{2}{3} \\sqrt{2}}{\\frac{4}{3} \\sqrt{2}}=\\frac{1}{2}$ which means $45^{\\circ}-\\theta=30^{\\circ}$ or $\\theta=15^{\\circ}$.\n\nThus the minimum value for $\\theta$ is $15^{\\circ}$.\n\n<img_3977>'
+ 'Let $A B=B C=C D=D A=1$.\n\nJoin $D$ to $B$. Let $\\angle A D P=\\theta$. Therefore, $\\angle P D B=45-\\theta$.\n\nLet $P D=a$ and $P B=b$ and $P Q=\\frac{a}{2}$.\n\n\n\nWe now establish a relationship between $a$ and $b$.\n\nIn $\\triangle P D B, b^{2}=a^{2}+2-2(a)(\\sqrt{2}) \\cos (45-\\theta)$\n\n$$\n\\text { or, } \\quad \\cos (45-\\theta)=\\frac{a^{2}-b^{2}+2}{2 \\sqrt{2} a}\n$$\n\n<img_3307>\n\nIn $\\triangle P D R,\\left(\\frac{a}{2}\\right)^{2}=a^{2}+\\left(\\frac{\\sqrt{2}}{2}\\right)^{2}-2 a \\frac{\\sqrt{2}}{2} \\cos (45-\\theta)$\n\nor, $\\cos (45-\\theta)=\\frac{\\frac{3}{4} a^{2}+\\frac{1}{2}}{a \\sqrt{2}}$\n\nComparing (1) and (2) gives, $\\frac{a^{2}-b^{2}+2}{2 \\sqrt{2} a}=\\frac{\\frac{3}{4} a^{2}+\\frac{1}{2}}{a \\sqrt{2}}$.\n\nSimplifying this, $a^{2}+2 b^{2}=2$\n\n$$\n\\text { or, } \\quad b^{2}=\\frac{2-a^{2}}{2}\n$$\n\nNow $\\cos (45-\\theta)=\\frac{a^{2}+2-\\left(\\frac{2-a^{2}}{2}\\right)}{2 a \\sqrt{2}}=\\frac{1}{4 \\sqrt{2}}\\left(3 a+\\frac{2}{a}\\right)$.\n\nNow considering $3 a+\\frac{2}{a}$, we know $\\left(\\sqrt{3 a}-\\sqrt{\\frac{2}{a}}\\right)^{2} \\geq 0$\n\n$$\n\\text { or, } \\quad 3 a+\\frac{2}{a} \\geq 2 \\sqrt{6}\n$$\n\nThus, $\\cos (45-\\theta) \\geq \\frac{1}{4 \\sqrt{2}}(2 \\sqrt{6})=\\frac{\\sqrt{3}}{2}$\n\n$$\n\\cos (45-\\theta) \\geq \\frac{\\sqrt{3}}{2}\n$$\n\n$\\cos (45-\\theta)$ has a minimum value for $45^{\\circ}-\\theta=30^{\\circ}$ or $\\theta=15^{\\circ}$.'
+ 'Join $B D$. Let $B D$ meet $P Q$ at $M$. Let $\\angle A D P=\\theta$.\n\nBy interior alternate angles, $\\angle P=\\angle Q$ and $\\angle P D M=\\angle Q B M$.\n\nThus $\\triangle P D M \\cong \\triangle Q B M$ by A.S.A., so $P M=Q M$ and $D M=B M$.\n\nSo $M$ is the midpoint of $B D$ and the centre of the square.\n\n\n\nWithout loss of generality, let $P M=1$. Then $P D=2$.\n\nSince $\\theta+\\alpha=45^{\\circ}$ (see diagram), $\\theta$ will be minimized when $\\alpha$ is maximized.\n\n<img_3749>\n\nConsider $\\triangle P M D$.\n\nUsing the sine law, $\\frac{\\sin \\alpha}{1}=\\frac{\\sin (\\angle P M D)}{2}$.\n\nTo maximize $\\alpha$, we maximize $\\sin \\alpha$.\n\nBut $\\sin \\alpha=\\frac{\\sin (\\angle P M D)}{2}$, so it is maximized when $\\sin (\\angle P M D)=1$.\n\nIn this case, $\\sin \\alpha=\\frac{1}{2}$, so $\\alpha=30^{\\circ}$.\n\nTherefore, $\\theta=45^{\\circ}-30^{\\circ}=15^{\\circ}$, and so the minimum value of $\\theta$ is $15^{\\circ}$.'
+ 'We place the diagram on a coordinate grid, with $D(0,0)$, $C(1,0), B(0,1), A(1,1)$.\n\nLet $P D=P Q=Q B=a$, and $\\angle A D P=\\theta$.\n\nDrop a perpendicular from $P$ to $A D$, meeting $A D$ at $X$.\n\nThen $P X=a \\sin \\theta, D X=a \\cos \\theta$.\n\nTherefore the coordinates of $P$ are $(a \\sin \\theta, a \\cos \\theta)$.\n\nSince $P D \\| B Q$, then $\\angle Q B C=\\theta$.\n\nSo by a similar argument (or by using the fact that $P Q$ are symmetric through the centre of the square), the coordinates of $Q$ are $(1-a \\sin \\theta, 1+a \\cos \\theta)$.\n\n<img_3706>\n\nNow $(P Q)^{2}=a^{2}$, so $(1-2 a \\sin \\theta)^{2}+(1-2 a \\cos \\theta)^{2}=a^{2}$\n\n$$\n2+4 a^{2} \\sin ^{2} \\theta+4 a^{2} \\cos ^{2} \\theta-4 a(\\sin \\theta+\\cos \\theta)=a^{2}\n$$\n\n\n\n$$\n\\begin{aligned}\n2+4 a^{2}-a^{2} & =4 a(\\sin \\theta+\\cos \\theta) \\\\\n\\frac{2+3 a^{2}}{4 a} & =\\sin \\theta+\\cos \\theta \\\\\n\\frac{2+3 a^{2}}{4 \\sqrt{2} a} & =\\frac{1}{\\sqrt{2}} \\sin \\theta+\\frac{1}{\\sqrt{2}} \\cos \\theta=\\cos \\left(45^{\\circ}\\right) \\sin \\theta+\\sin \\left(45^{\\circ}\\right) \\cos \\theta \\\\\n\\frac{2+3 a^{2}}{4 \\sqrt{2} a} & =\\sin \\left(\\theta+45^{\\circ}\\right)\n\\end{aligned}\n$$\n\n$$\n\\text { Now } \\begin{aligned}\n\\left(a-\\sqrt{\\frac{2}{3}}\\right)^{2} & \\geq 0 \\\\\na^{2}-2 a \\sqrt{\\frac{2}{3}}+\\frac{2}{3} & \\geq 0 \\\\\n3 a^{2}-2 a \\sqrt{6}+2 & \\geq 0 \\\\\n3 a^{2}+2 & \\geq 2 a \\sqrt{6} \\\\\n\\frac{3 a^{2}+2}{4 \\sqrt{2} a} & \\geq \\frac{\\sqrt{3}}{2}\n\\end{aligned}\n$$\n\nand equality occurs when $a=\\sqrt{\\frac{2}{3}}$.\n\nSo $\\sin \\left(\\theta+45^{\\circ}\\right) \\geq \\frac{\\sqrt{3}}{2}$ and thus since $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$, then $\\theta+45^{\\circ} \\geq 60^{\\circ}$ or $\\theta \\geq 15^{\\circ}$.\n\nTherefore the minimum possible value of $\\angle A D P$ is $15^{\\circ}$.']"	['$15$']		['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+4	"In the diagram, $\angle E A D=90^{\circ}, \angle A C D=90^{\circ}$, and $\angle A B C=90^{\circ}$. Also, $E D=13, E A=12$, $D C=4$, and $C B=2$. Determine the length of $A B$.
+
+<image_1>"	['By the Pythagorean Theorem in $\\triangle E A D$, we have $E A^{2}+A D^{2}=E D^{2}$ or $12^{2}+A D^{2}=13^{2}$, and so $A D=\\sqrt{169-144}=5$, since $A D>0$.\n\nBy the Pythagorean Theorem in $\\triangle A C D$, we have $A C^{2}+C D^{2}=A D^{2}$ or $A C^{2}+4^{2}=5^{2}$, and so $A C=\\sqrt{25-16}=3$, since $A C>0$.\n\n(We could also have determined the lengths of $A D$ and $A C$ by recognizing 3-4-5 and 5-12-13 right-angled triangles.)\n\nBy the Pythagorean Theorem in $\\triangle A B C$, we have $A B^{2}+B C^{2}=A C^{2}$ or $A B^{2}+2^{2}=3^{2}$, and so $A B=\\sqrt{9-4}=\\sqrt{5}$, since $A B>0$.']	['$\\sqrt{5}$']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+5	"In the diagram, $A B C D$ is a quadrilateral with $A B=B C=C D=6, \angle A B C=90^{\circ}$, and $\angle B C D=60^{\circ}$. Determine the length of $A D$.
+
+<image_1>"	"['Join $B$ to $D$.\n\n<img_3655>\n\nConsider $\\triangle C B D$.\n\nSince $C B=C D$, then $\\angle C B D=\\angle C D B=\\frac{1}{2}\\left(180^{\\circ}-\\angle B C D\\right)=\\frac{1}{2}\\left(180^{\\circ}-60^{\\circ}\\right)=60^{\\circ}$.\n\nTherefore, $\\triangle B C D$ is equilateral, and so $B D=B C=C D=6$.\n\nConsider $\\triangle D B A$.\n\nNote that $\\angle D B A=90^{\\circ}-\\angle C B D=90^{\\circ}-60^{\\circ}=30^{\\circ}$.\n\nSince $B D=B A=6$, then $\\angle B D A=\\angle B A D=\\frac{1}{2}\\left(180^{\\circ}-\\angle D B A\\right)=\\frac{1}{2}\\left(180^{\\circ}-30^{\\circ}\\right)=75^{\\circ}$.\n\nWe calculate the length of $A D$.\n\nMethod 1\n\nBy the Sine Law in $\\triangle D B A$, we have $\\frac{A D}{\\sin (\\angle D B A)}=\\frac{B A}{\\sin (\\angle B D A)}$.\n\nTherefore, $A D=\\frac{6 \\sin \\left(30^{\\circ}\\right)}{\\sin \\left(75^{\\circ}\\right)}=\\frac{6 \\times \\frac{1}{2}}{\\sin \\left(75^{\\circ}\\right)}=\\frac{3}{\\sin \\left(75^{\\circ}\\right)}$.\n\nMethod 2\n\nIf we drop a perpendicular from $B$ to $P$ on $A D$, then $P$ is the midpoint of $A D$ since $\\triangle B D A$ is isosceles. Thus, $A D=2 A P$.\n\nAlso, $B P$ bisects $\\angle D B A$, so $\\angle A B P=15^{\\circ}$.\n\nNow, $A P=B A \\sin (\\angle A B P)=6 \\sin \\left(15^{\\circ}\\right)$.\n\nTherefore, $A D=2 A P=12 \\sin \\left(15^{\\circ}\\right)$.\n\nMethod 3\n\nBy the Cosine Law in $\\triangle D B A$,\n\n$$\n\\begin{aligned}\nA D^{2} & =A B^{2}+B D^{2}-2(A B)(B D) \\cos (\\angle A B D) \\\\\n& =6^{2}+6^{2}-2(6)(6) \\cos \\left(30^{\\circ}\\right) \\\\\n& =72-72\\left(\\frac{\\sqrt{3}}{2}\\right) \\\\\n& =72-36 \\sqrt{3}\n\\end{aligned}\n$$\n\nTherefore, $A D=\\sqrt{36(2-\\sqrt{3})}=6 \\sqrt{2-\\sqrt{3}}$ since $A D>0$.'
+ 'Drop perpendiculars from $D$ to $Q$ on $B C$ and from $D$ to $R$ on $B A$.\n\n<img_3835>\n\nThen $C Q=C D \\cos (\\angle D C Q)=6 \\cos \\left(60^{\\circ}\\right)=6 \\times \\frac{1}{2}=3$.\n\nAlso, $D Q=C D \\sin (\\angle D C Q)=6 \\sin \\left(60^{\\circ}\\right)=6 \\times \\frac{\\sqrt{3}}{2}=3 \\sqrt{3}$.\n\nSince $B C=6$, then $B Q=B C-C Q=6-3=3$.\n\nNow quadrilateral $B Q D R$ has three right angles, so it must have a fourth right angle and so must be a rectangle.\n\nThus, $R D=B Q=3$ and $R B=D Q=3 \\sqrt{3}$.\n\nSince $A B=6$, then $A R=A B-R B=6-3 \\sqrt{3}$.\n\nSince $\\triangle A R D$ is right-angled at $R$, then using the Pythagorean Theorem and the fact that $A D>0$, we obtain\n\n$$\nA D=\\sqrt{R D^{2}+A R^{2}}=\\sqrt{3^{2}+(6-3 \\sqrt{3})^{2}}=\\sqrt{9+36-36 \\sqrt{3}+27}=\\sqrt{72-36 \\sqrt{3}}\n$$\n\nwhich we can rewrite as $A D=\\sqrt{36(2-\\sqrt{3})}=6 \\sqrt{2-\\sqrt{3}}$.']"	['$6\\sqrt{2-\\sqrt{3}}$']		['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+6	"A triangle has vertices $A(0,3), B(4,0)$, $C(k, 5)$, where $0<k<4$. If the area of the triangle is 8 , determine the value of $k$.
+
+<image_1>"	"['We ""complete the rectangle"" by drawing a horizontal line through $C$ which meets the $y$-axis at $P$ and the vertical line through $B$ at $Q$.\n\n<img_3215>\n\n\n\nSince $C$ has $y$-coordinate 5 , then $P$ has $y$-coordinate 5 ; thus the coordinates of $P$ are $(0,5)$.\n\nSince $B$ has $x$-coordinate 4 , then $Q$ has $x$-coordinate 4 .\n\nSince $C$ has $y$-coordinate 5 , then $Q$ has $y$-coordinate 5 .\n\nTherefore, the coordinates of $Q$ are $(4,5)$, and so rectangle $O P Q B$ is 4 by 5 and so has area $4 \\times 5=20$.\n\nNow rectangle $O P Q B$ is made up of four smaller triangles, and so the sum of the areas of these triangles must be 20 .\n\nLet us examine each of these triangles:\n\n- $\\triangle A B C$ has area 8 (given information)\n- $\\triangle A O B$ is right-angled at $O$, has height $A O=3$ and base $O B=4$, and so has area $\\frac{1}{2} \\times 4 \\times 3=6$.\n- $\\triangle A P C$ is right-angled at $P$, has height $A P=5-3=2$ and base $P C=k-0=k$, and so has area $\\frac{1}{2} \\times k \\times 2=k$.\n- $\\triangle C Q B$ is right-angled at $Q$, has height $Q B=5-0=5$ and base $C Q=4-k$, and so has area $\\frac{1}{2} \\times(4-k) \\times 5=10-\\frac{5}{2} k$.\n\nSince the sum of the areas of these triangles is 20 , then $8+6+k+10-\\frac{5}{2} k=20$ or $4=\\frac{3}{2} k$ and so $k=\\frac{8}{3}$.']"	['$\\frac{8}{3}$']		['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+7	"A helicopter hovers at point $H$, directly above point $P$ on level ground. Lloyd sits on the ground at a point $L$ where $\angle H L P=60^{\circ}$. A ball is droppped from the helicopter. When the ball is at point $B, 400 \mathrm{~m}$ directly below the helicopter, $\angle B L P=30^{\circ}$. What is the distance between $L$ and $P$ ?
+
+<image_1>"	"['Since $\\angle H L P=60^{\\circ}$ and $\\angle B L P=30^{\\circ}$, then $\\angle H L B=\\angle H L P-\\angle B L P=30^{\\circ}$.\n\nAlso, since $\\angle H L P=60^{\\circ}$ and $\\angle H P L=90^{\\circ}$, then $\\angle L H P=180^{\\circ}-90^{\\circ}-60^{\\circ}=30^{\\circ}$.\n\n<img_3808>\n\nTherefore, $\\triangle H B L$ is isosceles and $B L=H B=400 \\mathrm{~m}$.\n\nIn $\\triangle B L P, B L=400 \\mathrm{~m}$ and $\\angle B L P=30^{\\circ}$, so $L P=B L \\cos \\left(30^{\\circ}\\right)=400\\left(\\frac{\\sqrt{3}}{2}\\right)=200 \\sqrt{3}$ m.\n\nTherefore, the distance between $L$ and $P$ is $200 \\sqrt{3} \\mathrm{~m}$.'
+ 'Since $\\angle H L P=60^{\\circ}$ and $\\angle B L P=30^{\\circ}$, then $\\angle H L B=\\angle H L P-\\angle B L P=30^{\\circ}$.\n\nAlso, since $\\angle H L P=60^{\\circ}$ and $\\angle H P L=90^{\\circ}$, then $\\angle L H P=180^{\\circ}-90^{\\circ}-60^{\\circ}=30^{\\circ}$. Also, $\\angle L B P=60^{\\circ}$.\n\nLet $L P=x$.\n\n<img_3794>\n\nSince $\\triangle B L P$ is $30^{\\circ}-60^{\\circ}-90^{\\circ}$, then $B P: L P=1: \\sqrt{3}$, so $B P=\\frac{1}{\\sqrt{3}} L P=\\frac{1}{\\sqrt{3}} x$.\n\n\n\nSince $\\triangle H L P$ is $30^{\\circ}-60^{\\circ}-90^{\\circ}$, then $H P: L P=\\sqrt{3}: 1$, so $H P=\\sqrt{3} L P=\\sqrt{3} x$.\n\nBut $H P=H B+B P$ so\n\n$$\n\\begin{aligned}\n\\sqrt{3} x & =400+\\frac{1}{\\sqrt{3}} x \\\\\n3 x & =400 \\sqrt{3}+x \\\\\n2 x & =400 \\sqrt{3} \\\\\nx & =200 \\sqrt{3}\n\\end{aligned}\n$$\n\nTherefore, the distance from $L$ to $P$ is $200 \\sqrt{3} \\mathrm{~m}$.']"	['$200 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+8	"In the diagram, $A B C D$ is a quadrilateral in which $\angle A+\angle C=180^{\circ}$. What is the length of $C D$ ?
+
+<image_1>"	['In order to determine $C D$, we must determine one of the angles (or at least some information about one of the angles) in $\\triangle B C D$.\n\nTo do this, we look at $\\angle A$ use the fact that $\\angle A+\\angle C=180^{\\circ}$.\n\n\n\n<img_3524>\n\nUsing the cosine law in $\\triangle A B D$, we obtain\n\n$$\n\\begin{aligned}\n7^{2} & =5^{2}+6^{2}-2(5)(6) \\cos (\\angle A) \\\\\n49 & =61-60 \\cos (\\angle A) \\\\\n\\cos (\\angle A) & =\\frac{1}{5}\n\\end{aligned}\n$$\n\nSince $\\cos (\\angle A)=\\frac{1}{5}$ and $\\angle A+\\angle C=180^{\\circ}$, then $\\cos (\\angle C)=-\\cos \\left(180^{\\circ}-\\angle A\\right)=-\\frac{1}{5}$.\n\n(We could have calculated the actual size of $\\angle A$ using $\\cos (\\angle A)=\\frac{1}{5}$ and then used this to calculate the size of $\\angle C$, but we would introduce the possibility of rounding error by doing this.)\n\nThen, using the cosine law in $\\triangle B C D$, we obtain\n\n$$\n\\begin{aligned}\n7^{2} & =4^{2}+C D^{2}-2(4)(C D) \\cos (\\angle C) \\\\\n49 & =16+C D^{2}-8(C D)\\left(-\\frac{1}{5}\\right) \\\\\n0 & =5 C D^{2}+8 C D-165 \\\\\n0 & =(5 C D+33)(C D-5)\n\\end{aligned}\n$$\n\nSo $C D=-\\frac{33}{5}$ or $C D=5$. (We could have also determined these roots using the quadratic formula.)\n\nSince $C D$ is a length, it must be positive, so $C D=5$.\n\n(We could have also proceeded by using the sine law in $\\triangle B C D$ to determine $\\angle B D C$ and then found the size of $\\angle D B C$, which would have allowed us to calculate $C D$ using the sine law. However, this would again introduce the potential of rounding error.)']	['5']		['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+9	"In the diagram, the parabola
+
+$$
+y=-\frac{1}{4}(x-r)(x-s)
+$$
+
+intersects the axes at three points. The vertex of this parabola is the point $V$. Determine the value of $k$ and the coordinates of $V$.
+
+<image_1>"	['From the diagram, the $x$-intercepts of the parabola are $x=-k$ and $x=3 k$.\n\n\n\n<img_3883>\n\nSince we are given that $y=-\\frac{1}{4}(x-r)(x-s)$, then the $x$-intercepts are $r$ and $s$, so $r$ and $s$ equal $-k$ and $3 k$ in some order.\n\nTherefore, we can rewrite the parabola as $y=-\\frac{1}{4}(x-(-k))(x-3 k)$.\n\nSince the point $(0,3 k)$ lies on the parabola, then $3 k=-\\frac{1}{4}(0+k)(0-3 k)$ or $12 k=3 k^{2}$ or $k^{2}-4 k=0$ or $k(k-4)=0$.\n\nThus, $k=0$ or $k=4$.\n\nSince the two roots are distinct, then we cannot have $k=0$ (otherwise both $x$-intercepts would be 0 ).\n\nThus, $k=4$.\n\nThis tells us that the equation of the parabola is $y=-\\frac{1}{4}(x+4)(x-12)$ or $y=-\\frac{1}{4} x^{2}+$ $2 x+12$.\n\nWe still have to determine the coordinates of the vertex, $V$.\n\nSince the $x$-intercepts of the parabola are -4 and 12 , then the $x$-coordinate of the vertex is the average of these intercepts, or 4.\n\n(We could have also used the fact that the $x$-coordinate is $-\\frac{b}{2 a}=-\\frac{2}{2\\left(-\\frac{1}{4}\\right)}$.)\n\nTherefore, the $y$-coordinate of the vertex is $y=-\\frac{1}{4}\\left(4^{2}\\right)+2(4)+12=16$.\n\nThus, the coordinates of the vertex are 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+10	"A school has a row of $n$ open lockers, numbered 1 through $n$. After arriving at school one day, Josephine starts at the beginning of the row and closes every second locker until reaching the end of the row, as shown in the example below. Then on her way back, she closes every second locker that is still open. She continues in this manner along the row, until only one locker remains open. Define $f(n)$ to be the number of the last open locker. For example, if there are 15 lockers, then $f(15)=11$ as shown below:
+
+<image_1>
+Determine $f(50)$."	['We proceed directly.\n\nOn the first pass from left to right, Josephine closes all of the even numbered lockers, leaving the odd ones open.\n\nThe second pass proceeds from right to left. Before the pass, the lockers which are open are $1,3, \\ldots, 47,49$.\n\nOn the second pass, she shuts lockers 47, 43, 39, .., 3 .\n\nThe third pass proceeds from left to right. Before the pass, the lockers which are open are $1,5, \\ldots, 45,49$.\n\nOn the third pass, she shuts lockers $5,13, \\ldots, 45$.\n\nThis leaves lockers 1, 9, 17, 25, 33, 41, 49 open.\n\nOn the fourth pass, from right to left, lockers 41, 25 and 9 are shut, leaving 1, 17, 33, 49.\n\nOn the fifth pass, from left to right, lockers 17 and 49 are shut, leaving 1 and 33 open.\n\nOn the sixth pass, from right to left, locker 1 is shut, leaving 33 open.\n\nThus, $f(50)=33$.']	['33']		['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+11	"In the diagram, $P Q R S$ is a quadrilateral. What is its perimeter?
+
+<image_1>"	['The length of $P Q$ is equal to $\\sqrt{(0-5)^{2}+(12-0)^{2}}=\\sqrt{(-5)^{2}+12^{2}}=13$.\n\nIn a similar way, we can see that $Q R=R S=S P=13$.\n\nTherefore, the perimeter of $P Q R S$ is $4 \\cdot 13=52$.\n\n(We can also see that if $O$ is the origin, then $\\triangle P O Q, \\triangle P O S, \\triangle R O Q$, and $\\triangle R O S$ are congruent because $O Q=O S$ and $O P=O R$, which means that $P Q=Q R=R S=S P$.)']	['52']		['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+12	"In the diagram, $A$ has coordinates $(0,8)$. Also, the midpoint of $A B$ is $M(3,9)$ and the midpoint of $B C$ is $N(7,6)$. What is the slope of $A C$ ?
+
+<image_1>"	"['Suppose that $B$ has coordinates $(r, s)$ and $C$ has coordinates $(t, u)$.\n\nSince $M(3,9)$ is the midpoint of $A(0,8)$ and $B(r, s)$, then 3 is the average of 0 and $r$ (which gives $r=6)$ and 9 is the average of 8 and $s$ (which gives $s=10$ ).\n\nSince $N(7,6)$ is the midpoint of $B(6,10)$ and $C(t, u)$, then 7 is the average of 6 and $t$ (which gives $t=8$ ) and 6 is the average of 10 and $u$ (which gives $u=2$ ).\n\nThe slope of the line segment joining $A(0,8)$ and $C(8,2)$ is $\\frac{8-2}{0-8}$ which equals $-\\frac{3}{4}$.'
+ 'Since $M$ is the midpoint of $A B$ and $N$ is the midpoint of $B C$, then $M N$ is parallel to $A C$. Therefore, the slope of $A C$ equals the slope of the line segment joining $M(3,9)$ to $N(7,6)$, which is $\\frac{9-6}{3-7}$ or $-\\frac{3}{4}$.']"	['$-\\frac{3}{4}$']		['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+13	"In the diagram, $A B D E$ is a rectangle, $\triangle B C D$ is equilateral, and $A D$ is parallel to $B C$. Also, $A E=2 x$ for some real number $x$.
+
+<image_1>
+Determine the length of $A B$ in terms of $x$."	['We begin by determining the length of $A B$ in terms of $x$.\n\nSince $A B D E$ is a rectangle, $B D=A E=2 x$.\n\nSince $\\triangle B C D$ is equilateral, $\\angle D B C=60^{\\circ}$.\n\nJoin $A$ to $D$.\n\n<img_3330>\n\nSince $A D$ and $B C$ are parallel, $\\angle A D B=\\angle D B C=60^{\\circ}$.\n\nConsider $\\triangle A D B$. This is a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle since $\\angle A B D$ is a right angle.\n\nUsing ratios of side lengths, $\\frac{A B}{B D}=\\frac{\\sqrt{3}}{1}$ and so $A B=\\sqrt{3} B D=2 \\sqrt{3} x$']	['$2 \\sqrt{3} x$']		['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+K2k+I38I3L6RpeBDAb22EjEZO7HmYHzEd+iigD13SNOh0fR7LTYB+5tIEhTjsqgf0rjPjLr8Oi/DfUo2kAuL8fZIlzy27734bd36etaM+veLJl2WfgmZHPR73UbdVB9TsZz+lULHwBd6p4gh8Q+M7yDUb63/wCPSyt1ItLX3UNy5yAcnH0OBQBL8JPDk3hj4eWFrdRmO7uN11OjDBVnPAI7EKFB9wa7mmgHOT6U6gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigD/2Q==']	Multimodal	Competition	False		Expression		Open-ended	Geometry	Math	English
+14	"In the diagram, $A B D E$ is a rectangle, $\triangle B C D$ is equilateral, and $A D$ is parallel to $B C$. Also, $A E=2 x$ for some real number $x$.
+
+<image_1>
+Determine positive integers $r$ and $s$ for which
+
+$$
+\frac{A C}{A D}=\sqrt{\frac{r}{s}}
+$$"	['We begin by determining the length of $A B$ in terms of $x$.\n\nSince $A B D E$ is a rectangle, $B D=A E=2 x$.\n\nSince $\\triangle B C D$ is equilateral, $\\angle D B C=60^{\\circ}$.\n\nJoin $A$ to $D$.\n\n<img_3330>\n\nSince $A D$ and $B C$ are parallel, $\\angle A D B=\\angle D B C=60^{\\circ}$.\n\nConsider $\\triangle A D B$. This is a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle since $\\angle A B D$ is a right angle.\n\nUsing ratios of side lengths, $\\frac{A B}{B D}=\\frac{\\sqrt{3}}{1}$ and so $A B=\\sqrt{3} B D=2 \\sqrt{3} x$\n\nNext, we determine $\\frac{A C}{A D}$.\n\nNow, $\\frac{A D}{B D}=\\frac{2}{1}$ and so $A D=2 B D=4 x$.\n\nSuppose that $M$ is the midpoint of $A E$ and $N$ is the midpoint of $B D$.\n\nSince $A E=B D=2 x$, then $A M=M E=B N=N D=x$.\n\nJoin $M$ to $N$ and $N$ to $C$ and $A$ to $C$.\n\n<img_3952>\n\nSince $A B D E$ is a rectangle, then $M N$ is parallel to $A B$ and so $M N$ is perpendicular to both $A E$ and $B D$.\n\nAlso, $M N=A B=2 \\sqrt{3} x$.\n\nSince $\\triangle B C D$ is equilateral, its median $C N$ is perpendicular to $B D$.\n\nSince $M N$ and $N C$ are perpendicular to $B D, M N C$ is actually a straight line segment and so $M C=M N+N C$.\n\nNow $\\triangle B N C$ is also a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle, and so $N C=\\sqrt{3} B N=\\sqrt{3} x$.\n\nThis means that $M C=2 \\sqrt{3} x+\\sqrt{3} x=3 \\sqrt{3} x$.\n\n\n\nFinally, $\\triangle A M C$ is right-angled at $M$ and so\n\n$$\nA C=\\sqrt{A M^{2}+M C^{2}}=\\sqrt{x^{2}+(3 \\sqrt{3} x)^{2}}=\\sqrt{x^{2}+27 x^{2}}=\\sqrt{28 x^{2}}=2 \\sqrt{7} x\n$$\n\nsince $x>0$.\n\nThis means that $\\frac{A C}{A D}=\\frac{2 \\sqrt{7} x}{4 x}=\\frac{\\sqrt{7}}{2}=\\sqrt{\\frac{7}{4}}$, which means that the integers $r=7$ and $s=4$ satisfy the conditions.']	['7,4']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAPQBrAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiikPvQAZHrS5rLuddsrbXbLRmlLX12jyRxrzhE6sfQZIHufocaa0ALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRQaACkyKw/FPiF/DGjz6o2mzXltbo0k5idFKKPZiM/hXPaV8SLrWNDTW7Lwdrc+nMGKyxPbs7YJBxH5gY8gjgc0Ad91orB8K+L9G8Y6e97o9yZFjbZLFIpWSJvRl7fXocH0reoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKCcdaAEJA6ms3XtbttA0mS9uA8hBEcMMf355WOFjQd2J4H59ATV+aSOOF5JHVI0BZmY4Cgc5PpivM4PEsmreJzrlxoOu3On2oZNHEFkWjYMMNcE5GSwOF9Fz3bgA5zwAuqXfx+1+bWZ1nvLfTzvKfciYmL92v8AsruZffBPevchmvF/hFP/AGv8TvHmr+VLHmYRhJhh0DSPgMOxwnTtXtNABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRSN0oA8x+O2stY/D46dbnNzqtwlsqjqVB3MR+QH/Aq6K2vtD+HngmxtdR1C3tobC0RCC4DSMF52r1YscnA9a4PxhDF41+Ouh+HJoxPp2lW7XN3GwypLAPg+oOIh+NS/E/wDoOh+FJ/E3h+1XSdV0x45Y5bP5A2XVSCAcfxZ9ePQmgBfgloeppqPiHxPeWktla6tLutreRSpK7mbO30AYAHvz26+x1yXw08RXXirwHpurXyqLqRXSUqu0OVYruA7Zx24zmutoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKRjgZpaD0oA808TeIp9c8Raf4cGia7/YEkhOo3w0ucLIACVhHyZ2MwAZscg4zgk13eoahBpFh5zwXUiKMLHZ2skzZA6bYwcdO+B0q9g5oxjGaAPFvgsb/AEvVPEJ1nRNZsbjVLlZonn06YIwy5ILbcA/N3IFe15zTc84B5680o60ALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABVW/uZrS1MsFhPfSA48mBo1Y/i7KP1q1RQB4t4Q0rxponjPxB4k1TwZcXNxqj4i8rULbMMe4nacvzwEH/Aa0PFHh/wAdfEcR6Ve2dt4c0ESB5w9ytxPNjpwny++M9cHJwK9ZzikyKAM/Q9GtPD+i2mk2MZS2tYxGmTkn1J9yck+5rRozRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFIRkUtFAHj/xnutXHiLwXpWla1faUdTupLaSW1ndPvPCoZlUjdjef8aB8IfF/wD0VfXPriX/AOP0fF7/AJKH8Mv+wqf/AEdb16+KAPIP+FQeMP8AorGuflL/APH6P+FQeMP+isa5+Uv/AMfr2CigDx//AIVB4w/6Kxrn5S//AB+j/hUHjD/orGuflL/8fr2CigDx/wD4VB4w/wCisa5+Uv8A8fo/4VB4w/6Kxrn5S/8Ax+vYKKAPH/8AhUHjD/orGuflL/8AH6P+FQeMP+isa5+Uv/x+vYKKAPH/APhUHjD/AKKxrn5S/wDx+j/hUHjD/orGuflL/wDH69gooA8f/wCFQeMP+isa5+Uv/wAfo/4VB4w/6Kxrn5S//H69gooA8f8A+FQeMP8AorGuflL/APH6P+FQeMP+isa5+Uv/AMfr2CigDx//AIVB4w/6Kxrn5S//AB+j/hUHjD/orGuflL/8fr2CigDx/wD4VB4w/wCisa5+Uv8A8fo/4VB4w/6Kxrn5S/8Ax+vYKKAPH/8AhUHjD/orGuflL/8AH6P+FQeMP+isa5+Uv/x+vYKKAPH/APhUHjD/AKKxrn5S/wDx+j/hUHjD/orGuflL/wDH69gooA8f/wCFQeMP+isa5+Uv/wAfo/4VB4w/6Kxrn5S//H69gooA8f8A+FQeMP8AorGuflL/APH6P+FQeMP+isa5+Uv/AMfr2CigDwDxt4G8X+DvCN9rx+JmuXgtfL/cb5Y926RU+95xx97PSr/hz4k+JPCWj6WfG2mz3Wj3VvFJbazAN52OoKiX1bBwScNwfvZzXYfG3/kkWuf9u/8A6Pjrb8GW8V18N/D0E8aSwyaRbK8cihlYGFcgg8EfWgDU0fW9N13T0v8AS72G6tX6SRNkA+hHUH2PNaGR615VrHwmudI1CTW/h7qbaNqBOXs2b/RpvUYwcDngEFR2C9Q/Q/i21lqC6L49019A1TgLOw/0aX3B529euSvH3u1AHqeaKZHLHLGskbq6MMhlOQR2NPBB6UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAHj/xe/5KH8Mv+wqf/R1vXr4ryD4vf8lD+GX/AGFT/wCjrevXxQAtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAHn/xt/5JDrn/AGw/9Hx10HgT/knnhr/sFWv/AKKWuf8Ajb/ySHXP+2H/AKPjroPAn/JPPDX/AGCrX/0UtAG+azNc8O6V4jsGstXsILyA5wsi8qemVYcqeTyCDWpRQB45L4M8Z/DeV7vwRfNq2jA7n0a8bLKO+w8DPXpg9OGrqPB/xT0PxTP9gmL6VrKnZJp978r7h1Ckgbu/HB46V3TDIrk/F/w78P8AjSEnU7TZdgfJeQYWVcdBnoR7HNAHWZHrS14ydR8f/CvjVEfxT4aT/l6jz9ogX1bOTgc9dw+6Ny9K9F8LeMtD8YWZutHvkm2j95C3yyxH/aXqPr09CaAOhopMjOKXNABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAeP/ABe/5KH8Mv8AsKn/ANHW9evivIPi9/yUP4Zf9hU/+jrevXxQAtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAHn/wAbf+SQ65/2w/8AR8ddB4E/5J54a/7BVr/6KWuf+Nv/ACSHXP8Ath/6PjroPAn/ACTzw1/2CrX/ANFLQB0FFFFABRRRQAjAkYH615z4n+Eem6pef2x4fuH0DXUJdLi0+RHb/bVcYPqVxnPOa9HoNAHj9n8R/EPgm5j0z4i6UwhJ2R6xZpujk92A7nBPAB/2e9eq6dqdjq1lHe6fdw3VrIPklhcMp/EU69sbXUbSS0vbaK5tpBteKVAysPcHivLdS+Fuq+F72TWvhxqbWM2d82lzvm3nwPujP44DdN3DLigD1rOTilrzHw38XLVtQ/sTxjZv4e1pcAi4yIZT6qx+6D2ySPRjXpiurKrKwIYZBB6igB1FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAHj/xe/wCSh/DL/sKn/wBHW9evivIPi9/yUP4Zf9hU/wDo63r18UALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQB5/8bf8AkkOuf9sP/R8ddB4E/wCSeeGv+wVa/wDopa5/42/8kh1z/th/6PjroPAn/JPPDX/YKtf/AEUtAHQUUUUAFFFFABRRRQAUhGRS0UAYviLwro3ivT/sWs2EVzEMlGIw8ZPdWHKn6V5o/hvxz8Md03he5fxD4eXLNpd1zLEvU7Mdf+A9z9017LTWGQMdQc0AcZ4N+Jnh/wAYH7NDO1nqi5WTT7sbJQR129m6HpzxyBXabgO9cZ4x+GegeMV8+4gNnqa48u/tflkUjpuxw348jsRXHL4k8cfDFhD4pt38ReH0O1dUt/8AXQr0+cHr/wAC7n7xoA9lzRWL4b8UaN4qsBe6Nfx3UYA3qOHjJ7Mp5U8d+vatkEHoc/SgBaKM5ooAKKCcUmQe9AC0Um4UZFAC0UUUAFFFFAHj/wAXv+Sh/DL/ALCp/wDR1vXr4ryD4vf8lD+GX/YVP/o63r18UALRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQB5/8AG3/kkOuf9sP/AEfHXQeBP+SeeGv+wVa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ZIGT7kCvGfEOieMta+K2k+I38I3L6RpeBDAb22EjEZO7HmYHzEd+iigD13SNOh0fR7LTYB+5tIEhTjsqgf0rjPjLr8Oi/DfUo2kAuL8fZIlzy27734bd36etaM+veLJl2WfgmZHPR73UbdVB9TsZz+lULHwBd6p4gh8Q+M7yDUb63/wCPSyt1ItLX3UNy5yAcnH0OBQBL8JPDk3hj4eWFrdRmO7uN11OjDBVnPAI7EKFB9wa7mmgHOT6U6gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigD/2Q==']	Multimodal	Competition	True		Numerical		Open-ended	Geometry	Math	English
+15	"Five distinct integers are to be chosen from the set $\{1,2,3,4,5,6,7,8\}$ and placed in some order in the top row of boxes in the diagram. Each box that is not in the top row then contains the product of the integers in the two boxes connected to it in the row directly above. Determine the number of ways in which the integers can be chosen and placed in the top row so that the integer in the bottom box is 9953280000 .
+
+<image_1>"	['Suppose that the integers in the first row are, in order, $a, b, c, d, e$.\n\nUsing these, we calculate the integer in each of the boxes below the top row in terms of these variables, using the rule that each integer is the product of the integers in the two boxes above:\n\n$a$\n\n| $b$ | $c$ | $c$ |  | $d$ |\n| :---: | :---: | :---: | :---: | :---: |\n| $a b^{2} c$ | $b c$ | $c d$ | $c$ |  |\n|  | $a b^{3} c^{3} d$ | $b c^{2} d$ | $c d^{2} e$ |  |$\\quad d e$\n\nTherefore, $a b^{4} c^{6} d^{4} e=9953280000$.\n\n\n\nNext, we determine the prime factorization of the integer 9953280000 :\n\n$$\n\\begin{aligned}\n9953280000 & =10^{4} \\cdot 995328 \\\\\n& =2^{4} \\cdot 5^{4} \\cdot 2^{3} \\cdot 124416 \\\\\n& =2^{7} \\cdot 5^{4} \\cdot 2^{3} \\cdot 15552 \\\\\n& =2^{10} \\cdot 5^{4} \\cdot 2^{3} \\cdot 1944 \\\\\n& =2^{13} \\cdot 5^{4} \\cdot 2^{3} \\cdot 243 \\\\\n& =2^{16} \\cdot 5^{4} \\cdot 3^{5} \\\\\n& =2^{16} \\cdot 3^{5} \\cdot 5^{4}\n\\end{aligned}\n$$\n\nThus, $a b^{4} c^{6} d^{4} e=2^{16} \\cdot 3^{5} \\cdot 5^{4}$.\n\nSince the right side is not divisible by 7 , none of $a, b, c, d$, $e$ can equal 7 .\n\nThus, $a, b, c, d, e$ are five distinct integers chosen from $\\{1,2,3,4,5,6,8\\}$.\n\nThe only one of these integers divisible by 5 is 5 itself.\n\nSince $2^{16} \\cdot 3^{5} \\cdot 5^{4}$ includes exactly 4 factors of 5 , then either $b=5$ or $d=5$. No other placement of the 5 can give exactly 4 factors of 5 .\n\nCase 1: $b=5$\n\nHere, $a c^{6} d^{4} e=2^{16} \\cdot 3^{5}$ and $a, c, d, e$ are four distinct integers chosen from $\\{1,2,3,4,6,8\\}$. Since $a c^{6} d^{4} e$ includes exactly 5 factors of 3 and the possible values of $a, c, d$, e that are divisible by 3 are 3 and 6 , then either $d=3$ and one of $a$ and $e$ is 6 , or $d=6$ and one of $a$ and $e$ is 3 . No other placements of the multiples of 3 can give exactly 5 factors of 3 .\n\nCase 1a: $b=5, d=3, a=6$\n\nHere, $a \\cdot c^{6} \\cdot d^{4} \\cdot e=6 \\cdot c^{6} \\cdot 3^{4} \\cdot e=2 \\cdot 3^{5} \\cdot c^{6} \\cdot e$.\n\nThis gives $c^{6} e=2^{15}$ and $c$ and $e$ are distinct integers from $\\{1,2,4,8\\}$.\n\nTrying the four possible values of $c$ shows that $c=4$ and $e=8$ is the only solution in this case. Here, $(a, b, c, d, e)=(6,5,4,3,8)$.\n\nCase 1b: $b=5, d=3, e=6$ We obtain $(a, b, c, d, e)=(8,5,4,3,6)$.\n\nCase 1c: $b=5, d=6, a=3$\n\nHere, $a \\cdot c^{6} \\cdot d^{4} \\cdot e=3 \\cdot c^{6} \\cdot 6^{4} \\cdot e=2^{4} \\cdot 3^{5} \\cdot c^{6} \\cdot e$.\n\nThis gives $c^{6} e=2^{12}$ and $c$ and $e$ are distinct integers from $\\{1,2,4,8\\}$.\n\nTrying the four possible values of $c$ shows that $c=4$ and $e=1$ is the only solution in this case. Here, $(a, b, c, d, e)=(3,5,4,6,1)$.\n\nCase 1d: $b=5, d=6, e=3$ We obtain $(a, b, c, d, e)=(1,5,4,6,3)$.\n\nCase 2: $d=5$ : A similar analysis leads to 4 further quintuples $(a, b, c, d, e)$.\n\nTherefore, there are 8 ways in which the integers can be chosen and placed in the top row to obtain the desired integer in the bottom box.']	['8']		['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Theory	Math	English
+16	"In the diagram, eleven circles of four different radius 1, each circle labelled $X$ has radius 2, the circle labelled $Y$ has radius 4 , and the circle labelled $Z$ has radius $r$. Each of the circles labelled $W$ or $X$ is tangent to three other circles. The circle labelled $Y$ is tangent to all ten of the other circles. The circle labelled $Z$ is tangent to three other circles. Determine positive integers $s$ and $t$ for which $r=\frac{s}{t}$.
+
+
+<image_1>"	['We label the centres of the outer circles, starting with the circle labelled $Z$ and proceeding clockwise, as $A, B, C, D, E, F, G, H, J$, and $K$, and the centre of the circle labelled $Y$ as $L$.\n\n<img_3893>\n\nJoin $L$ to each of $A, B, C, D, E, F, G, H, J$, and $K$. Join $A$ to $B, B$ to $C, C$ to $D, D$ to $E, E$ to $F, F$ to $G, G$ to $H, H$ to $J, J$ to $K$, and $K$ to $A$.\n\nWhen two circles are tangent, the distance between their centres equals the sum of their radii.\n\nThus,\n\n$$\n\\begin{array}{r}\nB C=C D=D E=E F=F G=G H=H J=J K=2+1=3 \\\\\nB L=D L=F L=H L=K L=2+4=6 \\\\\nC L=E L=G L=J L=1+4=5 \\\\\nA B=A K=r+2 \\\\\nA L=r+4\n\\end{array}\n$$\n\nBy side-side-side congruence, the following triangles are congruent:\n\n$$\n\\triangle B L C, \\triangle D L C, \\triangle D L E, \\triangle F L E, \\triangle F L G, \\triangle H L G, \\triangle H L J, \\triangle K L J\n$$\n\nSimilarly, $\\triangle A L B$ and $\\triangle A L K$ are congruent by side-side-side.\n\nLet $\\angle A L B=\\theta$ and let $\\angle B L C=\\alpha$.\n\n\n\nBy congruent triangles, $\\angle A L K=\\theta$ and\n\n$$\n\\angle B L C=\\angle D L C=\\angle D L E=\\angle F L E=\\angle F L G=\\angle H L G=\\angle H L J=\\angle K L J=\\alpha\n$$\n\nThe angles around $L$ add to $360^{\\circ}$ and so $2 \\theta+8 \\alpha=360^{\\circ}$ which gives $\\theta+4 \\alpha=180^{\\circ}$ and so $\\theta=180^{\\circ}-4 \\alpha$.\n\nSince $\\theta=180^{\\circ}-4 \\alpha$, then $\\cos \\theta=\\cos \\left(180^{\\circ}-4 \\alpha\\right)=-\\cos 4 \\alpha$.\n\nConsider $\\triangle A L B$ and $\\triangle B L C$.\n\n<img_3240>\n\nBy the cosine law in $\\triangle A L B$,\n\n$$\n\\begin{aligned}\nA B^{2} & =A L^{2}+B L^{2}-2 \\cdot A L \\cdot B L \\cdot \\cos \\theta \\\\\n(r+2)^{2} & =(r+4)^{2}+6^{2}-2(r+4)(6) \\cos \\theta \\\\\n12(r+4) \\cos \\theta & =r^{2}+8 r+16+36-r^{2}-4 r-4 \\\\\n\\cos \\theta & =\\frac{4 r+48}{12(r+4)} \\\\\n\\cos \\theta & =\\frac{r+12}{3 r+12}\n\\end{aligned}\n$$\n\nBy the cosine law in $\\triangle B L C$,\n\n$$\n\\begin{aligned}\nB C^{2} & =B L^{2}+C L^{2}-2 \\cdot B L \\cdot C L \\cdot \\cos \\alpha \\\\\n3^{2} & =6^{2}+5^{2}-2(6)(5) \\cos \\alpha \\\\\n60 \\cos \\alpha & =36+25-9 \\\\\n\\cos \\alpha & =\\frac{52}{60} \\\\\n\\cos \\alpha & =\\frac{13}{15}\n\\end{aligned}\n$$\n\nSince $\\cos \\alpha=\\frac{13}{15}$, then\n\n$$\n\\begin{aligned}\n\\cos 2 \\alpha & =2 \\cos ^{2} \\alpha-1 \\\\\n& =2 \\cdot \\frac{169}{225}-1 \\\\\n& =\\frac{338}{225}-\\frac{225}{225} \\\\\n& =\\frac{113}{225}\n\\end{aligned}\n$$\n\n\n\nand\n\n$$\n\\begin{aligned}\n\\cos 4 \\alpha & =2 \\cos ^{2} 2 \\alpha-1 \\\\\n& =2 \\cdot \\frac{113^{2}}{225^{2}}-1 \\\\\n& =\\frac{25538}{50625}-\\frac{50625}{50625} \\\\\n& =-\\frac{25087}{50625}\n\\end{aligned}\n$$\n\nFinally,\n\n$$\n\\begin{aligned}\n\\cos \\theta & =-\\cos 4 \\alpha \\\\\n\\frac{r+12}{3 r+12} & =\\frac{25087}{50625} \\\\\n\\frac{r+12}{r+4} & =\\frac{25087}{16875} \\\\\n\\frac{(r+4)+8}{r+4} & =\\frac{25087}{16875} \\\\\n1+\\frac{8}{r+4} & =\\frac{25087}{16875} \\\\\n\\frac{8}{r+4} & =\\frac{8212}{16875} \\\\\n\\frac{2}{r+4} & =\\frac{2053}{16875} \\\\\n\\frac{r+4}{2} & =\\frac{16875}{2053} \\\\\nr+4 & =\\frac{33750}{2053} \\\\\nr & =\\frac{25538}{2053}\n\\end{aligned}\n$$\n\nTherefore, the positive integers $s=25538$ and $t=2053$ satisfy the required conditions.']	['$25538$,$2053$']		['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0paKAPOPi54Zu9R0S28Q6NuTWtCk+1QMg5aMYLr74wGA74I711Hg3xPa+L/DFnrNqVHmriaMH/VSj76n6Hp6gg963m+6cV4sM/B/4jEnKeD9fk4PRLSf+QH5fKf9igD2qimgg8g5B7jvTqACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKQ8ihhkEVx3xG8bweB/C0t8dr39xmKzhP8UhHUj+6vU/gO9AHB/FPXrTxF4ptfB018tpotgRea5dF8BVGCqA92wQAAD8zqMfKRVqPx14p8XD+zfhtocdppUAEA1W9UJGgAwAikYwMdMMcEZArmPhn8JLnxL/xUvi15TZ3b/aYrUsQ10xJPmSei8nA6tuzkD730JbWsNpDHBbQpDBGoRIo1CqoHQADgCgDy2L4LNq8qXXjPxPqmszht/ko/lwg+gBycfTbXRWfwj8CWK4j8O28mepnd5f8A0NjjpXb0UAcbc/CnwNdR+XJ4aswvrFujP5qQf1rnrz4G6LDO934b1XVdBvMYRra5ZkB9Tk7v/Hq9TooA+ZPHnhL4g6Depr9yw1GezAMetaeuyZFA481VAJAGfmIPB5bGBXf/AA2+NFh4lWHS9eaOy1fhUkPyxXP07K3seCenXA9bPSvG/iZ8FrbW4p9Z8NwR22q8vLaqAsdye5APCufyJ68nNAHsmR1zS182fD/x9400TT5lNpLrunWL+Xd2bZ+2WWM8gH5ig5HIIG0g7K9m8K/Efwz4wVBpuoot0RlrO4xHMOP7pPP1Uke9AHW0UZooAKKKKACiiigAoopDQAtITxWD4k8ZeH/Cdv5us6nDbkjckWd0rj/ZQcke+MV4f4n+NHiLxbO+k+CdOvII3BBkijMl049QFyEH0yfcUAepeOPibp3hWUaZYxNqniCUhYdPgySGPTfjOOucDk8djmsrwh8PNRu9bXxh45nF3rhwba1BBisQDwABkEjsOQCSeW5ri/BPhP4i6F5k2k+FtJsrqcZfUdVl8y4OTk9G+Ue2zJ4yT1HYfYPjSJi41Pw0VySI9jYA9P8AV5/WgD1NRgnrmvJtMlHhH4/6nYzMI7LxLbLPbk5wZlzxk8ZyJeP9pfWpG8Q/F7Rn3X/hTS9WtkHzNYSlHP0BYn8krlvGXjbRfG2nw2V/Be+F/Femyi5sft6bVEnXZvIG0HA5YKMhTnrQB7rq07W+jX08YLPHbyMAoySQpOAB1NcT8JPAv/CGeGFe7T/ia3wEtyTyYx1WP8M5PuT7UfDb4k2XjPT1sbxkt9dhTFxatx5gA++meoPUjt9ME+gcZ60AeY6N4LbVPjFr3i7UoWFvbSR29hHIPvusSK0mO4Hb3J9K9PHXNA4NB6UABryPSJB4v+P99qcR8yw8OWn2SKRT8pmbIIx/wKUZ/wBkVo/EX4gvpkg8M+Gc3nii9PlRxxAN9mBGd7dg2MkA9PvHgDPQfD3wbF4I8NRaaHE13ITNdzgf6yU9fwAwB9M4yTQB1feloooAKKKKACiiigAooooAKKKKAA1i+KPDWn+K/D1zpGpRloJV+V1+9Gw6OvuDz79K2qQ9DQB5R8NfEuoaHq8nw88VNt1KyGNPuWPy3UIHygHvgDj2yCAVOfV8g9xXF/EPwHF400mMwSfZdYsm8yxvASpRuDtJHOCQOeoIBFZ/w78fzaxPL4Z8SILPxTYfu5o3AUXAH8SgcZxgkDg9Rx0APRaKMj1ooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKOgyarX9/a6ZYT3t7OkFtAheSR2wFAoAh1nWLHQtIutU1GdYbS2QvI59B2A7knAA7kgV4V4Tsbr4y+P7jxRrUJGhac4jtrRxuRmHKxnsezP65A6HjC+JfifVPHtg2p2yS2/hq2u1tdPgKnde3B3fNj2XP04A5Y4+gPBPhuLwn4R07R41XzIYgZmH8cp5c5/3s49sCgDeUEEcU+iigAoyKQ9K4rxN8SLHwnrFpp+paNq7SXr+XaywpC0cpyBgEyAjkjg460AdtmiqMl5LFpjXbWFx5irvNtvjEg9sl9n/j1YPhHxvF4xiW5sND1iCxbO27u44o42/3cSFm6YyARmgDrKQ9KWigDyP4l6Je+GNZt/iP4ejIurUhNUt1GBcQnALH6cA+gAP8NbGoeBfBnxJ0q211LTyJbxBNHfWZEUuTj72BgsCMHdkjnkV313bQ3tnNa3MaywTI0ckbDIdSMEH2IOK8r+Es0vh/W/EfgG6dmOmXBuLLecsbdyPy6o3HeQ0AIPC3xP8ACRx4e8S2+u2Shttpqi4cZPADE5P/AH2o9qVfiz4g0TcnizwFqtqEHzXNiPOjJ/H5R/32a6Dx78RLXwhHDY2ts2oa7d82djGCS2SQGYAdMg4HUkHGOSOYg+GniTxq4vviD4guFhcBk0mwcLHHzkBjyvqDwT0+bigDXsvjr4EvEUyajcWjH+Ge1ckfigI/WtuL4o+B5l3r4m08D/bcqfyIzVS1+D/gO0Xanh2F89WmlkkJ/wC+mOPwrL1b4EeCNRiItbS406XORJbTsc/VXLDH0xQB0LfE7wQqlj4m08gf3Zcn8hWRd/G7wDaBgusyTsP4YrWU5+hKgfrXnD+HJPhReL/wkvhzS/EPhqR9v9opZIZoMnA35Hr2Oc5wG7V7V4f03wtJYQahoOnaWtrOu+OW1tkTdn6AfQg88c9KAOG/4XPdavhfC3gjW9TBOPNlXyox7llDDH1Irk/G/ir4k2mnRtrWpWPh9rrKQabp3z3c2eAQQW29fvB19hnivZPGXi2y8F+GrjWLz5wg2wwhsGaQ52oD2zjrzgAntXG/DvwVe396fHPjFRNrl6N9tbsPltIiPlAU9Dg9Ow6/MTQBx/gf4G3WrSLrXjWe4AmPmCy8w+a5PO6V85H0HPPUdD7ppGiaZoVktnpVhBZ24/ghQKD7n1PueavjOe9LQAd6KKKAEPIIrL1rw9pPiKyNnrGnQXkB6CVOV91PVT7jFatFAHzl44+B+o6E51nwdPc3EcLeZ9lBPnwY5zGw+9z26/Wrvgf4hfEW70tZrS0sfFFtAdsyCTy7uHj5Q+cZzjqFbODzmvfyAQQRkGvI/H/hS98MasfiF4RQx3tv82pWSj5LqI/ebA7/AN76bhyOQCQfFLxo0nkj4War52MkmZwnb+Lysfhmo5ovi74x3W8i2XhPTnPzOj77gr7FSTkexSvQ/C/iKw8VeH7TWNOfME68qT80bDgow7EH+hHBraoA5DwX8PNG8FRPJaJJc6hMP399c4aWTJyRn+EZ5wPQZJ611wFLRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAhzg461wvj/wCHkXi2GLUrCf8As/xFZfPaXqfKTjkI5HbPfnbyR1IPd0h6UAebeBPiPNf6i3hfxbB/Zvia3Pl7ZPlS6HYrzgMeuBkHqvoPSsj1rkvG3gDSPHGniK9QwXkI/wBGvYlHmQn+q552/qDXFaT4713wBqMWgfEOJ5LZvltNciBdZB2Eh7n3xkdSDndQB7FRUNrcwXltFcW00c0EihkkjYMrA9wamoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACikJ7d65Pxp4/wBE8F2g+3Sme+lH7ixgOZZT0HH8Iz3P4ZPFAG/q+r6foelz6jql1HbWkC7pJJDwPoOpOegHJPArx+OHWvjZqSzTLPpXgi2lzGn3Zb5lOM+nHcjgdBk5xa0vwX4g+JGpR6/498y10pDvstDjJUAdi/ccevzHP8IAFeu28EVtbxQQQpDDEoRI0UKEAAwABwABxgcUAeSeMLGzn+K/w/8AClrCkFlZB7wW8S4UKuWTj0zCw/E+tewjoP615XeqrftK6Z5jkMuhkoPU7pB/Isa9VoAKKKKAEPSvM/jnpL33w+bUIMi50q5jukdeGAztbB/4Fu/4CK9MJ+U4P61wupLN8Q4bvTLGZ7fw6A0U96n3r1xkbIj/AM81P3m/iIwOMmgCPTNQn+JWkWjJFJbeH5IUa7cgo15Jj5oUHURg8M38RBUcZNd1DDHBHHFFGscaLtVEGFUDoAOwrzL4FajI/g250O6OLvR7yS3dCeQpJYf+Pbx+FepUAFFFFACHoa8W+IWqp4F+MGkeKJB/ot3pk8E6KOZWRWKj6kmIfhXtVeH/AB908at4g8C6cWKi8uprcsOwdoFz+tAG58KPC89wk3jvxCGn17V8yRPIP9RCcBQo7ZAGPRcDjnO5rnxJtPD/AIktNBu9C1l729YLa+SkLLPltowfNGOf72Md67OGGOCFIolCRooRUHRQBgCvJvFH+nftG+ErU4KW1g87exxMR/6CtAHY+J/G0XhDSF1TU9E1X7IdokkhEL+UxOAGHm9enIyORzV/R/Eba3oiapb6NqUUMsImgjm8lXnVhkbQJDgkf3ivWvJ/jpq1xqXhueO2mKaVZXqWshXpcXJDMy+6xgc+rnH8Jr0bUHuLfStK8J6bI0WoXNsI3lTra26KqyS/XkKvqzA/wmgBfDfiu08eWNyIdC1OPTn3wyS3scSxSdmUYclvTIBHBGa4vwp53w0+JUng2aWZ9A1cGfSnkORFJ1KZ7dxjrnYf4q9Z0+xt9MsLexs4VhtrdBHFGowFUDivMvjtA9t4a0rxFaqBe6RqMUsbkdAc8f8AfQSgClPH/wALK+M0lrLmTQfC330x8s1znofxXH0jP96vYgMYHYDpXnXwS0dtN+HsF9PuN5qsr3szPyx3HC8+hVQ3/AjXo9ABRRSEZFABketLkDrXF/EXWvEfhnw3PrmiJp8yWoVriC5gkZmXIG5WV1xjPIIPAPPatDwdq95rnhLT9Yvrizd7uBZsW0LRpHxypy7ZwcgnI6dKAOkorgfDer+NfEGo3U6S6JHoMcxS2vDZS+ZdqOrIvnYC5yA2eeoFd4AcjJoAdTXUMjKehGDTqKAPHvDsZ+G/xbufDYDR6B4gBubBR92KYdUHbsVwMnHl+tewd68v+OOnyjwlaeIrPCX2iXsdzHJjkKWC/wDoRQ/hXoumX0Wp6baahbnMN1Ck8Zx/Cyhh/OgC5RRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUABqlqml2Os6fNY6laRXVrKuGilXIP+B9xyKu0UAeOXHgfxb8OrmXUPAV41/pRO+XRLti2PXyyep+mG4H3uldJ4S+LOg+Ipxp96X0fWFbZJZXx2Hf8A3VY9T7cH2rvj0rmvFPgPw94xhK6vpyPNjCXUfyTJ9GHX6HI9qAOmorx8eFPiP4F58MayniDSU6adqHEir0AViccAdmUf7NXNO+Nmlw3K2Pi3StR8O33dbiJmjPuCBux/wHHvQB6pRWdpWt6VrUPn6XqVrex92glV8fXHT8a0aACiiigAooooAKKKKACikJABJIArkNf+J3g/w4GF7rdvJMuR5FqfOkyOxC52n64oA7CqGr6zpmhae99qt7BaWydZJnCgnrgep46Dn0rzA+P/ABz4yBi8FeF3sbRjgapqvygLjqq9Cfpv+lW9K+DkV3erqvjfVrjxDqIO4RuxWBO+AOpGfoMfw0AUrv4jeI/HFzJpnw50x1gDbJdavV2xxj1UEde/ILY/hrofB3wt07w7enV9Tnk1nXpMtJfXWTtJ4+QHOOOMkk/TOK7i2tYLOCO3toY4YIxtSOJAqqPQAcD8KnoAQZpT0ooPSgDybxgV0j49eC9VkfZDe28ticngt84Uf99SrXrHevNfjbpNzc+C4tZsNy32iXSXsbKMsADhsfT5W/4DXb+HtZtvEXh+w1i0P7m7iEgAOdp6FT7g5H4UAalB6UUUAcf4z8M6/wCKLOawtPEkel2EoxIkVgZJXGOVL+YPlPoFBI4yc10FpZ3FppEdpG9rHNFF5aNFblYVIGBiPfnHTjd+IrQooA838O/DnXPDnizU9etvE9s51OVpLu1bSyImJYt8uJsggk4PPXvXowB4z1706igAooooAD0rwb40a7DZ/FDwXFKwCadLHeSEngBpl6/hF+te8MQFJPSvn218ON8WtR8ea+jfIyix0ls/KTGVYEem7Ymf+ujUAfQPvmvC9WudQuv2i7210pCb1dOW2hmK5W33KpaRh/sq7EDucDvXdfCrxePE/hSK3u22axpoFrewvw+5eA5H+1jn3DDtWzonhW30rX9b1tyJb7VJVLSY+5GqhVQflk//AFhQB5j8bLaz0Xwv4U0OCOU2ovw20fPJIEGGJ7s5MhPuSa9P8LaZcwpPq+qKBq2olZJlznyIxny4QfRQevdix703WfClvrnijRdWvDvi0lZnhhI6yvs2uf8AdCnA9cGuiUYPSgBxryn9oG9S3+Gwt2I3XV5FGB34yx/9Br1U4II/Cvmr4v6rP451PUv7IxLo/heEGadASsk0siIQCOuO3bCMRxigD33wnaf2f4R0Wz7wWEMZ9chAK2ayfDN2t/4W0e8XG2eyhlGOg3ID/WtagAooprsEQszBVAySTgAUAU9Z0+DVtDvtOuT+4urd4ZD6KykE/rXi/wAHxfeJvC8vhq9uI10nSbhluUifL3e4lhFnoI8hiSPvZA4Gc+gNM3j6Z4bd3j8LRMUmmU4OpMDgohHSEYIZv4jkDjJPI6CIvB/7QWqaQqpBZa7ZrcW8agKA65OMdOqzce4oA9fijWJEjjQIiDaqquAB6AdqlozmigAooooA5n4iW63Xw38SRuoIGnTyAH1VCw/UCqnwrumu/hh4fkZslbUR/ghKAfktXfiDMkHw58SO54Om3Cfi0ZUfqaofCi2Nr8LvD8ZXaTbeZj/eYtn8c5oA7OiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigBD0qnqGl2Oq2pttQsre7gPWOeJXUn1wRV2igDzTU/gd4Su7hbrTlvdGuVJZZLC4KgHHowOPwxVH/hBvidoahdC8ex30YOfL1SHJx6biHJ/SvWaKAPKBrnxj0x9lz4V0fU4lH+stJxGW/76f/2Wkf4oeM7ORUvfhdqxGTlraRpf0EZH616xRQB5XF8XtT+bz/hz4nQjoFtmb+ajFQv8XtfZG+z/AAy8RyMPuho5AD9SIzivWcc5/pRQB5X/AMJ38SdQjP8AZ3w3e3fsbu7AH4hgn8xTPJ+NWsxruuPD+hg4yEXzHA/ESDP416xRQB5P/wAKc1DWWLeLPHGsamjEMbeA+TGD6YJYY+gFdXoXw18IeHCj6foVt5y8iedfOkB9QXzj8MV1tFADRninUUUAFFFFABRRRQBFcQRXVtLbzxrJDKhjkRujKRgg/hXkHgS9k+HXje78AarK39n3UhuNFuJOAwbqhPTJxjt8wP8AeFext901yXj3wRbeN9E+yyMLe+tz5tldqPmgk/ng4GfoD1AoA66ivLfA3xCu7bUv+EP8cf6Fr8B8uK4lOI7wdiD03HjHZu3PFepZHrQAUUUUAFFFFABSEgig4IPf2rhvHnxGsfCEa2dtH9v164wtrYRfMxY8KXA5C+3U9vUAGX8WfFFxDaQeDtDzLr2tjyVRP+WcLZDMfTIBGT0G49q7Hwl4bt/Cfhew0W3O4W8fzyY/1kh5Zse7E8emPSuV+HPgS80u5ufFHidzc+J9Q+aUsQwtkPRF9+gOOAAFHA59GAoA8w8beB9Xste/4TbwRIsOtIuLuyYDy7xeOMevHI4zgEEMMmz4Z+L/AIf1ZzZ6yToGrRfLNbah+6UHvtZsD8GwfbvXozcjisLX/B3h/wAUR41rSba7fGBIy4kUegdcMPzoA24pop4xJFIkiMMhkYEH8aztY8R6LoFuZdW1S0s1xkCaYKzf7o6k/QV57P8AADwg8u+3uNXs+CCILlec/wC8hP61oaT8D/A+lyLI2nTX0inIN5MXH4qMKfxBoA5/U/Guu/Eqd9A8B28ttpb5S71qdCihOhCemR/wI+ijJrrovh5Y6V8MtQ8K6cgL3FpIrTOuGlnK8OfxA+gAHauxtbWCygjt7WCOCCMbUjiQKqD0AHQfSpm+6eM/WgDz74La1/a/w1sIpGJuNPZrKZSMFdh+Uf8AfBX8c16FXj1jL/wrr4zXlhOSmi+KT59u7Z2x3OeV9OWYjA/vR+levjrQA49KyNc8Oad4jtDaaolxLbkbWiju5YlcejBGG78c1r0UAZOh6Bp/h3T00/S4pYbROUie4klC+y72OB7DjmsfUPhr4W1XV01a+s7ua/QgpO2o3IaPByNpEny4JPAwBXXUUAQ21ulrBHDGZGSMYBkkaRvxZiST7kmpqKKACjpRTSwK8EUAeafHHUXh8BjR7YFrzWLqK0hjU/M2GDHA79AP+BD1rvtG05NH0Ww0yI5js7eOBM9wihc/pXlllMPiN8avtcZ8zQfC6FYWHKy3BOCefcHkdo1P8Vewj9aAFooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKQ8ilooA5rxb4J0Xxrpos9XtSzLnybiPAlhJ7qcfocg9xXBRz/ABF+GR8q4tn8W+HUwFliz9phX3HJx7fMMAcr0r2Kg9KAOA0L4x+C9aGw6mNOnz80GoKISP8AgWdn612tpqNjfxiWzvLe4jPR4ZVcH8Qayta8F+G/ELO+q6HZ3MrDBmMYEuP98Yb9a5G6+A3gW4YGKzvLYAdIrpyD/wB97qAPSJriCBC800cajkl3AArkta+KPgvQ0b7Tr9pLIM/urVvPfPoQmQD9cVgRfAHwRHIHeK/lA/he5OD+QBrptH+Gvg7QWD6f4fsxIG3CSZTM6n1DSFiD9MUAcRJ438cePf8AR/BWhSaXpsowdW1H5Tg90HI/LefpXUeCfhnpvhOV9RuZpdV12UlptRuvmYk9duc7frkk8844ruACDnNOoAaBjHX+dOoooAKKKKACiiigAooooA5P4geDIPG/hiXTnIiu4z5tpOR/q5R0z32nofrntWH8NfHcuqM/hfxHutfFGnjypY5cD7SFH317FscnHX7wyOno7Zxx1rhvHnw7g8XRw6jZztp3iGy+e0vo+DkHIV8ckZ6HseR3BAO5yPWlrybQ/ijd6Ffp4f8AiJZtpt+vyx6iF/cXPbdkcDtyOOudvSvUrW5t7y3jubWeOeCUbklicMrD2I4IoAnoooyPWgAooyPWq97eWtjZy3N5cRW9vGuZJZXCqo9STwKAJyRg815h8SPGl6LiPwV4ULT+I9QGx2iP/HpGc5YsPutjPuB83B25o6x8TNU8W6g3h/4cWr3Mx+W41aRSsNuD3XI+vzH0+UMcV1fgL4fWPgm1lk8xr3Vrs7ry/l+/ITzgZJwM89eSck9MAGh4K8JWngrw3baPafOVzJPNjHnSnG5j+QA9gB2ro6KKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoPTiiigDN1jQ9N8QWL2Oq2MN1bNyUlTOD6g9QfcHNecS/CDUdBne68CeKrzSGYlvsdwfNgJPbnt/vKxr1migDycal8ZtJZI59D0XWox1lgmETH82X/wBBqX/hN/iURsHw2Pmbsbvty7f8++a9TooA8ma9+NOsmSKHSNE0KM/cmmkEjr+TOP8Axylg+Dc2s3aXnjjxNf63IpDC2jYwwL6jA7f7uyvWKKAKOl6TYaNZR2Wm2UNnbJ92KFAoHucdT796vUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAf/Z']	Multimodal	Competition	True		Numerical		Open-ended	Geometry	Math	English
+17	"A circular disc is divided into 36 sectors. A number is written in each sector. When three consecutive sectors contain $a, b$ and $c$ in that order, then $b=a c$. If the number 2 is placed in one of the sectors and the number 3 is placed in one of the adjacent sectors, as shown, what is the sum of the 36 numbers on the disc?
+
+<image_1>"	['We are told that when $a, b$ and $c$ are the numbers in consecutive sectors, then $b=a c$. This means that if $a$ and $b$ are the numbers in consecutive sectors, then the number in the next sector is $c=\\frac{b}{a}$. (That is, each number is equal to the previous number divided by the one before that.)\n\nStarting with the given 2 and 3 and proceeding clockwise, we obtain\n\n$$\n2,3, \\quad \\frac{3}{2}, \\frac{3 / 2}{3}=\\frac{1}{2}, \\frac{1 / 2}{3 / 2}=\\frac{1}{3}, \\frac{1 / 3}{1 / 2}=\\frac{2}{3}, \\frac{2 / 3}{1 / 3}=2, \\frac{2}{2 / 3}=3, \\quad \\frac{3}{2}, \\ldots\n$$\n\nAfter the first 6 terms, the first 2 terms ( 2 and 3) reappear, and so the first 6 terms will repeat again. (This is because each term comes from the previous two terms, so when two consecutive terms reappear, then the following terms are the same as when these two consecutive terms appeared earlier.)\n\nSince there are 36 terms in total, then the 6 terms repeat exactly $\\frac{36}{6}=6$ times.\n\nTherefore, the sum of the 36 numbers is $6\\left(2+3+\\frac{3}{2}+\\frac{1}{2}+\\frac{1}{3}+\\frac{2}{3}\\right)=6(2+3+2+1)=48$.']	['48']		['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+18	"In the diagram, $A C D F$ is a rectangle with $A C=200$ and $C D=50$. Also, $\triangle F B D$ and $\triangle A E C$ are congruent triangles which are right-angled at $B$ and $E$, respectively. What is the area of the shaded region?
+
+<image_1>"	['Join $B E$.\n\n<img_3698>\n\nSince $\\triangle F B D$ is congruent to $\\triangle A E C$, then $F B=A E$.\n\nSince $\\triangle F A B$ and $\\triangle A F E$ are each right-angled, share a common side $A F$ and have equal hypotenuses $(F B=A E)$, then these triangles are congruent, and so $A B=F E$.\n\nNow $B A F E$ has two right angles at $A$ and $F$ (so $A B$ and $F E$ are parallel) and has equal sides $A B=F E$ so must be a rectangle.\n\nThis means that $B C D E$ is also a rectangle.\n\nNow the diagonals of a rectangle partition it into four triangles of equal area. (Diagonal $A E$ of the rectangle splits the rectangle into two congruent triangles, which have equal area. The diagonals bisect each other, so the four smaller triangles all have equal area.) Since $\\frac{1}{4}$ of rectangle $A B E F$ is shaded and $\\frac{1}{4}$ of rectangle $B C D E$ is shaded, then $\\frac{1}{4}$ of the total area is shaded. (If the area of $A B E F$ is $x$ and the area of $B C D E$ is $y$, then the total shaded area is $\\frac{1}{4} x+\\frac{1}{4} y$, which is $\\frac{1}{4}$ of the total area $x+y$.)\n\nSince $A C=200$ and $C D=50$, then the area of rectangle $A C D F$ is $200(50)=10000$, so the total shaded area is $\\frac{1}{4}(10000)=2500$.']	['2500']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+19	"In the diagram, $\triangle X Y Z$ is isosceles with $X Y=X Z=a$ and $Y Z=b$ where $b<2 a$. A larger circle of radius $R$ is inscribed in the triangle (that is, the circle is drawn so that it touches all three sides of the triangle). A smaller circle of radius $r$ is drawn so that it touches $X Y, X Z$ and the larger circle. Determine an expression for $\frac{R}{r}$ in terms of $a$ and $b$.
+
+<image_1>"	['Suppose that $M$ is the midpoint of $Y Z$.\n\nSuppose that the centre of the smaller circle is $O$ and the centre of the larger circle is $P$. Suppose that the smaller circle touches $X Y$ at $C$ and $X Z$ at $D$, and that the larger circle touches $X Y$ at $E$ and $X Z$ at $F$.\n\nJoin $O C, O D$ and $P E$.\n\nSince $O C$ and $P E$ are radii that join the centres of circles to points of tangency, then $O C$ and $P E$ are perpendicular to $X Y$.\n\nJoin $X M$. Since $\\triangle X Y Z$ is isosceles, then $X M$ (which is a median by construction) is an altitude (that is, $X M$ is perpendicular to $Y Z)$ and an angle bisector (that is, $\\angle M X Y=\\angle M X Z$ ).\n\nNow $X M$ passes through $O$ and $P$. (Since $X C$ and $X D$ are tangents from $X$ to the same circle, then $X C=X D$. This means that $\\triangle X C O$ is congruent to $\\triangle X D O$ by side-side-side. This means that $\\angle O X C=\\angle O X D$ and so $O$ lies on the angle bisector of $\\angle C X D$, and so $O$ lies on $X M$. Using a similar argument, $P$ lies on $X M$.)\n\n<img_3625>\nDraw a perpendicular from $O$ to $T$ on $P E$. Note that $O T$ is parallel to $X Y$ (since each is perpendicular to $P E$ ) and that $O C E T$ is a rectangle (since it has three right angles).\n\nConsider $\\triangle X M Y$ and $\\triangle O T P$.\n\nEach triangle is right-angled (at $M$ and at $T$ ).\n\nAlso, $\\angle Y X M=\\angle P O T$. (This is because $O T$ is parallel to $X Y$, since both are perpendicular to $P E$.)\n\nTherefore, $\\triangle X M Y$ is similar to $\\triangle O T P$.\n\nThus, $\\frac{X Y}{Y M}=\\frac{O P}{P T}$.\n\nNow $X Y=a$ and $Y M=\\frac{1}{2} b$.\n\nAlso, $O P$ is the line segment joining the centres of two tangent circles, so $O P=r+R$.\n\nLastly, $P T=P E-E T=R-r$, since $P E=R, E T=O C=r$, and $O C E T$ is a rectangle. Therefore,\n\n$$\n\\begin{aligned}\n\\frac{a}{b / 2} & =\\frac{R+r}{R-r} \\\\\n\\frac{2 a}{b} & =\\frac{R+r}{R-r} \\\\\n2 a(R-r) & =b(R+r) \\\\\n2 a R-b R & =2 a r+b r \\\\\nR(2 a-b) & =r(2 a+b) \\\\\n\\frac{R}{r} & =\\frac{2 a+b}{2 a-b} \\quad(\\text { since } 2 a>b \\text { so } 2 a-b \\neq 0, \\text { and } r>0)\n\\end{aligned}\n$$\n\nTherefore, $\\frac{R}{r}=\\frac{2 a+b}{2 a-b}$.']	['$\\frac{2 a+b}{2 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+20	"In the diagram, what is the area of figure $A B C D E F$ ?
+
+<image_1>"	['Because all of the angles in the figure are right angles, then $B C=D E=4$.\n\nThus, we can break up the figure into a 4 by 8 rectangle and a 4 by 4 square, by extending $B C$ to hit $F E$. Therefore, the area of the figure is $(8)(4)+(4)(4)=48$.']	['48']		['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+21	"In the diagram, $A B C D$ is a rectangle with $A E=15, E B=20$ and $D F=24$. What is the length of $C F$ ?
+
+<image_1>"	['By the Pythagorean Theorem in triangle $A B E$, $A B^{2}=15^{2}+20^{2}=625$, so $A B=25$.\n\nSince $A B C D$ is a rectangle, $C D=A B=25$, so by the Pythagorean Theorem in triangle $C F D$, we have $625=25^{2}=24^{2}+C F^{2}$, so $C F^{2}=625-576=49$, or $C F=7$.']	['7']		['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+22	"In the diagram, $A B C D$ is a square of side length 6. Points $E, F, G$, and $H$ are on $A B, B C, C D$, and $D A$, respectively, so that the ratios $A E: E B, B F: F C$, $C G: G D$, and $D H: H A$ are all equal to $1: 2$.
+
+What is the area of $E F G H$ ?
+
+<image_1>"	"['Since $A B C D$ is a square of side length 6 and each of $A E: E B, B F: F C, C G: G D$, and $D H: H A$ is equal to $1: 2$, then $A E=B F=C G=D H=2$ and $E B=F C=G D=H A=4$.\n\nThus, each of the triangles $H A E, E B F, F C G$, and $G D H$ is right-angled, with one leg of length 2 and the other of length 4.\n\nThen the area of $E F G H$ is equal to the area of square $A B C D$ minus the combined area of the four triangles, or $6^{2}-4\\left[\\frac{1}{2}(2)(4)\\right]=36-16=20$ square units.'
+ 'Since $A B C D$ is a square of side length 6 and each of $A E: E B, B F: F C, C G: G D$, and $D H: H A$ is equal to $1: 2$, then $A E=B F=C G=D H=2$ and $E B=F C=G D=H A=4$.\n\nThus, each of the triangles $H A E, E B F, F C G$, and $G D H$ is right-angled, with one leg of length 2 and the other of length 4.\n\nBy the Pythagorean Theorem,\n\n$E F=F G=G H=H E=\\sqrt{2^{2}+4^{2}}=\\sqrt{20}$.\n\nSince the two triangles $H A E$ and $E B F$ are congruent (we know the lengths of all three sides of each), then $\\angle A H E=\\angle B E F$. But $\\angle A H E+\\angle A E H=90^{\\circ}$, so $\\angle B E F+\\angle A E H=90^{\\circ}$, so $\\angle H E F=90^{\\circ}$.\n\nIn a similar way, we can show that each of the four angles of $E F G H$ is a right-angle, and so $E F G H$ is a square of side length $\\sqrt{20}$.\n\nTherefore, the area of $E F G H$ is $(\\sqrt{20})^{2}=20$ square units.']"	['20']		['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+23	"In the diagram, line $A$ has equation $y=2 x$. Line $B$ is obtained by reflecting line $A$ in the $y$-axis. Line $C$ is perpendicular to line $B$. What is the slope of line $C$ ?
+
+<image_1>"	['When line $A$ with equation $y=2 x$ is reflected in the $y$-axis, the resulting line (line $B$ ) has equation $y=-2 x$. (Reflecting a line in the $y$-axis changes the sign of the slope.)\n\nSince the slope of line $B$ is -2 and line $C$ is perpendicular to line $B$, then the slope of line $C$ is $\\frac{1}{2}$ (the slopes of perpendicular lines are negative reciprocals).']	['$\\frac{1}{2}$']		['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+24	"Three squares, each of side length 1 , are drawn side by side in the first quadrant, as shown. Lines are drawn from the origin to $P$ and $Q$. Determine, with explanation, the length of $A B$.
+
+<image_1>"	"['Consider the line through $O$ and $P$. To get from $O$ to $P$, we go right 2 and up 1. Since $B$ lies on this line and to get from $O$ to $B$ we go over 1, then we must go up $\\frac{1}{2}$, to keep the ratio constant.\n\nConsider the line through $O$ and $Q$. To get from $O$ to $Q$, we go right 3 and up 1. Since $A$ lies on this line and to get from $O$ to $A$ we go over 1, then we must go up $\\frac{1}{3}$, to keep the ratio constant.\n\nTherefore, since $A$ and $B$ lie on the same vertical line, then $A B=\\frac{1}{2}-\\frac{1}{3}=\\frac{1}{6}$.'
+ 'Since the line through $P$ passes through the origin, then its equation is of the form $y=m x$. Since it passes through the point $(2,1)$, then $1=2 m$, so the line has equation $y=\\frac{1}{2} x$. Since $B$ has $x$-coordinate 1, then $y=\\frac{1}{2}(1)=\\frac{1}{2}$, so $B$ has coordinates $\\left(1, \\frac{1}{2}\\right)$. Similarly, we can determine that the equation of the line through $Q$ is $y=\\frac{1}{3} x$, and so $A$ has coordinates $\\left(1, \\frac{1}{3}\\right)$.\n\nTherefore, since $A$ and $B$ lie on the same vertical line, then $A B=\\frac{1}{2}-\\frac{1}{3}=\\frac{1}{6}$.']"	['$\\frac{1}{6}$']		['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+25	"In the diagram, the parabola with equation $y=x^{2}+t x-2$ intersects the $x$-axis at points $P$ and $Q$.
+
+Also, the line with equation $y=3 x+3$ intersects the parabola at points $P$ and $R$. Determine the value of $t$ and the area of triangle $P Q R$.
+
+<image_1>"	['Point $P$ is the point where the line $y=3 x+3$ crosses the $x$ axis, and so has coordinates $(-1,0)$.\n\nTherefore, one of the roots of the parabola $y=x^{2}+t x-2$ is $x=-1$, so\n\n$$\n\\begin{aligned}\n0 & =(-1)^{2}+t(-1)-2 \\\\\n0 & =1-t-2 \\\\\nt & =-1\n\\end{aligned}\n$$\n\nThe parabola now has equation $y=x^{2}-x-2=(x+1)(x-2)$ (we already knew one of the roots so this helped with the factoring) and so its two $x$-intercepts are -1 and 2 , ie. $P$ has coordinates $(-1,0)$ and $Q$ has coordinates $(2,0)$.\n\nWe now have to find the coordinates of the point $R$. We know that $R$ is one of the two points of intersection of the line and the parabola, so we equate their equations:\n\n$$\n\\begin{aligned}\n3 x+3 & =x^{2}-x-2 \\\\\n0 & =x^{2}-4 x-5 \\\\\n0 & =(x+1)(x-5)\n\\end{aligned}\n$$\n\n(Again, we already knew one of the solutions to this equation $(x=-1)$ so this made factoring easier.) Since $R$ does not have $x$-coordinate -1 , then $R$ has $x$-coordinate $x=5$. Since $R$ lies on the line, then $y=3(5)+3=18$, so $R$ has coordinates $(5,18)$.\n\nWe can now calculate the area of triangle $P Q R$. This triangle has base of length 3 (from $P$ to $Q$ ) and height of length 18 (from the $x$-axis to $R$ ), and so has area $\\frac{1}{2}(3)(18)=27$.\n\nThus, $t=-1$ and the area of triangle $P Q R$ is 27 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+26	"In the diagram, $A C=B C, A D=7, D C=8$, and $\angle A D C=120^{\circ}$. What is the value of $x$ ?
+
+<image_1>"	['We first calculate the length of $A C$ using the cosine law:\n\n$$\n\\begin{aligned}\nA C^{2} & =7^{2}+8^{2}-2(7)(8) \\cos \\left(120^{\\circ}\\right) \\\\\nA C^{2} & =49+64-112\\left(-\\frac{1}{2}\\right) \\\\\nA C^{2} & =169 \\\\\nA C & =13\n\\end{aligned}\n$$\n\nSince triangle $A B C$ is right-angled and isosceles, then $x=A B=\\sqrt{2}(A C)=13 \\sqrt{2}$.']	['$13 \\sqrt{2}$']		['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+27	"Donna has a laser at $C$. She points the laser beam at the point $E$. The beam reflects off of $D F$ at $E$ and then off of $F H$ at $G$, as shown, arriving at point $B$ on $A D$. If $D E=E F=1 \mathrm{~m}$, what is the length of $B D$, in metres?
+
+<image_1>"	['First, we note that a triangle with one right angle and one angle with measure $45^{\\circ}$ is isosceles.\n\nThis is because the measure of the third angle equals $180^{\\circ}-90^{\\circ}-45^{\\circ}=45^{\\circ}$ which means that the triangle has two equal angles.\n\nIn particular, $\\triangle C D E$ is isosceles with $C D=D E$ and $\\triangle E F G$ is isosceles with $E F=F G$. Since $D E=E F=1 \\mathrm{~m}$, then $C D=F G=1 \\mathrm{~m}$.\n\nJoin $C$ to $G$.\n\n<img_3379>\n\nConsider quadrilateral $C D F G$. Since the angles at $D$ and $F$ are right angles and since $C D=G F$, it must be the case that $C D F G$ is a rectangle.\n\nThis means that $C G=D F=2 \\mathrm{~m}$ and that the angles at $C$ and $G$ are right angles.\n\nSince $\\angle C G F=90^{\\circ}$ and $\\angle D C G=90^{\\circ}$, then $\\angle B G C=180^{\\circ}-90^{\\circ}-45^{\\circ}=45^{\\circ}$ and $\\angle B C G=90^{\\circ}$.\n\nThis means that $\\triangle B C G$ is also isosceles with $B C=C G=2 \\mathrm{~m}$.\n\nFinally, $B D=B C+C D=2 \\mathrm{~m}+1 \\mathrm{~m}=3 \\mathrm{~m}$.']	['3']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+28	"An L shape is made by adjoining three congruent squares. The L is subdivided into four smaller L shapes, as shown. Each of the resulting L's is subdivided in this same way. After the third round of subdivisions, how many L's of the smallest size are there?
+<image_1>"	"[""After each round, each L shape is divided into 4 smaller $\\mathrm{L}$ shapes.\n\nThis means that the number of $\\mathrm{L}$ shapes increases by a factor of 4 after each round.\n\nAfter 1 round, there are $4 \\mathrm{~L}$ shapes.\n\nAfter 2 rounds, there are $4^{2}=16$ L's of the smallest size.\n\nAfter 3 rounds, there are $4^{3}=64$ L's of the smallest size.""]"	['64']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Number Theory	Math	English
+29	"Jimmy is baking two large identical triangular cookies, $\triangle A B C$ and $\triangle D E F$. Each cookie is in the shape of an isosceles right-angled triangle. The length of the shorter sides of each of these triangles is $20 \mathrm{~cm}$. He puts the cookies on a rectangular baking tray so that $A, B, D$, and $E$ are at the vertices of the rectangle, as shown. If the distance between parallel sides $A C$ and $D F$ is $4 \mathrm{~cm}$, what is the width $B D$ of the tray?
+
+<image_1>"	['We note that $B D=B C+C D$ and that $B C=20 \\mathrm{~cm}$, so we need to determine $C D$.\n\nWe draw a line from $C$ to $P$ on $F D$ so that $C P$ is perpendicular to $D F$.\n\nSince $A C$ and $D F$ are parallel, then $C P$ is also perpendicular to $A C$.\n\nThe distance between $A C$ and $D F$ is $4 \\mathrm{~cm}$, so $C P=4 \\mathrm{~cm}$.\n\nSince $\\triangle A B C$ is isosceles and right-angled, then $\\angle A C B=45^{\\circ}$.\n\n<img_3723>\n\nThus, $\\angle P C D=180^{\\circ}-\\angle A C B-\\angle P C A=180^{\\circ}-45^{\\circ}-90^{\\circ}=45^{\\circ}$.\n\nSince $\\triangle C P D$ is right-angled at $P$ and $\\angle P C D=45^{\\circ}$, then $\\triangle C P D$ is also an isosceles right-angled triangle.\n\nTherefore, $C D=\\sqrt{2} C P=4 \\sqrt{2} \\mathrm{~cm}$.\n\nFinally, $B D=B C+C D=(20+4 \\sqrt{2}) \\mathrm{cm}$.']	['$(20+4 \\sqrt{2})$']		['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+30	"In the diagram, $\angle A C B=\angle A D E=90^{\circ}$. If $A B=75, B C=21, A D=20$, and $C E=47$, determine the exact length of $B D$.
+
+<image_1>"	['We use the cosine law in $\\triangle A B D$ to determine the length of $B D$ :\n\n$$\nB D^{2}=A B^{2}+A D^{2}-2(A B)(A D) \\cos (\\angle B A D)\n$$\n\nWe are given that $A B=75$ and $A D=20$, so we need to determine $\\cos (\\angle B A D)$.\n\nNow\n\n$$\n\\begin{aligned}\n\\cos (\\angle B A D) & =\\cos (\\angle B A C+\\angle E A D) \\\\\n& =\\cos (\\angle B A C) \\cos (\\angle E A D)-\\sin (\\angle B A C) \\sin (\\angle E A D) \\\\\n& =\\frac{A C}{A B} \\frac{A D}{A E}-\\frac{B C}{A B} \\frac{E D}{A E}\n\\end{aligned}\n$$\n\nsince $\\triangle A B C$ and $\\triangle A D E$ are right-angled.\n\nSince $A B=75$ and $B C=21$, then by the Pythagorean Theorem,\n\n$$\nA C=\\sqrt{A B^{2}-B C^{2}}=\\sqrt{75^{2}-21^{2}}=\\sqrt{5625-441}=\\sqrt{5184}=72\n$$\n\nsince $A C>0$.\n\nSince $A C=72$ and $C E=47$, then $A E=A C-C E=25$.\n\nSince $A E=25$ and $A D=20$, then by the Pythagorean Theorem,\n\n$$\nE D=\\sqrt{A E^{2}-A D^{2}}=\\sqrt{25^{2}-20^{2}}=\\sqrt{625-400}=\\sqrt{225}=15\n$$\n\nsince $E D>0$.\n\nTherefore,\n\n$$\n\\cos (\\angle B A D)=\\frac{A C}{A B} \\frac{A D}{A E}-\\frac{B C}{A B} \\frac{E D}{A E}=\\frac{72}{75} \\frac{20}{25}-\\frac{21}{75} \\frac{15}{25}=\\frac{1440-315}{75(25)}=\\frac{1125}{75(25)}=\\frac{45}{75}=\\frac{3}{5}\n$$\n\n\n\nFinally,\n\n$$\n\\begin{aligned}\nB D^{2} & =A B^{2}+A D^{2}-2(A B)(A D) \\cos (\\angle B A D) \\\\\n& =75^{2}+20^{2}-2(75)(20)\\left(\\frac{3}{5}\\right) \\\\\n& =5625+400-1800 \\\\\n& =4225\n\\end{aligned}\n$$\n\nSince $B D>0$, then $B D=\\sqrt{4225}=65$, as required.']	['65']		['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+31	"A circle, with diameter $A B$ as shown, intersects the positive $y$-axis at point $D(0, d)$. Determine $d$.
+
+<image_1>"	"['The centre of the circle is $(3,0)$ and the circle has a radius of 5.\n\nThus $\\sqrt{d^{2}+3^{2}}=5$\n\n$$\n\\begin{aligned}\n& d^{2}=5^{2}-3^{2} \\\\\n& d^{2}=16\n\\end{aligned}\n$$\n\nTherefore $d=4$, since $d>0$.'
+ 'Since $A B$ is a diameter of the circle, $\\angle A D B=90^{\\circ}$ and $\\angle A O D=90^{\\circ}$.\n\n$\\triangle A D O \\sim \\triangle D B O$\n\nTherefore, $\\frac{O D}{A O}=\\frac{B O}{O D}$\n\nand $d^{2}=2(8)$\n\n$$\n\\begin{aligned}\nd^{2} & =16 \\\\\nd & =4, \\text { since } d>0 .\n\\end{aligned}\n$$'
+ '$\\angle A D B=\\angle A O D=\\angle B O D=90^{\\circ}$\n\nIn $\\triangle A O D, A D^{2}=4+d^{2}$.\n\nIn $\\triangle B O D, D B^{2}=64+d^{2}$.\n\nIn $\\Delta A D B,\\left(4+d^{2}\\right)+\\left(64+d^{2}\\right)=100$\n\n$$\n\\begin{aligned}\n2 d^{2} & =32 \\\\\nd & =4, d>0\n\\end{aligned}\n$$']"	['4']		['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UImKXFrqcbRuvVSUfp+IH5CgDrPiXp01z4JvL60by9Q0r/iY2ko6o8XzH813Lj3rc8M63F4i8MabrEShFu7dZSgOdjEfMue+DkfhQ8i6t4VZ2TC3diWK9gHj6frXC/AS6Nx8MYI2ORBdTRgegJD/APs1AHpVzCtzbTQPykiFGHqCMGvJ/wBn24kj8JarpE5/0iw1F1Kf3QVXj/vpWr133rxvwKR4e+O3jPQiNqX4F7ETwCch8D8JW/75oA9lFFA6UUAFFFFABRRRQAUUUUAFJ3paTvQB454Y/wCTnfFv/YNH8ras74n6hceM/ifo/wAPreVksY5EkvQh5Yld5z/ux9Pdj6CtHwx/yc54t/7Bq/ytqp2tn/ZP7UVw96Cq39s0lm5/jJiAOPpscfhQB7VbwRWlrFbwRrFDEgSNF4CqBgAewFeG/tGXX2k+HNGt1Ml1LLJKEXqc7UUfiSfyr26+vLbTbKa9vJkgtoFLyyyHAUDkmvHvCmi3fxD+JD+PdSgeLRrMhNJilGDNtztfHoDl/wDePGQKAPaIVZIUV23MFAZvU9zT6B0ooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAEJA61474z/wCJx+0J4N0ojdFZwm7OOzDe/P8A36T869iNeOeCj/wkPx88X60Pmg06P7EmecMNqZH/AH6f86APZK4T4keO73wBp0OpDS7e9tZphAoN0Y3DFS2cbCCPlPf0rusgV4T8a72HV/HvhXwxKZDao4ubsRxs52s2D8qgkkKjngfxUAdxd+M/EuleGU8Q3/ha2Nh5KzzR2+pF54oiASSjRKpIBGcN61W8X/EO3b4WjXdAlZ5tTZbSzJ+VklYkHjsy4b8QPWqXi7x5ZeI9Pu/B/hiOS41fUYGgAuVNokcbDax/e7SeM4Cg5rkdc8Lv4RHwv8LzTrO0msG4uSvCs/mRcDPoGIoA9v8AD+jQeHtAsdKtx8ltEELd3f8Aic+7MST7muesPFGpeKNU1ODw6LOKy02Y2sl7eK8vmzAfMqRqV4Hdi3OeARzXZ9B06V4TNqeo/BTxlem6t3vfC2t3Dzo0TDzIpOpwCQMgHBBwCADkYNAHpOg+K7qfxXfeFdYtIINTtoVuopLZyYriEnG5QeVIOAQc855rrh0rzrwHbL4j1q9+IMtzbyyX0AtLS2gk3C1iBBKOcf6wkAkfw5OOteijOBnrQAucVxbeLbzWfGF74e8OR2x/s5VN/f3QZ44nOcRqikF245O4AYI6111zMLe2lnYZEaFyB3AGa8j/AGe9934Y1zVZ233N3qjeax/iIRWz+bmgDsNM8XXUXi+Twnr9vBDqTw/aLSeBm8m7j74DZKsMHK5PQ1Q+KrS6LpFl4vslzeaJco74ODLBIRHJGfY5U+2M1zPxmmfSvGvgPWISVljvHRiD95Q0XH0ILD6Gu7+JcCz/AA08QqwyBYyOPqo3D9RQB01rcRXdpDcwsGimQSIw7qwyD+RqauQ+Fl0bv4X+HpWJJFoI+fRCVH/oNdfQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAY/ibXrfw3oNzqdwpk8oARwr96aQ8IijuSeK5rRPAmpW1o1zdeKdYt9RvW+1Xy2v2fy/OYDIXdExwOg56AYrT8T+B4PFOo2F5davqsH2CZbi2gt3iEayqchyGjbJ+pIHpyc9IkLrbrG0zu4XaZWC7if72ANufwx7UAeK+BIG8OftAeJtGkuppxdW3miWcrvlY7JMnaACcM/QetbHi2JrH9oHwVqUgIhubWa1DDuwWTj85VrcPwtsm8Yf8JUdf1v+18/63fBtxt2bdvlYxt4/XrzU/xH0C61XRrTUtNTzNX0W6TULWPjMpQ5ZP8AgQ/MgUAdi4d428ttrMvDYzg9jiuKHhjx5j/kogH/AHBIf/iq6zSNUtda0i11Kyk8y3uIw6nuM9QR2IOQR2IIq7QBw3/CL+PMn/i4g+v9iQf/ABVddpsF1a6dBBfXn2y6RcSXHlCPzD67RwKt1ma9pH9uaPc6ab26tEuE2PLasqybT1ALKcZHGcZ9KAMG+QeOZ2sUOfDkMgFzID/x/up/1an/AJ5gj5m/ixgcbs9OLu1guotP8xFnMRkSIDoikAnjoMkCvMl+AXh5VCjXPEYUDAAu48Af9+66nwZ8PtK8DvePp9xfXU12EEkl7KrsqrnAUhRxz/L0oA6iK6t7guIZ45DG2xwjAlW9D6H2qXrzXmXw/wDBNl4a8YavdaJd30ulPD5DtcsGElxvJOwgDIQAgtzyxGTg16avIoA8f+G9s+gfF/x3o067Wu3W9h4+8m9mGPoJgPwqX4+CTUPD+haDajdeajqiLEg/iAVl/m613mt+E7TWNTtdWinnsNWtAVhvbbbu2HqjBgQ6n0I78YqK08HwL4gTXtTu59T1KFDHbvMqKlup6+WigAE5PJyeetAF/UGi0jwrdNnEVpZOc9gEjP8AhXF/Amxez+FtlI6bTczyzDPcbtoP/jtanxHnubjQF8Nad82qa432SIDny4v+W0jf7KoSD7stdTo+mW+i6LZaXa58i0gSCPPUhQBk+/FAF3tXjPxQb/hFvip4Q8YAAW7t9iu3PCqvIyffbI5/4AK9mrhfi54c/wCEk+HepQRoXubUC8gAGTujByAO+VLD8aAO6HSiuP8Ahh4kHij4faXfPJvuY4/s1ySefMT5ST7sMN/wKuwHIoAKKKKACiiigAooooAKTvS0negDxzwx/wAnO+Lf+waP5W1eleIPC2keJ4YU1S08x7dvMt5kcxywv2ZHXDA8A+nA64rzXwx/yc74t/7Bo/lbV7IOlAHLnwHpNw8R1SS/1VIiGSO/u3ljVh0JTO1j7sCfeulWMIqoihVUYAUYAHp7U+igBBnHPWloooAKKKKACiiigAooooAKKKKACiiigAooooAKKKM80AZfiPWItA8O6jq0wylnbvLtzjcQMhfqTgfjXAfAfR5bPwJJqtzk3OrXT3DOw+ZlB2jP4hz/AMCqv8dNTnuNK0rwhYEG91u7RSpPARWGM9xlyvP+ya9O0fTYNG0Wy0y2GILSBIU+igAZ9+KALZx0zXiPgMjxb8d/E3iM/Pb6cpt7dgeAf9WpH1VHP/Aq9k1KylvrR4Ir25smcY8622bx9N6sB+VcT4d+Etl4Ta4bQ/Eev2n2gDzgJLdw+M4OHhIzyeaAOT/aJazXSdDMYA1f7WTA0ZxIIwp3YxyPmMf41Y+Jcd5byfDTWtRBEtnqEC3jEYw7eWzZ9P8AVtXc2Pw80S11tdZuzd6rqiY8u71GfzWj/wBxRhV9eBwTxUvj3wx/wlnhC90tWEdyQJbaT+7Mpyv0ycg+xNAHTH+teVfDbyfGs3ifWdehS6upL57BYLhQwgtlAIjCnoCSc+pXPau68K65/wAJB4dtruRDFdqPJvIDw0M68OpHbnOPUEHvVBvBENrrl3rOiajdaTeXuDdpCEeGdh/EyOD83J5BHU0AcBoGjt8O/jfDoOkyyHRNXs3uXgdtwgwG559CmM+j4JOM17Lb3VvcwiW3mjliPAeNgy/mK5mPwRpyS6ne6leXV7qF/atazXszqjRwkHKxhQAg78DPrWP8LfCkPheHWP7Onu5NHuZ0az+1Y3OFXDSABRgMTgeoUHoRkA9AljWaN43GVcFWHqDXknwGjl0iy8S+HLoYu9P1MtIpGCQyhc/Q+XXr2K5u/wDB8E2vnX9Ou5tN1V4xFNNAqlbhR0EiMCGx68HtnHFAHBfFa3bXviX4E0GAb3Sd7qdRyVi3pk/lG/5V2XxRuVtPhj4gkc4DWhiHuXIQD82q7pPhK10vV7nWp7ia+1a5UJJd3O3KIOiIFACr9Bz3zWD8Q7dvFNxpvgq2YkXcq3WpMh/1NrGc/gXYAL9D2FAGx8NrF9O+G/h+2kXa4s0cr6bvm/8AZq6mmxokcSpGAEUAKB0A7U6gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACmkHJIFOooAoafpFppct09nD5X2qXzZVVjt3nqwXOAT1OAMnJ6mrw6UtFABRRRQAU1gT04p1FADdvPrTh0oooAKKKKAKEek2kWsXGqLFuvJ0WNpWckhF6KuThRnkgAZPJzV4DApaKACmsM9s06igDxPwQT8Pvi5rHgyY7dO1b/S9OyflBwSB+QZfcxj1Fe1jpXmPxo8NXOoaDbeJNKLJq2gyfaUdPveWMFvxXaG+gb1rr/BXii28YeFLLWINqvKm2eJTnypRwy/n09QQe9AHQUUdaKACiiigAooooAKTvS0negDxzwx/yc74t/wCwaP5W1eyDpXjfhj/k53xb/wBg0fytq9kHSgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACmswUEscADNKSB1rzX4zeK5NF8Lro2nb31fWm+zQRxcvtJAYjHc5CjvlvagDn/BefiD8YtW8YON+l6QPsmnkjhjggEfgXf2Lr7V7UOlcz4C8KReDfCNnpKAGZR5tzIv/LSVvvH+QHsorp6ACiiigAppGadRQBnQaRaW2qXOoQw+XcXKqJyrfLIV+6SOhYdN2M4wCeAK0BwKWigBpHOR1pVBCgGlooAKKKKAGkHdkVRstItNOmu57aLE13J5s8ruWZz0AJJzgDgDoPStCigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAGSIHVlZQysMEHpXiGlufhD8UZdIuMr4X1599q55SCTIwPbaTtP+zsJ6V7lXL+PfB1r438MT6VORHOP3lrOR/qpR0P0OcH2J74oA6YMMDnqM06vKvhN40urnzvBniPdBr+lfuwJTzPGvHXuyjH1GCM8mvVAeBmgBaKAcjIooAKKKKACk70tJ3oA8c8Mf8nO+Lf+waP5W1eyDpXjfhj/AJOd8W/9g0fytq9kHSgAooooAKKKKACi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+32	"A square $P Q R S$ with side of length $x$ is subdivided into four triangular regions as shown so that area (A) + area $(B)=\text{area}(C)$. If $P T=3$ and $R U=5$, determine the value of $x$.
+
+<image_1>"	['Since the side length of the square is $x, T S=x-3$ and $V S=x-5$\n\nArea of triangle $A=\\frac{1}{2}(3)(x)$.\n\nArea of triangle $B=\\frac{1}{2}(5)(x)$\n\nArea of triangle $C=\\frac{1}{2}(x-5)(x-3)$.\n\nFrom the given information, $\\frac{1}{2}(3 x)+\\frac{1}{2}(5 x)=\\frac{1}{2}(x-5)(x-3)$. Labelled diagram\n\n$3 x+5 x=x^{2}-8 x+15$\n\n$x^{2}-16 x+15=0$\n\n<img_3393>\n\nThus $x=15$ or $x=1$.\n\nTherefore $x=15$ since $x=1$ is inadmissible.']	['15']		['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+33	"In the diagram, $A D=D C, \sin \angle D B C=0.6$ and $\angle A C B=90^{\circ}$. What is the value of $\tan \angle A B C$ ?
+
+<image_1>"	['Let $D B=10$.\n\nTherefore, $D C=A D=6$.\n\nBy the theorem of Pythagoras, $B C^{2}=10^{2}-6^{2}=64$.\n\nTherefore, $B C=8$.\n\n\n\nThus, $\\tan \\angle A B C=\\frac{12}{8}=\\frac{3}{2}$.']	['$\\frac{3}{2}$']		['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+34	"On a cross-sectional diagram of the Earth, the $x$ and $y$-axes are placed so that $O(0,0)$ is the centre of the Earth and $C(6.40,0.00)$ is the location of Cape Canaveral. A space shuttle is forced to land on an island at $A(5.43,3.39)$, as shown. Each unit represents $1000 \mathrm{~km}$.
+
+Determine the distance from Cape Canaveral to the island, measured on the surface of the earth, to the nearest $10 \mathrm{~km}$.
+
+<image_1>"	['$\\tan \\angle A O C=\\frac{3.39}{5.43}$\n\n$\\angle A O C=\\tan ^{-1}\\left(\\frac{3.39}{5.43}\\right)=31.97^{\\circ}$\n\nThe arc length $\\overparen{A C}=\\frac{31.97}{360^{\\circ}}[(2 \\pi)(6.40)]=3.57$ units\n\nThe distance is approximately $3570 \\mathrm{~km}$.']	['3570']		['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+35	"The parabola $y=-x^{2}+4$ has vertex $P$ and intersects the $x$-axis at $A$ and $B$. The parabola is translated from its original position so that its vertex moves along the line $y=x+4$ to the point $Q$. In this position, the parabola intersects the $x$-axis at $B$ and $C$. Determine the coordinates of $C$.
+
+<image_1>"	"['The parabola $y=-x^{2}+4$ has vertex $P(0,4)$ and intersects the $x$-axis at $A(-2,0)$ and $B(2,0)$. The intercept $B(2,0)$ has its pre-image, $B^{\\prime}$ on the parabola $y=-x^{2}+4$. To find $B^{\\prime}$, we find the point of intersection of the line passing through $B(2,0)$, with slope 1 , and the parabola $y=-x^{2}+4$.\n\nThe equation of the line is $y=x-2$.\n\nIntersection points, $x-2=-x^{2}+4$\n\n$$\n\\begin{array}{r}\nx^{2}+x-6=0 \\\\\n(x+3)(x-2)=0 .\n\\end{array}\n$$\n\nTherefore, $x=-3$ or $x=2$.\n\nFor $x=-3, y=-3-2=-5$. Thus $B^{\\prime}$ has coordinates $(-3,-5)$.\n\nIf $(-3,-5) \\rightarrow(2,0)$ then the required general translation mapping $y=-x^{2}+4$ onto the parabola with vertex $Q$ is $(x, y) \\rightarrow(x+5, y+5)$.\n\nPossibility 1\n\nUsing the general translation, we find the coordinates of $Q$ to be, $P(0,4) \\rightarrow Q(0+5,4+5)=Q(5,9)$.\n\nIf $C$ is the reflection of $B$ in the axis of symmetry of the parabola, i.e. $x=5, C$ has coordinates $(8,0)$.\n\nPossibility 2\n\nIf $B^{\\prime}$ has coordinates $(-3,-5)$ then $C^{\\prime}$ is the reflection of $B^{\\prime}$ in the $y$-axis. Thus $C^{\\prime}$ has coordinates $(3,-5)$.\n\nIf we apply the general translation then $C$ has coordinates $(3+5,-5+5)$ or $(8,0)$.\n\nThus $C$ has coordinates $(8,0)$.\n\nPossibility 3\n\nUsing the general translation, we find the coordinates of $Q$ to be, $P(0,4) \\rightarrow Q(0+5,4+5)=Q(5,9)$.\n\nThe equation of the image parabola is $y=-(x-5)^{2}+9$.\n\n\n\nTo find its intercepts, $-(x-5)^{2}+9=0$\n\n$$\n\\begin{aligned}\n(x-5)^{2} & =9 \\\\\nx-5 & = \\pm 3 .\n\\end{aligned}\n$$\n\nTherefore $x=8$ or $x=2$.\n\nThus $C$ has coordinates $(8,0)$.'
+ 'The translation moving the parabola with equation $y=-x^{2}+4$ onto the parabola with vertex $Q$ is $T(t, t)$ because the slope of the line $y=x+4$ is 1 .\n\nThe pre-image of $B^{\\prime}$ is $(2-t,-t)$.\n\nSince $B^{\\prime}$ is on the parabola with vertex $P$, we have\n\n$$\n\\begin{aligned}\n-t & =-(2-t)^{2}+4 \\\\\n-t & =-4+4 t-t^{2}+4 \\\\\nt^{2}-5 t & =0 \\\\\nt(t-5) & =0\n\\end{aligned}\n$$\n\nTherefore, $t=0$ or $t=5$.\n\nThus $B^{\\prime}$ is $(-3,-5)$.\n\nLet $C$ have coordinates $(c, 0)$.\n\nThe pre-image of $C$ is $(c-5,-5)$.\n\nTherefore, $-5=-(c-5)^{2}+4$.\n\nOr, $(c-5)^{2}=9$.\n\nTherefore $c-5=3$ or $c-5=-3$.\n\n$$\nc=8 \\text { or } \\quad c=2\n$$\n\nThus $C$ has coordinates $(8,0)$.']"	['$(8,0)$']		['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+36	"In the isosceles trapezoid $A B C D$, $A B=C D=x$. The area of the trapezoid is 80 and the circle with centre $O$ and radius 4 is tangent to the four sides of the trapezoid. Determine the value of $x$.
+
+<image_1>"	['Using the tangent properties of a circle, the lengths of line segments are as shown on the diagram.\n\nArea of trapezoid $A B C D=\\frac{1}{2}(8)(B C+A D)$\n\n$$\n\\begin{aligned}\n& =4(2 b+2 x-2 b) \\\\\n& =8 x .\n\\end{aligned}\n$$\n\n<img_3854>\n\nThus, $8 x=80$.\n\nTherefore, $x=10$.']	['10']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+37	"In the diagram, points $P(p, 4), B(10,0)$, and $O(0,0)$ are shown. If $\triangle O P B$ is right-angled at $P$, determine all possible values of $p$.
+
+<image_1>"	"['Since $\\angle O P B=90^{\\circ}$, then $O P$ and $P B$ are perpendicular, so the product of their slopes is -1 .\n\nThe slope of $O P$ is $\\frac{4-0}{p-0}=\\frac{4}{p}$ and the slope of $P B$ is $\\frac{4-0}{p-10}=\\frac{4}{p-10}$.\n\nTherefore, we need\n\n$$\n\\begin{aligned}\n\\frac{4}{p} \\cdot \\frac{4}{p-10} & =-1 \\\\\n16 & =-p(p-10) \\\\\np^{2}-10 p+16 & =0 \\\\\n(p-2)(p-8) & =0\n\\end{aligned}\n$$\n\nand so $p=2$ or $p=8$. Since each these steps is reversible, then $\\triangle O P B$ is right-angled precisely when $p=2$ and $p=8$.'
+ 'Since $\\triangle O P B$ is right-angled at $P$, then $O P^{2}+P B^{2}=O B^{2}$ by the Pythagorean Theorem. Note that $O B=10$ since $O$ has coordinates $(0,0)$ and $B$ has coordinates $(10,0)$.\n\nAlso, $O P^{2}=(p-0)^{2}+(4-0)^{2}=p^{2}+16$ and $P B^{2}=(10-p)^{2}+(4-0)^{2}=p^{2}-20 p+116$. Therefore,\n\n$$\n\\begin{aligned}\n\\left(p^{2}+16\\right)+\\left(p^{2}-20 p+116\\right) & =10^{2} \\\\\n2 p^{2}-20 p+32 & =0 \\\\\np^{2}-10 p+16 & =0\n\\end{aligned}\n$$\n\n\n\nand so $(p-2)(p-8)=0$, or $p=2$ or $p=8$. Since each these steps is reversible, then $\\triangle O P B$ is right-angled precisely when $p=2$ and 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+38	"A snail's shell is formed from six triangular sections, as shown. Each triangle has interior angles of $30^{\circ}, 60^{\circ}$ and $90^{\circ}$. If $A B$ has a length of $1 \mathrm{~cm}$, what is the length of $A H$, in $\mathrm{cm}$ ?
+
+<image_1>"	"['In a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle, the ratio of the side opposite the $90^{\\circ}$ to the side opposite the $60^{\\circ}$ angle is $2: \\sqrt{3}$.\n\nNote that each of $\\triangle A B C, \\triangle A C D, \\triangle A D E, \\triangle A E F, \\triangle A F G$, and $\\triangle A G H$ is a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle.\n\nTherefore, $\\frac{A H}{A G}=\\frac{A G}{A F}=\\frac{A F}{A E}=\\frac{A E}{A D}=\\frac{A D}{A C}=\\frac{A C}{A B}=\\frac{2}{\\sqrt{3}}$.\n\nThus, $A H=\\frac{2}{\\sqrt{3}} A G=\\left(\\frac{2}{\\sqrt{3}}\\right)^{2} A F=\\left(\\frac{2}{\\sqrt{3}}\\right)^{3} A E=\\left(\\frac{2}{\\sqrt{3}}\\right)^{4} A D=\\left(\\frac{2}{\\sqrt{3}}\\right)^{5} A C=\\left(\\frac{2}{\\sqrt{3}}\\right)^{6} A B$.\n\n(In other words, to get from $A B=1$ to the length of $A H$, we multiply by the ""scaling factor"" $\\frac{2}{\\sqrt{3}}$ six times.)\n\nTherefore, $A H=\\left(\\frac{2}{\\sqrt{3}}\\right)^{6}=\\frac{64}{27}$.']"	['$\\frac{64}{27}$']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+39	"In rectangle $A B C D$, point $E$ is on side $D C$. Line segments $A E$ and $B D$ are perpendicular and intersect at $F$. If $A F=4$ and $D F=2$, determine the area of quadrilateral $B C E F$.
+
+<image_1>"	"['Since $\\triangle A F D$ is right-angled at $F$, then by the Pythagorean Theorem,\n\n$$\nA D=\\sqrt{A F^{2}+F D^{2}}=\\sqrt{4^{2}+2^{2}}=\\sqrt{20}=2 \\sqrt{5}\n$$\n\nsince $A D>0$.\n\nLet $\\angle F A D=\\beta$.\n\nSince $A B C D$ is a rectangle, then $\\angle B A F=90^{\\circ}-\\beta$.\n\nSince $\\triangle A F D$ is right-angled at $F$, then $\\angle A D F=90^{\\circ}-\\beta$.\n\nSince $A B C D$ is a rectangle, then $\\angle B D C=90^{\\circ}-\\left(90^{\\circ}-\\beta\\right)=\\beta$.\n\n\n\n<img_3979>\n\nTherefore, $\\triangle B F A, \\triangle A F D$, and $\\triangle D F E$ are all similar as each is right-angled and has either an angle of $\\beta$ or an angle of $90^{\\circ}-\\beta$ (and hence both of these angles).\n\nTherefore, $\\frac{A B}{A F}=\\frac{D A}{D F}$ and so $A B=\\frac{4(2 \\sqrt{5})}{2}=4 \\sqrt{5}$.\n\nAlso, $\\frac{F E}{F D}=\\frac{F D}{F A}$ and so $F E=\\frac{2(2)}{4}=1$.\n\nSince $A B C D$ is a rectangle, then $B C=A D=2 \\sqrt{5}$, and $D C=A B=4 \\sqrt{5}$.\n\nFinally, the area of quadrilateral $B C E F$ equals the area of $\\triangle D C B$ minus the area $\\triangle D F E$. Thus, the required area is\n\n$$\n\\frac{1}{2}(D C)(C B)-\\frac{1}{2}(D F)(F E)=\\frac{1}{2}(4 \\sqrt{5})(2 \\sqrt{5})-\\frac{1}{2}(2)(1)=20-1=19\n$$'
+ 'Since $\\triangle A F D$ is right-angled at $F$, then by the Pythagorean Theorem,\n\n$$\nA D=\\sqrt{A F^{2}+F D^{2}}=\\sqrt{4^{2}+2^{2}}=\\sqrt{20}=2 \\sqrt{5}\n$$\n\nsince $A D>0$.\n\nLet $\\angle F A D=\\beta$.\n\nSince $A B C D$ is a rectangle, then $\\angle B A F=90^{\\circ}-\\beta$. Since $\\triangle B A F$ is right-angled at $F$, then $\\angle A B F=\\beta$.\n\nSince $\\triangle A F D$ is right-angled at $F$, then $\\angle A D F=90^{\\circ}-\\beta$.\n\nSince $A B C D$ is a rectangle, then $\\angle B D C=90^{\\circ}-\\left(90^{\\circ}-\\beta\\right)=\\beta$.\n\n<img_3412>\n\nLooking at $\\triangle A F D$, we see that $\\sin \\beta=\\frac{F D}{A D}=\\frac{2}{2 \\sqrt{5}}=\\frac{1}{\\sqrt{5}}, \\cos \\beta=\\frac{A F}{A D}=\\frac{4}{2 \\sqrt{5}}=\\frac{2}{\\sqrt{5}}$, and $\\tan \\beta=\\frac{F D}{A F}=\\frac{2}{4}=\\frac{1}{2}$.\n\nSince $A F=4$ and $\\angle A B F=\\beta$, then $A B=\\frac{A F}{\\sin \\beta}=\\frac{4}{\\frac{1}{\\sqrt{5}}}=4 \\sqrt{5}$.\n\nSince $F D=2$ and $\\angle F D E=\\beta$, then $F E=F D \\tan \\beta=2 \\cdot \\frac{1}{2}=1$.\n\nSince $A B C D$ is a rectangle, then $B C=A D=2 \\sqrt{5}$, and $D C=A B=4 \\sqrt{5}$.\n\nFinally, the area of quadrilateral $E F B C$ equals the area of $\\triangle D C B$ minus the area $\\triangle D F E$. Thus, the required area is\n\n$$\n\\frac{1}{2}(D C)(C B)-\\frac{1}{2}(D F)(F E)=\\frac{1}{2}(4 \\sqrt{5})(2 \\sqrt{5})-\\frac{1}{2}(2)(1)=20-1=19\n$$']"	['19']		['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+40	"In the diagram, points $B, P, Q$, and $C$ lie on line segment $A D$. The semi-circle with diameter $A C$ has centre $P$ and the semi-circle with diameter $B D$ has centre $Q$. The two semi-circles intersect at $R$. If $\angle P R Q=40^{\circ}$, determine the measure of $\angle A R D$.
+
+<image_1>"	['Suppose $\\angle P A R=x^{\\circ}$ and $\\angle Q D R=y^{\\circ}$.\n\n<img_3198>\n\nSince $P R$ and $P A$ are radii of the larger circle, then $\\triangle P A R$ is isosceles.\n\nThus, $\\angle P R A=\\angle P A R=x^{\\circ}$.\n\nSince $Q D$ and $Q R$ are radii of the smaller circle, then $\\triangle Q R D$ is isosceles.\n\nThus, $\\angle Q R D=\\angle Q D R=y^{\\circ}$.\n\nIn $\\triangle A R D$, the sum of the angles is $180^{\\circ}$, so $x^{\\circ}+\\left(x^{\\circ}+40^{\\circ}+y^{\\circ}\\right)+y^{\\circ}=180^{\\circ}$ or $2 x+2 y=140$ or $x+y=70$.\n\nTherefore, $\\angle C P D=x^{\\circ}+40^{\\circ}+y^{\\circ}=(x+y+40)^{\\circ}=110^{\\circ}$.']	['$110$']		['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']	Multimodal	Competition	False	^{\circ}	Numerical		Open-ended	Geometry	Math	English
+41	"In the diagram, a line is drawn through points $P, Q$ and $R$. If $P Q=Q R$, what are the coordinates of $R$ ?
+
+<image_1>"	"['To get from $P$ to $Q$, we move 3 units right and 4 units up.\n\nSince $P Q=Q R$ and $R$ lies on the line through $Q$, then we must use the same motion to get from $Q$ to $R$.\n\nTherefore, to get from $Q(0,4)$ to $R$, we move 3 units right and 4 units up, so the coordinates of $R$ are $(3,8)$.'
+ 'The line through $P(-3,0)$ and $Q(0,4)$ has slope $\\frac{4-0}{0-(-3)}=\\frac{4}{3}$ and $y$-intercept 4 , so has equation $y=\\frac{4}{3} x+4$.\n\nThus, $R$ has coordinates $\\left(a, \\frac{4}{3} a+4\\right)$ for some $a>0$.\n\nSince $P Q=Q R$, then $P Q^{2}=Q R^{2}$, so\n\n$$\n\\begin{aligned}\n(-3)^{2}+4^{2} & =a^{2}+\\left(\\frac{4}{3} a+4-4\\right)^{2} \\\\\n25 & =a^{2}+\\frac{16}{9} a^{2} \\\\\n\\frac{25}{9} a^{2} & =25 \\\\\na^{2} & =9\n\\end{aligned}\n$$\n\nso $a=3$ since $a>0$.\n\nThus, $R$ has coordinates $\\left(3, \\frac{4}{3}(3)+4\\right)=(3,8)$.']"	['(3,8)']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAWsCNAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAK8u8Xa7rvhT4heH0GrzzaPqdxtkgljixH84BAIUNgB16knivUa8u+Odi8nhGz1KEfvrG9Vt391WBB/wDHgtAHZ+LbmWx8I6rd2961nNBbPLHOqqdrKMgYYEcnj8fpVL4e3Ws33gqxvddnMl7cBpQzIEOwk7Mge2D+NYOrXw+IOr6X4ftSTpQhi1HVnXgMrANFDn3PJ9vcV6QqhUVVUKoGAAOgoAUdKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACuO+JxtX+H2qw3JJaZVigRRlnm3AooHc5Aq74um8RQQafN4bsheSx3Qa4hMyxB4trArluOpB/Cs/TNF1rW9attb8UxW9sLP5rLTIJPMWJz1kkbozemOB160ATfD/wAJr4P8KwWkoBvpgJbt85+cjG0H0UcfgT3rrh0oHQUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABXOeONc/4R7wjqF8r7JvL8uE5wfMbgH8M5/A10deJfG7XPNvrHQ4mysI+0zAf3jwv5Lu/76oA774b63/b3gqykd909sPs031Xpn6rtP41146V4N8Ftd+xeIbnSJH/dXse6Mf8ATRATj8Vz+Qr3kdKQBRRRTAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAimlSCGSaVwkcalnYnoBySfwryC08PPrGk+KPHGrRN5t5ZXTWEMgxsj8tgrkf7uAPbJ7jHrOo2MepWUlnNnyZcLKo/iTPzL9GHB9iaz/FahPBOuKoAUadOAB2HltQwPItO8MTp4B0Hxjo0WdS05pJpox1njWZ/zIHHuPoAfcLG7hv7G3vLd98M8ayI2MZVhkfoa5P4Vgf8ACtdJz/02/wDRz10+nWEOmWv2W2G23VmMadkBO7H0BJx6DikgLtFFFMAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArG8Xf8iXr3/YPuP8A0W1bNY3i7/kS9e/7B9x/6LagDG+FX/JNdJ/7bf8Ao567KuN+FX/JNdJ/7bf+jnrsqFsAUUUUAFFFFABRRRQAUUUUAFYHjDxNa+EvDs+q3UZmCFUjhDBTIxPC5/MnrwDW/XkXx3uLKTw/ZWRnJ1AXInjt0OcphlLMOnU8Z9/fABgWnx/1EakzXejWzae33Y43YSoPdjw3rwor3GyvYNR0+3vrVt1vcRLLGx7qRkfjz3r45tdD1S+1A6fbaddy3gO1oEhbev1B5A7kmvrXwloreH/Cem6VIxaSCACQls/OeWAPpknHtQM5nW/igvh3U7fTtR8Oakl1cf6hFeF/M5wMbX4yfUVNd/EltLh8/VPCXiG1tl+/OYFdV9zhq43xF/xPP2itJsvvLYLFkehVWm/qK9lvpLaGwuJLto1tVjYyl/uhMc59sUCKWg6/pviTTUv9LukuLdyQSOqn0YdjWsOgrxP4ARzBdfmQOLF3iWPd3YbuPqARn6ivbKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKxvF3/ACJevf8AYPuP/RbVs1jeLv8AkS9e/wCwfcf+i2oAxvhV/wAk10n/ALbf+jnrsq434Vf8k10n/tt/6OeuyoWwBRRRQAUUUUAFFFFABRRXL+NPGll4P03zZSJr2UH7PbA4LH1Poo9aAIvHPja08IaaekuozKRbwZz7b2x0UH8T0HtzPgPwTe3OoHxd4qJl1GY+bDBKPuejsOgP91eij3xiHwR4MvNa1L/hMPFJMtxMwntoH5A/uuR2AH3Vr1kdKAGBQCWwMnqcdaXP606sTxH4m0rw1ZNNqN9BbyNG7QpI2GkKjoB35I/OgDxrwzaap4o+L/iPUdK1VdPlt3l23BthP8u7Yq7WIxkDOfbFTam+t6j8SI/BfjLxBdy6ZNgxtbRpbrPkZTcoByCRt5zhhUnwL1PS7N9UF5fwRahfTRpDDK2HcAE8euSx/Ktr446HI+mWHiayyl1p0oSR14IQnKt/wF8Y92oA9M0fRrDQdMj07TbZbe2jGAi85J6knufetCuZ8H+MtM8WaXbyW91C18IVe5tgfmiboePTOcGumoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArG8Xf8iXr3/YPuP/RbVs1jeLv+RL17/sH3H/otqAMb4Vf8k10n/tt/6OeuyrjfhV/yTXSf+23/AKOeuyoWwBRRRQAUUUUAFFFYHizxVY+EtIe+uzvlb5YIA2Glb0HoPU9h68AgEPjLxbY+ENKN1dYkuJcrb2+cGRh/JR3Pv34B4fwZ4RvfFWpnxh4sBk80h7a1dcAgfdJH9zHQd+p46x+EPC2oeNNXPi3xWrSQHBtLZhhXHUHb2Qdh/EeTx19gAwoHtQAo6UUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVjeLv8AkS9e/wCwfcf+i2rZrG8Xf8iXr3/YPuP/AEW1AGN8Kv8Akmuk/wDbb/0c9dlXG/Cr/kmuk/8Abb/0c9dlQtgCiiigAoorG8ReIbDwzpcuo6jNtjThI1+9I3ZVHcn/APXxzQAzxN4lsPCukyX98/ciKIH5pX7KP6nt1rzbwt4b1L4ga4PFPihc2CnNtasPlkA6AD+4O/8AeP40zQNA1H4l64fEviIPHpCNi2tgcCQA42jvtyOW6noPb2ZEWONURQqKAFVRgAegoAVQFUADAAwBS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFY3i7/kS9e/7B9x/wCi2rZrG8Xf8iXr3/YPuP8A0W1AGN8Kv+Sa6T/22/8ARz12Vcb8Kv8Akmuk/wDbb/0c9dlQtgCiis3W9XstC0ybUdQl8uCJcnHVj2AHcnsKAGa9r1h4c0uXUdQl2Qx8BR96Ru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+42	"In the diagram, $O A=15, O P=9$ and $P B=4$. Determine the equation of the line through $A$ and $B$. Explain how you got your answer.
+
+<image_1>"	['Since $O P=9$, then the coordinates of $P$ are $(9,0)$.\n\nSince $O P=9$ and $O A=15$, then by the Pythagorean Theorem,\n\n$$\nA P^{2}=O A^{2}-O P^{2}=15^{2}-9^{2}=144\n$$\n\nso $A P=12$.\n\nSince $P$ has coordinates $(9,0)$ and $A$ is 12 units directly above $P$, then $A$ has coordinates $(9,12)$.\n\nSince $P B=4$, then $B$ has coordinates $(13,0)$.\n\nThe line through $A(9,12)$ and $B(13,0)$ has slope $\\frac{12-0}{9-13}=-3$ so, using the point-slope form, has equation $y-0=-3(x-13)$ or $y=-3 x+39$.']	['$y=-3 x+39$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAW4BvAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAopCQoJJAA6k00SIxIVgSOuOcUAPophlRSAWAz05p2QTQAtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBHPDFcW8kE0aSxSKUeN1BVlPBBB4IPpXkHhS2h8MfHPWtGgjSCzvLbfbxoMKDhXwB7fOB6Yr2OvH/iNNF4c+KfhTxJK3l28im3mfsFBIYn/AIDL+lAGj8WtOg8RHQfDltGkmqXd0XjfGWhhCnzHz2H3frt7kV6TbwrBDFCuSsaBVJPYDFcZ4Gsp9Vu7zxnqUZS51QBbOJ+sFoPuD6t94/h713AFAC0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFB5FADDKm4puG4DJGecev6GvPPF+n2vjrxbpnh1UEtvpzfbNRmH/LNSMJD/ALzdx1wM1ra74Q1fUfEh1jTfFE2lbrZbaSKO1WTcoYtwSeDlj2rU0bSNM8J6clvHI+Z5h5lxcPuluJnOMs3diaANlIwgVVAVV4VQOAKfSZpaACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAQ9K8c+M3iSW2v9K0y0l2SW7C+cjswJEf8mP4ivYpHWONndgqKMlj0A9a+UvFGrvr/iS/1Vw3lzzMI8jogwFX6hdufrSbA+ntD1SPWtEstSh+5cwq+PQkcj8DkfhWjXlfwW1iR9MvtAudyT2UhkSN+GCE4YY9mBz/AL1eqZpgFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABSHgUtIRkYoA5nxxJeS+H20rTP+P/VH+yx+iqeXY+ihA2T7j1ryv4p+H7PwxonhnTLIDYgud8mMGRz5WWPvx+AwO1e6+QnnrOUUyKpVXPUA4JA9jgfXA9BXkPx34/4R/p/y8H/0XQB0+uaKNC8ZWfi+wXbDKwg1SNRxsbgS/QHBb/dB7Gu7U5OKbJEksbRSKrRuCrKwyGB7H2pY4xGgUEkAAcnJ/OgB9FFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFJmloAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAA9K8b+PH/Mv/APbx/wC0q9kPSvG/jx/zL/8A28f+0qAPY+9LSd6WgAooooAKKKKACiiigAooooAKKKKACkbAU5paq6jf2+m6bc311J5dvBGZHc9gBmgDiviL8RW8Gm3gsrJL26ZfOnRmIEUWQoY46ZYgCuo8L6z/AMJD4Z07Vtio11AHdVOQrdGA/EGvPE0G41/4feKPEWown+0dbtmuII26xQRjfDGPrtB98jOK0PgdqH2zwALYtlrK6kiA9A2H/mx/KgZ6VRRRQIKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAD0rxv48f8AMv8A/bx/7Sr2Q9K8b+PH/Mv/APbx/wC0qAPY+9LSd6WgAooooAKKKKACiiigAooooAKKKCcDNAAeleZ/EW6v/EOrWng7RYFunwt7qKNL5amFWG2Mt23H+QruNc1ldF05rn7FfXjnISGztnlZjjP8IO0e5rkfhhFO8eq6jqtlfwa5f3BmujdWckSquSERGYAMAB29fagC01947a3MB8LaT5ZUoV/tA4xjGMbfSuJ+BU0un634i0G4+WWMq+zOdrIxR+ffK/lXtF1Otray3DrI6xqWKxoXYgeijJJ9hXhmhtqOjfGG/wBej8P69/ZF40u9/wCzZt2H+bO3b/fx70Ae80UyOQSxq6hgGAIDDB59QeRT6ACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAA9K8b+PH/Mv/8Abx/7Sr2Q9K8b+PH/ADL/AP28f+0qAPY+9LSd6WgAooooAKKKKACiiigAooooAKD0oooAaFwc55oxg5pxOKwPGHia38K+HbjUJsNLjZBGf+Wkh6D6dz7A0Abuc4Azzz0ox0/LrXg3wx8a3sXjA2uq3k0kGpMy4kbKpOTlSAemclePUele9Bske/egBAuDTqKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAA9K8b+PH/Mv/wDbx/7Sr2Q9K8b+PH/Mv/8Abx/7SoA9j70tJ3paACiiigAooooAKKKKACiiigAoPAzRSHpQAySREid5GCooJZicAD1zXjEfmfFfx/u+b/hHdLPA6Bxnj8Xxn2VfXrsfFLxJcTzQeD9GzJf3xVZwh5Cnon/Aup/2fY8dr4S8N23hbQINOhIeQDfPLjmSQ9T9Ow9gKQHh2o+HmvbvxvqVmCk+kaj5iiPjEZkl3YHbG1W+imvbvBXiBfEvhez1AsDPt8u4HpKo+b6Z4P0Irkfh1Gk3i/4gRSqrxvf7WVhkMDJPkEVmeDZH8DfEu+8LTuwsL47rUt68lDn3GUPqQPSgD2KikBz2NLTAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAD0rxv48f8y//ANvH/tKvZD0rxv48f8y//wBvH/tKgD2PvS0neloAKKKKACiiigAooooAKKKCcCgA6CsLxb4ltvCvh+fUZwHcDZDET/rJD0X6dz7A1tSSKkbMx2qoJJJxgCvGD5nxX+IGxSw8O6Yc5AIEgz1Hu5Hsdq+opAbHwu8NXFzJP4w1ktLfXxYwFxghG+8/tnoPRR6NXqABHXFIkSRKixoqIo2qqjAA9BT6YHmfw158aePh/wBRAf8AoyenfF7w/JdaLB4gswUvdMYMWXg+WSP/AEFsH6bqb8Nf+R18e/8AYQH/AKMnr0e5gjubaW3mRXilQo6sMhgRggigDH8I69H4l8NWeqKR5kiYmC/wyDhhj9R7EVu14/4DuJPBnxC1PwjdSE2ty3mWrMepxlT77k6/7S4r1/POKAFooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAPSvG/jx/zL/8A28f+0q9kPSvG/jx/zL//AG8f+0qAPY+9LSd6WgAooooAKKKKACiiigApD0NKTgZrD8V+Jbbwt4fn1KcBnX5IIicebIfur/U+gBoA4n4p+JZj5PhPSCZL+/KrMsZ+YIx4T2LfyHoa7Pwf4ag8K+H4NOiKvNjfcTDrLIRy307D2Arivhd4bubuafxjrOZb69Zjblx0U9X9s9B6D616mBikAtFFFMDzT4af8jr49/7CI/8ARk9elnpXmnw0/wCR18e/9hEf+jJ69LPIoA8w+L2hzNYWniWxcx3mmyAtIvUKWG1v+Atg46ck12/hjXYvEfh+z1SMBTOn7xAc7HHDLn2Oa0by0hvbOe2uEDwzI0cikfeUjBFeT/Du6m8I+N9T8GXzkxSOZLVmPVgM5/4EmCfdcUgPX6KTNLTAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAPSvG/jx/zL/8A28f+0q9kPSvG/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+43	"In the diagram, $\triangle A B C$ is right-angled at $B$ and $A B=10$. If $\cos (\angle B A C)=\frac{5}{13}$, what is the value of $\tan (\angle A C B)$ ?
+
+<image_1>"	['Since $\\cos (\\angle B A C)=\\frac{A B}{A C}$ and $\\cos (\\angle B A C)=\\frac{5}{13}$ and $A B=10$, then $A C=\\frac{13}{5} A B=26$.\n\nSince $\\triangle A B C$ is right-angled at $B$, then by the Pythagorean Theorem, $B C^{2}=A C^{2}-A B^{2}=26^{2}-10^{2}=576$ so $B C=24$ since $B C>0$.\n\nTherefore, $\\tan (\\angle A C B)=\\frac{A B}{B C}=\\frac{10}{24}=\\frac{5}{12}$.']	['$\\frac{5}{12}$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAYEBHAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKDntSEmgBaKQHNLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAhOKxPE/inTvCejSalqUmEB2xxry8r/3VHc1tmvDPE5PjX442Ph+X5tNsCA8f8LBV8x8j3ICfhQB1mjT+N/G0KalLfR+HNKl+aCCCFZZ5U7Es3AGDwQB9K1LjwXrCxZsfHGuRT44a4EUyZ/3do/nXZKqqAoGABjHpS4GMYoA5H4eX+t32j34167S6u7XUJrUSrGqArGQvRQB94NXX01I0jBCIqgkkgDGSep+tOoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAQnArw/RkOl/tG6hFdZRrnzDCT0bcm8Y/DI+vHavcDXHeNPANt4qktr63un0/V7TH2e8jGehyAw7jJJ/P3oA7AdfwqnrGq22iaRdaneMVt7aMyPgZJx2HuTgD3NcvbXHxEtIhDcWWg6g4GBcJcvFu9yu08/TFUdS8LeMPFbRweIdX02z0wSLI9np8Tv5wU5Adnweo+ntQB6DFJ5iK21l3KDtbgjPqKfTVp1ABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFACHOOBTAyltuQcjOPWmXl3BYWc13dSLFBCheR26KoGSTXhnh7xHceL/jLY3srzRQEyrDErFdsaxuyg468gMe2fbgAHvIAznvS4pKWgAxRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABTWOBxj8aU8CuX8deLI/Cfh+S6BU3suY7WM/wAT46/RRyfXp3oA434ka1eeI9dtfA+jENI8itduM4BHIUn0UfMfwHUU2PRbTw/8ZPCemWYxFBpjLk9XO2clj7kkn09K1vhV4Vk07TpNf1Lc+p6kN4ZzkrGTnP1Y/Mfw75qPWePj74c9P7Pf/wBBuKQHpVLRRTAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopGOBmgBk80dvbyTzOqRRqXd2OAoAySa8a0uKX4p+P5NVuEb+wdNO2KN/4xzhSPU/ePtgGtP4oeILrUr228FaKTJd3bL9p2nHB5VCe3HzE9gB6mu98M+H7Xw1odvpdsAwjXMj7cGRzjLH64/IAdqQGuB/kV5trX/JfvDn/YOf/wBBuK9KrzXWv+S/eHP+wc//AKDcUwPS6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAQ1z/AIy8TQeFfD01/IUac/JbxN0kkxwPp3PsK35GCIXYgKvJJ6AV4uvmfFX4g+Zhv+Ee0zsDxIM8fi5B+ij8wDd+FXhmdY5vFerbpNR1DLRNIPmCNyX9i36DHrXpgAHSkRQqhVGFAwABwKdQAGvNNa/5L94c/wCwc/8A6DcV6Wa801r/AJL94c/7Bz/+g3FAHpdFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUh4oPArG8U+IrbwxoNxqVyRlBtijJ/1khHCj+vtmgDifip4mnYQeEdILyahf4WYR/eVGyAnsW5z/sg9jXY+EPDVv4W8PwafEA0v+suJB/HIRyfp2HsBXF/C7w9c393ceNNay97eMxtw4xgHgvj0IAC+w9CMep4pAGMUUUUwA15prX/JfvDn/YOf/wBBuK9LNeaa1/yX7w5/2Dn/APQbigD0uiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopD0oAGOBnt3rxfUZZfil8QU0+Bm/sHTDmR1PDjPzHI/vEYX2BI9uh+KniuWxso/DumFn1PUcIVj5ZIycYHu3Qfj7V0fgfwrF4S8OxWQ2tdP+8uZB/E57fQDgfn3pAdDBFHBCkMSLHHGoVUUYCgdAB7DFSUUUwCiiigANeaa1/wAl+8Of9g5//QbivSzXmmtf8l+8Of8AYOf/ANBuKAPS6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAENZev65a+HdEudTvGAjhXIUnlm7KPc1qN0rxnxFcz/Erx5D4esZGGkae5NxMp4Ygje3pn+Ffc56GgC98NNEutf1m68b60N0ssjC0UjgdtwHoB8o+h9Aa9YAA6VBaWsNlbRWtvGI4YUCRoOiqBgAflVigAooooAKKKKAA15prX/JfvDn/YOf/wBBuK9LNeaa1/yX7w5/2Dn/APQbigD0uiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACkPSlqhrGrW2iaTc6jePiCBNzYHLHoFHuTgD3NAHHfFHxdJoWkrpdg7HVL8bIwn3kQnBb6k/KPxPatH4eeEk8KeHkilQG/uQJblvQ9kHsufzJPeuO+H+lXfjDxRc+N9XXMaSFbOPHG4ccf7K9AfXJ6jNevKMflikAuADS0UUwCiiigAooooADXmmtf8AJfvDn/YOf/0G4r0s15prX/JfvDn/AGDn/wDQbigD0uiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKRjgZoAD06V454yv7j4geNLfwhpT/6DaybruZeVyOGP/ARlR6sa6v4l+MP+EZ0FoLaQDUrwFIAOqL/ABP+GcD3I9DTvhr4QHhrQfPukP8AaV6BJOWzlB/Cntjv7/QUgOt06wttM0+CxtIxHBBGERQOgFWsYoxiimAUUUUAFFFFABRRRQAGvNNa/wCS/eHP+wc//oNxXpZrzTWv+S/eHP8AsHP/AOg3FAHpdFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVV1K/t9L06e/u5BHbwIXdj6D09+wFWScCvIfH+q3fjHxRa+CdHf8AdpJuvJP4Qy8nOOyjr/tEDsKAIvBljc+PvGdz4v1SPFlayBbWBjkbl5Vfoucn/aNexKMHFUdG0u00TSrbTbKPZbwJtUHqx6kn1JJJJ9TWhSAKKKKYBRRRQAUUUUAFFFFAAa801r/kv3hz/sHP/wCg3FelmvNNa/5L94c/7Bz/APoNxQB6XRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUhpahvLqGys5rq5kWOCFC8jscBVHJJoA5b4heL18J+HmkiYf2hcfu7ZOpz3fHoM/nis/4XeEm0PSG1O/BbVNQAkk3nLRoTkDPqep98DtXL+HLWf4leO5vEeoRf8SfT22QROOGI+6hHrzubtyB04r2YdaQCgY6UUUUwCiiigAooooAKKKKACiiigANeaa1/yX7w5/2Dn/8AQbivSzXmmtf8l+8Of9g5/wD0G4oA9LooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopD0oAGOBXkvxL1y78Qa1a+CNFJeaWRTdMOgPUKT6AfMf+A+9dn458WReE/D0t3uU3kuY7WNv4n9T7Dqfy71gfCrwrJY2EniLU9z6nqQ3qznJWMnOT7t976YHrSA7Pw/otp4e0W20uyH7qBMbj1c92PuTzWpjFGKKYBRRRQAUUUUAFFFFABRRRQAUUUUABrzTWv8Akv3hz/sHP/6DcV6Wa801r/kv3hz/ALBz/wDoNxQB6XRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABUc8scEDzSuqRIpZ3Y4CqBkkk9qexwK8r+KPiG6v7u38FaNmS9u2UXG04wDyqZ7eregx60AZWnRS/FPx++pTo39g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+44	"In the diagram, $A B=B C=2 \sqrt{2}, C D=D E$, $\angle C D E=60^{\circ}$, and $\angle E A B=75^{\circ}$. Determine the perimeter of figure $A B C D E$. Explain how you got your answer.
+
+<image_1>"	['Since $\\triangle A B C$ is isosceles and right-angled, then $\\angle B A C=45^{\\circ}$.\n\nAlso, $A C=\\sqrt{2} A B=\\sqrt{2}(2 \\sqrt{2})=4$.\n\nSince $\\angle E A B=75^{\\circ}$ and $\\angle B A C=45^{\\circ}$, then $\\angle C A E=\\angle E A B-\\angle B A C=30^{\\circ}$.\n\nSince $\\triangle A E C$ is right-angled and has a $30^{\\circ}$ angle, then $\\triangle A E C$ is a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle.\n\nThus, $E C=\\frac{1}{2} A C=2$ (since $E C$ is opposite the $30^{\\circ}$ angle) and $A E=\\frac{\\sqrt{3}}{2} A C=2 \\sqrt{3}$ (since $A E$ is opposite the $60^{\\circ}$ angle).\n\nIn $\\triangle C D E, E D=D C$ and $\\angle E D C=60^{\\circ}$, so $\\triangle C D E$ is equilateral.\n\nTherefore, $E D=C D=E C=2$.\n\nOverall, the perimeter of $A B C D E$ is\n\n$$\nA B+B C+C D+D E+E A=2 \\sqrt{2}+2 \\sqrt{2}+2+2+2 \\sqrt{3}=4+4 \\sqrt{2}+2 \\sqrt{3}\n$$']	['$4+4 \\sqrt{2}+2 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+45	"In the diagram, the parabola intersects the $x$-axis at $A(-3,0)$ and $B(3,0)$ and has its vertex at $C$ below the $x$-axis. The area of $\triangle A B C$ is 54 . Determine the equation of the parabola. Explain how you got your answer.
+
+<image_1>"	"['From the diagram, the parabola has $x$-intercepts $x=3$ and $x=-3$.\n\nTherefore, the equation of the parabola is of the form $y=a(x-3)(x+3)$ for some real number $a$.\n\nTriangle $A B C$ can be considered as having base $A B$ (of length $3-(-3)=6$ ) and height $O C$ (where $O$ is the origin).\n\nSuppose $C$ has coordinates $(0,-c)$. Then $O C=c$.\n\nThus, the area of $\\triangle A B C$ is $\\frac{1}{2}(A B)(O C)=3 c$. But we know that the area of $\\triangle A B C$ is 54 , so $3 c=54$ or $c=18$.\n\nSince the parabola passes through $C(0,-18)$, then this point must satisfy the equation of the parabola.\n\nTherefore, $-18=a(0-3)(0+3)$ or $-18=-9 a$ or $a=2$.\n\nThus, the equation of the parabola is $y=2(x-3)(x+3)=2 x^{2}-18$.'
+ 'Triangle $A B C$ can be considered as having base $A B$ (of length $3-(-3)=6$ ) and height $O C$ (where $O$ is the origin).\n\nSuppose $C$ has coordinates $(0,-c)$. Then $O C=c$.\n\nThus, the area of $\\triangle A B C$ is $\\frac{1}{2}(A B)(O C)=3 c$. But we know that the area of $\\triangle A B C$ is 54 , so $3 c=54$ or $c=18$.\n\nTherefore, the parabola has vertex $C(0,-18)$, so has equation $y=a(x-0)^{2}-18$.\n\n(The vertex of the parabola must lie on the $y$-axis since its roots are equally distant from the $y$-axis, so $C$ must be the vertex.)\n\nSince the parabola passes through $B(3,0)$, then these coordinates satisfy the equation, so $0=3^{2} a-18$ or $9 a=18$ or $a=2$.\n\nTherefore, the equation of the parabola is $y=2 x^{2}-18$.']"	['$y=2 x^{2}-18$']		['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+46	"In the diagram, $A(0, a)$ lies on the $y$-axis above $D$. If the triangles $A O B$ and $B C D$ have the same area, determine the value of $a$. Explain how you got your answer.
+
+<image_1>"	['$\\triangle A O B$ is right-angled at $O$, so has area $\\frac{1}{2}(A O)(O B)=\\frac{1}{2} a(1)=\\frac{1}{2} a$.\n\nWe next need to calculate the area of $\\triangle B C D$.\n\nMethod 1: Completing the trapezoid\n\nDrop a perpendicular from $C$ to $P(3,0)$ on the $x$-axis.\n\n<img_3403>\n\nThen $D O P C$ is a trapezoid with parallel sides $D O$ of length 1 and $P C$ of length 2 and height $O P$ (which is indeed perpendicular to the parallel sides) of length 3.\n\nThe area of the trapezoid is thus $\\frac{1}{2}(D O+P C)(O P)=\\frac{1}{2}(1+2)(3)=\\frac{9}{2}$.\n\nBut the area of $\\triangle B C D$ equals the area of trapezoid $D O P C$ minus the areas of $\\triangle D O B$ and $\\triangle B P C$.\n\n$\\triangle D O B$ is right-angled at $O$, so has area $\\frac{1}{2}(D O)(O B)=\\frac{1}{2}(1)(1)=\\frac{1}{2}$.\n\n$\\triangle B P C$ is right-angled at $P$, so has area $\\frac{1}{2}(B P)(P C)=\\frac{1}{2}(2)(2)=2$.\n\nThus, the area of $\\triangle D B C$ is $\\frac{9}{2}-\\frac{1}{2}-2=2$.\n\n\n\n(A similar method for calculating the area of $\\triangle D B C$ would be to drop a perpendicular to $Q$ on the $y$-axis, creating a rectangle $Q O P C$.)\n\nMethod 2: $\\triangle D B C$ is right-angled\n\nThe slope of line segment $D B$ is $\\frac{1-0}{0-1}=-1$.\n\nThe slope of line segment $B C$ is $\\frac{2-0}{3-1}=1$.\n\nSince the product of these slopes is -1 (that is, their slopes are negative reciprocals), then $D B$ and $B C$ are perpendicular.\n\nTherefore, the area of $\\triangle D B C$ is $\\frac{1}{2}(D B)(B C)$.\n\nNow $D B=\\sqrt{(1-0)^{2}+(0-1)^{2}}=\\sqrt{2}$ and $B C=\\sqrt{(3-1)^{2}+(2-0)^{2}}=\\sqrt{8}$.\n\nThus, the area of $\\triangle D B C$ is $\\frac{1}{2} \\sqrt{2} \\sqrt{8}=2$.\n\nSince the area of $\\triangle A O B$ equals the area of $\\triangle D B C$, then $\\frac{1}{2} a=2$ or 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+47	"The Little Prince lives on a spherical planet which has a radius of $24 \mathrm{~km}$ and centre $O$. He hovers in a helicopter $(H)$ at a height of $2 \mathrm{~km}$ above the surface of the planet. From his position in the helicopter, what is the distance, in kilometres, to the furthest point on the surface of the planet that he can see?
+
+<image_1>"	['Suppose that $O$ is the centre of the planet, $H$ is the place where His Highness hovers in the helicopter, and $P$ is the furthest point on the surface of the planet that he can see.\n\n<img_3899>\n\nThen $H P$ must be a tangent to the surface of the planet (otherwise he could see further), so $O P$ (a radius) is perpendicular to $H P$ (a tangent).\n\nWe are told that $O P=24 \\mathrm{~km}$.\n\nSince the helicopter hovers at a height of $2 \\mathrm{~km}$, then $O H=24+2=26 \\mathrm{~km}$.\n\nTherefore, $H P^{2}=O H^{2}-O P^{2}=26^{2}-24^{2}=100$, so $H P=10 \\mathrm{~km}$.\n\nTherefore, the distance to the furthest point that he can see is $10 \\mathrm{~km}$.']	['10']		['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UUUUAFFFFABRRRQAUUUUAFFFFAH/2Q==']	Multimodal	Competition	False	km	Numerical		Open-ended	Geometry	Math	English
+48	"In the diagram, points $A$ and $B$ are located on islands in a river full of rabid aquatic goats. Determine the distance from $A$ to $B$, to the nearest metre. (Luckily, someone has measured the angles shown in the diagram as well as the distances $C D$ and $D E$.)
+
+<image_1>"	['Since we know the measure of $\\angle A D B$, then to find the distance $A B$, it is enough to find the distances $A D$ and $B D$ and then apply the cosine law.\n\nIn $\\triangle D B E$, we have $\\angle D B E=180^{\\circ}-20^{\\circ}-70^{\\circ}=90^{\\circ}$, so $\\triangle D B E$ is right-angled, giving $B D=100 \\cos \\left(20^{\\circ}\\right) \\approx 93.969$.\n\nIn $\\triangle D A C$, we have $\\angle D A C=180^{\\circ}-50^{\\circ}-45^{\\circ}=85^{\\circ}$.\n\nUsing the sine law, $\\frac{A D}{\\sin \\left(50^{\\circ}\\right)}=\\frac{C D}{\\sin \\left(85^{\\circ}\\right)}$, so $A D=\\frac{150 \\sin \\left(50^{\\circ}\\right)}{\\sin \\left(85^{\\circ}\\right)} \\approx 115.346$.\n\n\n\nFinally, using the cosine law in $\\triangle A B D$, we get\n\n$$\n\\begin{aligned}\nA B^{2} & =A D^{2}+B D^{2}-2(A D)(B D) \\cos (\\angle A D B) \\\\\nA B^{2} & \\approx(115.346)^{2}+(93.969)^{2}-2(115.346)(93.969) \\cos \\left(35^{\\circ}\\right) \\\\\nA B^{2} & \\approx 4377.379 \\\\\nA B & \\approx 66.16\n\\end{aligned}\n$$\n\nTherefore, the distance from $A$ to $B$ is approximately $66 \\mathrm{~m}$.']	['66']		['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+49	"In the $4 \times 4$ grid shown, three coins are randomly placed in different squares. Determine the probability that no two coins lie in the same row or column.
+
+<image_1>"	['We consider placing the three coins individually.\n\nPlace one coin randomly on the grid.\n\nWhen the second coin is placed (in any one of 15 squares), 6 of the 15 squares will leave two coins in the same row or column and 9 of the 15 squares will leave the two coins in different rows and different columns.\n\n<img_3640>\n\nTherefore, the probability that the two coins are in different rows and different columns is $\\frac{9}{15}=\\frac{3}{5}$.\n\nThere are 14 possible squares in which the third coin can be placed.\n\n\n\nOf these 14 squares, 6 lie in the same row or column as the first coin and an additional 4 lie the same row or column as the second coin. Therefore, the probability that the third coin is placed in a different row and a different column than each of the first two coins is $\\frac{4}{14}=\\frac{2}{7}$.\n\nTherefore, the probability that all three coins are placed in different rows and different columns is $\\frac{3}{5} \\times \\frac{2}{7}=\\frac{6}{35}$.']	['$\\frac{6}{35}$']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+50	"In the diagram, the area of $\triangle A B C$ is 1 . Trapezoid $D E F G$ is constructed so that $G$ is to the left of $F, D E$ is parallel to $B C$, $E F$ is parallel to $A B$ and $D G$ is parallel to $A C$. Determine the maximum possible area of trapezoid $D E F G$.
+
+<image_1>"	['Suppose that $A B=c, A C=b$ and $B C=a$.\n\nSince $D G$ is parallel to $A C, \\angle B D G=\\angle B A C$ and $\\angle D G B=\\angle A C B$, so $\\triangle D G B$ is similar to $\\triangle A C B$.\n\n(Similarly, $\\triangle A E D$ and $\\triangle E C F$ are also both similar to $\\triangle A B C$.)\n\nSuppose next that $D B=k c$, with $0<k<1$.\n\nThen the ratio of the side lengths of $\\triangle D G B$ to those of $\\triangle A C B$ will be $k: 1$, so $B G=k a$ and $D G=k b$.\n\nSince the ratio of the side lengths of $\\triangle D G B$ to $\\triangle A C B$ is $k: 1$, then the ratio of their areas will be $k^{2}: 1$, so the area of $\\triangle D G B$ is $k^{2}$ (since the area of $\\triangle A C B$ is 1 ).\n\nSince $A B=c$ and $D B=k c$, then $A D=(1-k) c$, so using similar triangles as before, $D E=(1-k) a$ and $A E=(1-k) b$. Also, the area of $\\triangle A D E$ is $(1-k)^{2}$.\n\nSince $A C=b$ and $A E=(1-k) b$, then $E C=k b$, so again using similar triangles, $E F=k c$, $F C=k a$ and the area of $\\triangle E C F$ is $k^{2}$.\n\nNow the area of trapezoid $D E F G$ is the area of the large triangle minus the combined areas of the small triangles, or $1-k^{2}-k^{2}-(1-k)^{2}=2 k-3 k^{2}$.\n\nWe know that $k \\geq 0$ by its definition. Also, since $G$ is to the left of $F$, then $B G+F C \\leq B C$ or $k a+k a \\leq a$ or $2 k a \\leq a$ or $k \\leq \\frac{1}{2}$.\n\nLet $f(k)=2 k-3 k^{2}$.\n\nSince $f(k)=-3 k^{2}+2 k+0$ is a parabola opening downwards, its maximum occurs at its vertex, whose $k$-coordinate is $k=-\\frac{2}{2(-3)}=\\frac{1}{3}$ (which lies in the admissible range for $k$ ).\n\nNote that $f\\left(\\frac{1}{3}\\right)=\\frac{2}{3}-3\\left(\\frac{1}{9}\\right)=\\frac{1}{3}$.\n\nTherefore, the maximum area of the trapezoid is $\\frac{1}{3}$.']	['$\\frac{1}{3}$']		['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+51	"In the diagram, $\triangle P Q R$ has $P Q=a, Q R=b, P R=21$, and $\angle P Q R=60^{\circ}$. Also, $\triangle S T U$ has $S T=a, T U=b, \angle T S U=30^{\circ}$, and $\sin (\angle T U S)=\frac{4}{5}$. Determine the values of $a$ and $b$.
+<image_1>"	['Using the cosine law in $\\triangle P Q R$,\n\n$$\n\\begin{aligned}\nP R^{2} & =P Q^{2}+Q R^{2}-2 \\cdot P Q \\cdot Q R \\cdot \\cos (\\angle P Q R) \\\\\n21^{2} & =a^{2}+b^{2}-2 a b \\cos \\left(60^{\\circ}\\right) \\\\\n441 & =a^{2}+b^{2}-2 a b \\cdot \\frac{1}{2} \\\\\n441 & =a^{2}+b^{2}-a b\n\\end{aligned}\n$$\n\nUsing the sine law in $\\triangle S T U$, we obtain $\\frac{S T}{\\sin (\\angle T U S)}=\\frac{T U}{\\sin (\\angle T S U)}$ and so $\\frac{a}{4 / 5}=\\frac{b}{\\sin \\left(30^{\\circ}\\right)}$. Therefore, $\\frac{a}{4 / 5}=\\frac{b}{1 / 2}$ and so $a=\\frac{4}{5} \\cdot 2 b=\\frac{8}{5} b$.\n\nSubstituting into the previous equation,\n\n$$\n\\begin{aligned}\n& 441=\\left(\\frac{8}{5} b\\right)^{2}+b^{2}-\\left(\\frac{8}{5} b\\right) b \\\\\n& 441=\\frac{64}{25} b^{2}+b^{2}-\\frac{8}{5} b^{2} \\\\\n& 441=\\frac{64}{25} b^{2}+\\frac{25}{25} b^{2}-\\frac{40}{25} b^{2} \\\\\n& 441=\\frac{49}{25} b^{2} \\\\\n& 225=b^{2}\n\\end{aligned}\n$$\n\nSince $b>0$, then $b=15$ and so $a=\\frac{8}{5} b=\\frac{8}{5} \\cdot 15=24$.']	['$24,15$']		['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+52	"A triangle of area $770 \mathrm{~cm}^{2}$ is divided into 11 regions of equal height by 10 lines that are all parallel to the base of the triangle. Starting from the top of the triangle, every other region is shaded, as shown. What is the total area of the shaded regions?
+
+<image_1>"	"['We make two copies of the given triangle, labelling them $\\triangle A B C$ and $\\triangle D E F$, as shown:\n<img_3909>\n\nThe combined area of these two triangles is $2 \\cdot 770 \\mathrm{~cm}^{2}=1540 \\mathrm{~cm}^{2}$, and the shaded area in each triangle is the same.\n\nNext, we rotate $\\triangle D E F$ by $180^{\\circ}$ :\n<img_3429>\n\nand join the two triangles together:\n\n<img_3440>\n\nWe note that $B C$ and $A E$ (which was $F E$ ) are equal in length (since they were copies of each other) and parallel (since they are $180^{\\circ}$ rotations of each other). The same is true for $A B$ and $E C$.\n\nTherefore, $A B C E$ is a parallelogram.\n\nFurther, $A B C E$ is divided into 11 identical parallelograms (6 shaded and 5 unshaded) by the horizontal lines. (Since the sections of the two triangles are equal in height, the horizontal lines on both sides of $A C$ align.)\n\nThe total area of parallelogram $A B C E$ is $1540 \\mathrm{~cm}^{2}$.\n\nThus, the shaded area of $A B C E$ is $\\frac{6}{11} \\cdot 1540 \\mathrm{~cm}^{2}=840 \\mathrm{~cm}^{2}$.\n\nSince this shaded area is equally divided between the two halves of the parallelogram, then the combined area of the shaded regions of $\\triangle A B C$ is $\\frac{1}{2} \\cdot 840 \\mathrm{~cm}^{2}=420 \\mathrm{~cm}^{2}$.'
+ 'We label the points where the horizontal lines touch $A B$ and $A C$ as shown:\n\n<img_3141>\n\nWe use the notation $|\\triangle A B C|$ to represent the area of $\\triangle A B C$ and use similar notation for the area of other triangles and quadrilaterals.\n\nLet $\\mathcal{A}$ be equal to the total area of the shaded regions.\n\nThus,\n\n$$\n\\mathcal{A}=\\left|\\triangle A B_{1} C_{1}\\right|+\\left|B_{2} B_{3} C_{3} C_{2}\\right|+\\left|B_{4} B_{5} C_{5} C_{4}\\right|+\\left|B_{6} B_{7} C_{7} C_{6}\\right|+\\left|B_{8} B_{9} C_{9} C_{8}\\right|+\\left|B_{10} B C C_{10}\\right|\n$$\n\nThe area of each of these quadrilaterals is equal to the difference of the area of two triangles. For example,\n\n$$\n\\left|B_{2} B_{3} C_{3} C_{2}\\right|=\\left|\\triangle A B_{3} C_{3}\\right|-\\left|\\triangle A B_{2} C_{2}\\right|=-\\left|\\triangle A B_{2} C_{2}\\right|+\\left|\\triangle A B_{3} C_{3}\\right|\n$$\n\nTherefore,\n\n$$\n\\begin{aligned}\n\\mathcal{A}=\\mid & \\triangle A B_{1} C_{1}|-| \\triangle A B_{2} C_{2}|+| \\triangle A B_{3} C_{3}|-| \\triangle A B_{4} C_{4}|+| \\triangle A B_{5} C_{5} \\mid \\\\\n& \\quad-\\left|\\triangle A B_{6} C_{6}\\right|+\\left|\\triangle A B_{7} C_{7}\\right|-\\left|\\triangle A B_{8} C_{8}\\right|+\\left|\\triangle A B_{9} C_{9}\\right|-\\left|\\triangle A B_{10} C_{10}\\right|+|\\triangle A B C|\n\\end{aligned}\n$$\n\nEach of $\\triangle A B_{1} C_{1}, \\triangle A B_{2} C_{2}, \\ldots, \\triangle A B_{10} C_{10}$ is similar to $\\triangle A B C$ because their two base angles are equal due.\n\nSuppose that the height of $\\triangle A B C$ from $A$ to $B C$ is $h$.\n\nSince the height of each of the 11 regions is equal in height, then the height of $\\triangle A B_{1} C_{1}$ is $\\frac{1}{11} h$, the height of $\\triangle A B_{2} C_{2}$ is $\\frac{2}{11} h$, and so on.\n\nWhen two triangles are similar, their heights are in the same ratio as their side lengths:\n\nTo see this, suppose that $\\triangle P Q R$ is similar to $\\triangle S T U$ and that altitudes are drawn from $P$ and $S$ to $V$ and $W$.\n<img_3919>\n\nSince $\\angle P Q R=\\angle S T U$, then $\\triangle P Q V$ is similar to $\\triangle S T W$ (equal angle; right angle), which means that $\\frac{P Q}{S T}=\\frac{P V}{S W}$. In other words, the ratio of sides is equal to the ratio of heights.\n\nSince the height of $\\triangle A B_{1} C_{1}$ is $\\frac{1}{11} h$, then $B_{1} C_{1}=\\frac{1}{11} B C$.\n\nTherefore, $\\left|\\triangle A B_{1} C_{1}\\right|=\\frac{1}{2} \\cdot B_{1} C_{1} \\cdot \\frac{1}{11} h=\\frac{1}{2} \\cdot \\frac{1}{11} B C \\cdot \\frac{1}{11} h=\\frac{1^{2}}{11^{2}} \\cdot \\frac{1}{2} \\cdot B C \\cdot h=\\frac{1^{2}}{11^{2}}|\\triangle A B C|$.\n\nSimilarly, since the height of $\\triangle A B_{2} C_{2}$ is $\\frac{2}{11} h$, then $B_{2} C_{2}=\\frac{2}{11} B C$.\n\n\n\nTherefore, $\\left|\\triangle A B_{2} C_{2}\\right|=\\frac{1}{2} \\cdot B_{2} C_{2} \\cdot \\frac{2}{11} h=\\frac{1}{2} \\cdot \\frac{2}{11} B C \\cdot \\frac{2}{11} h=\\frac{2^{2}}{11^{2}} \\cdot \\frac{1}{2} \\cdot B C \\cdot h=\\frac{2^{2}}{11^{2}}|\\triangle A B C|$.\n\nThis result continues for each of the triangles.\n\nTherefore,\n\n$$\n\\begin{aligned}\n\\mathcal{A} & =\\frac{1^{2}}{11^{2}}|\\triangle A B C|-\\frac{2^{2}}{11^{2}}|\\triangle A B C|+\\frac{3^{2}}{11^{2}}|\\triangle A B C|-\\frac{4^{2}}{11^{2}}|\\triangle A B C|+\\frac{5^{2}}{11^{2}}|\\triangle A B C| \\\\\n& \\quad-\\frac{6^{2}}{11^{2}}|\\triangle A B C|+\\frac{7^{2}}{11^{2}}|\\triangle A B C|-\\frac{8^{2}}{11^{2}}|\\triangle A B C|+\\frac{9^{2}}{11^{2}}|\\triangle A B C|-\\frac{10^{2}}{11^{2}}|\\triangle A B C|+\\frac{11^{2}}{11^{2}}|\\triangle A B C| \\\\\n& =\\frac{1}{11^{2}}|\\triangle A B C|\\left(11^{2}-10^{2}+9^{2}-8^{2}+7^{2}-6^{2}+5^{2}-4^{2}+3^{2}-2^{2}+1\\right) \\\\\n& =\\frac{1}{11^{2}}\\left(770 \\mathrm{~cm}^{2}\\right)((11+10)(11-10)+(9+8)(9-8)+\\cdots+(3+2)(3-2)+1) \\\\\n& =\\frac{1}{11^{2}}\\left(770 \\mathrm{~cm}^{2}\\right)(11+10+9+8+7+6+5+4+3+2+1) \\\\\n& =\\frac{1}{11}\\left(70 \\mathrm{~cm}^{2}\\right) \\cdot 66 \\\\\n& =420 \\mathrm{~cm}^{2}\n\\end{aligned}\n$$\n\nTherefore, the combined area of the shaded regions of $\\triangle A B C$ is $420 \\mathrm{~cm}^{2}$.']"	['$420$']		['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+53	"A square lattice of 16 points is constructed such that the horizontal and vertical distances between adjacent points are all exactly 1 unit. Each of four pairs of points are connected by a line segment, as shown. The intersections of these line segments are the vertices of square $A B C D$. Determine the area of square $A B C D$.
+
+<image_1>"	"['We label five additional points in the diagram:\n\n<img_3827>\n\nSince $P Q=Q R=R S=1$, then $P S=3$ and $P R=2$.\n\nSince $\\angle P S T=90^{\\circ}$, then $P T=\\sqrt{P S^{2}+S T^{2}}=\\sqrt{3^{2}+1^{2}}=\\sqrt{10}$ by the Pythagorean Theorem.\n\nWe are told that $A B C D$ is a square.\n\nThus, $P T$ is perpendicular to $Q C$ and to $R B$.\n\nThus, $\\triangle P D Q$ is right-angled at $D$ and $\\triangle P A R$ is right-angled at $A$.\n\nSince $\\triangle P D Q, \\triangle P A R$ and $\\triangle P S T$ are all right-angled and all share an angle at $P$, then these three triangles are similar.\n\nThis tells us that $\\frac{P A}{P S}=\\frac{P R}{P T}$ and so $P A=\\frac{3 \\cdot 2}{\\sqrt{10}}$. Also, $\\frac{P D}{P S}=\\frac{P Q}{P T}$ and so $P D=\\frac{1 \\cdot 3}{\\sqrt{10}}$.\n\nTherefore,\n\n$$\nD A=P A-P D=\\frac{6}{\\sqrt{10}}-\\frac{3}{\\sqrt{10}}=\\frac{3}{\\sqrt{10}}\n$$\n\nThis means that the area of square $A B C D$ is equal to $D A^{2}=\\left(\\frac{3}{\\sqrt{10}}\\right)^{2}=\\frac{9}{10}$.'
+ 'We add coordinates to the diagram as shown:\n\n<img_3409>\n\nWe determine the side length of square $A B C D$ by determining the coordinates of $D$ and $A$ and then calculating the distance between these points.\n\nThe slope of the line through $(0,3)$ and $(3,2)$ is $\\frac{3-2}{0-3}=-\\frac{1}{3}$.\n\nThis equation of this line can be written as $y=-\\frac{1}{3} x+3$.\n\nThe slope of the line through $(0,0)$ and $(1,3)$ is 3.\n\nThe equation of this line can be written as $y=3 x$.\n\nThe slope of the line through $(1,0)$ and $(2,3)$ is also 3.\n\nThe equation of this line can be written as $y=3(x-1)=3 x-3$.\n\nPoint $D$ is the intersection point of the lines with equations $y=-\\frac{1}{3} x+3$ and $y=3 x$.\n\nEquating expressions for $y$, we obtain $-\\frac{1}{3} x+3=3 x$ and so $\\frac{10}{3} x=3$ which gives $x=\\frac{9}{10}$.\n\nSince $y=3 x$, we get $y=\\frac{27}{10}$ and so the coordinates of $D$ are $\\left(\\frac{9}{10}, \\frac{27}{10}\\right)$.\n\nPoint $A$ is the intersection point of the lines with equations $y=-\\frac{1}{3} x+3$ and $y=3 x-3$.\n\nEquating expressions for $y$, we obtain $-\\frac{1}{3} x+3=3 x-3$ and so $\\frac{10}{3} x=6$ which gives $x=\\frac{18}{10}$.\n\nSince $y=3 x-3$, we get $y=\\frac{24}{10}$ and so the coordinates of $A$ are $\\left(\\frac{18}{10}, \\frac{24}{10}\\right)$. (It is easier to not reduce these fractions.)\n\nTherefore,\n\n$$\nD A=\\sqrt{\\left(\\frac{9}{10}-\\frac{18}{10}\\right)^{2}+\\left(\\frac{27}{10}-\\frac{24}{10}\\right)^{2}}=\\sqrt{\\left(-\\frac{9}{10}\\right)^{2}+\\left(\\frac{3}{10}\\right)^{2}}=\\sqrt{\\frac{90}{100}}=\\sqrt{\\frac{9}{10}}\n$$\n\nThis means that the area of square $A B C D$ is equal to $D A^{2}=\\left(\\sqrt{\\frac{9}{10}}\\right)^{2}=\\frac{9}{10}$.']"	['$\\frac{9}{10}$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAbQB0gMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAormPHHjKz8DeHX1W9RpmLCKGBDgyyEZAz2GATmvF7D9o/VxqanUNFsWsGYZW3LrKo7/MSQx/AUAfR9FU9M1C21bTLXULOTzLa5jWWNvVSMj+dXKACiiigAooooAKKKKACiiigAooooAKKKKACiivD/AB98dptD1y40jw9Y21y1o5jnubrcULjgqqqynAORuJ6g8dCQD3CivL/hf8WYvHN1Npl9aR2eqxoZVEbZjlUEZK55BGR8vPr649QoAKKKKACiiigAooooAKKKKACiiigAooooAKKK5vxr4usvBPhyXV71TJhhHDChwZpCOFz26E59AaAOkor5vtP2kNY/tUG80WwOnFhmOFnEqr/vE4J9tor3/R9Wtdd0iy1WxkL211EskZI5wR0PoR0PuKANGiiigAooooAKKKKACiiigAooooAKKKKACiijtQAUV4n8Qfjo/h7XbjR9Asra5mtW2T3FyWZFf+JAqkEkdDz17cVr/DH4vp43vm0jUrSOz1QIZIzE2Y5gOoUE5BA7c5GTkUAeq0UUUAFFFFABRRRQAUHpRRQB5R8d/DGo+IPB9tcabDJcS2E5lkgjBJZCMFgB1I4+gJr5hs7K51G8is7O3kubmVtscUSFmc+wHJr71PSoUt4I5WlSGNZH+8yqAT9TQB4r4c8QeN/hroNppOueDJrzTLYELd6fL5rorEsdwG7kEkfwjp7Z7Xw/8W/BniLYkOrJZ3LY/cXo8lgSem4/KT7BjXd1zeveA/C/ifLavolrNK3WdVMcn/fa4b9aAOhRldQ6sGUjIYHOR9RT68kb4Q6r4eZp/A/jC/0zkuLS6PmQsx9QOPzVjSf8Jt8RvCvyeKPCI1S0U4N9pLZOO7FRn9QlAHrlFcHoHxf8G6/tRNVWxuCOYL8eSRzjG4/KT7BjXco6uiujBlYZBByCPUYoAfRRRQAUUUUAFFFFABRRRQAHpXxZ8R9AvPDfjvVLW7GFmne4gfORJE7Eqc/ofcGvtM9K8f8Ah/FH4x+Jni3xbOizWULjTbHd8yMq4yRnvhVb/gZoA4L4B+FtRu/GUfiIxSxadZRuolKkLM7KU2A98Akn0wM9RX07TUVUUKqgKOAAMAU6gAooooAKKKKACiiigAooooAKKKKACiiigAry/wCOnhy/8QeBUk0+NppLC4Fy8S8lo9rBiB3IyD9M16hXA/GLXjoXw31Dy2Iub7FlFjqxf72P+AB6APke0tbi9u4ra0hee4lYLHFGpZmPYAd6+zPh1oNx4a8A6RpN22bmGNmlAOdrO5cr+BbH4VL4I8NweF/CWl6asMa3MNuondV5Mh+Z+f8AeJ/CuloAKKKKACiiigAooooAKKKKACiiigAooooAKD0oooA+Nvih4Y1Hw3451M3sDi2vLqW4tZzkrKjNuxu7sNwB7/gRnpfgN4Y1G/8AG9trwgkTTtPWQtOwIV3ZGTYp7n5sn0A56ivp+aGO4jKSxrIh6q65FORFRQiKFVegUYAoAfRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABSHoaWigDnNf8DeGfEwZtX0W1uJW6zBNkn/AH2uG/WuHf4P6joMpn8D+LdQ0o53fZLhvNhY9s47fVWr1uigDyMeMfiX4TO3xL4VTWLNODe6UctjuxUZ/UJ9a3dB+MXgzXSIv7S/s+5PBg1AeSV9t33Pw3Zrvj0Nc/r3gnw34mVv7X0a1uZCMecU2yj0/eLhv1oA3Y5ElRZI3V0YAqynII9R61JXkr/B290NjP4J8W6lpL7i4tp28yBj2BA7e5DU3/hLPif4T+XxB4Yi12zUgfbNLPz47koOf/HVHvQB65RXnuhfGXwbrZETagdMuc4MOoL5WD/vcp198130Usc8SywuskbAFXQ5DD1B6GgCSiiigDlPiLr/APwjPgLVtTRyk6wGKAqeRI+EUj6Eg/hVb4V+H/8AhHPh3pVq6BbieP7VPxg75PmwfcLtX/gNcv8AFQnxN4z8JeB0w0U9x9vvFPaJQR+oEv6V60BjAHAFADqKKKACiiigAooooAKKKKACiiigAooooAKKKKAA9K8i8YZ8W/Grw14aXLWmkIdSvAvOGzlQ3txGP+2letSypDC8sjhI0UszMcAAdSa8p+D8Umu6v4n8c3CMDql2YLTePmWFD0z6fcX6pQB6uB/P6U6iigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACig9K87+M08Nh8PL6/E1xDeoUjtZIJ3jIdnXP3SM8A0AdLr3gzw74nQjWdHtbtyoXzWTbKB7SDDD864aT4Nz6NI1x4K8V6nozli5t5H82Bj2GBj/wAe3Vb8E+Bbe68E6VdavfazJqVzAJnnTVbhCN/zKMBwowpUdO1ZWua9rnwp8Tab9u1a61jwrqL7N16wee2YHk+ZjLDBBGeoBGOMkAl/4Sf4peEsDX/Ddv4gs0wGu9LOJPclQM+/3FHv6bWh/GbwdrL+RNfPpV0CVaDUU8raQcHL8p+bA+1egK6vGGVtykZB9RXA/Fyz0GPwJq2qarpdrcXMVuY7eVowZEkfCJhuuASD+FAGH8OT/wAJX8TvFfjIsHtIXGnWLg5VlGMsD9FU8f8APQ167XhvhT4RaxY+F9O1PRPFeoaRqtxAs88ON0LMwyFK5HQEA53cj8tb/hIvir4RG3W/D1t4js0AButOYrIfcqBnp/sD60AeuUV51onxo8H6vL9nubuXSLsEq0Oox+Xgjr8wJUfiQfau/t7iG6gSe3mSWJxlZI2DBh7GgCaiiigAooooAKKKKACiiigAooooAKD0ooPSgDz/AOMuunRPhxfpEf8ASNQIsYgOp3/e/wDHA1dH4N0FfDXg/StIAAe2t1WXb0Mh+Zz+LFj+NcB4rP8Awlnxv8OeHkO600WM6ld7W6PwVB/KL8HNetjjgDAoAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACvFvjTI+v8Aifwl4KiY/wCl3IuLkDsudgb8B5p/Cvae1eJ+EnXxd+0Br+t532ujxG2tz1Ab/V8exxKfxoA9ojVY41RF2oqhQB29q8b/AGj5ol8HaVAT+9fUN6/RY3B/9CFeuajqVlpVm93f3cFrbx8tLM4VR+JryL+ybv4uePLLW57eWDwhpZ/0Xz1KtetnJYKeQpIGSeygdScAHqfhiGa18J6NBc/6+KxhSTP94IAf1rzn4sN/wkfi3wl4FjIZbu5+23i9xEmf5gS/kK9aA6cV5L8P8+Kviv4q8WtlrS0YabYnOVO37xU/Rc/SSgD1pVCgAdF4Ap1FFAGJrnhTQfEkezWNJtbzjAeSP5x9GHzD8K4Kb4L/ANkztd+DPE2paHKx3NDvMsL46AgkHH+9ur1iigDyM698V/CXGr6Fa+J7Jetxpx2yn0+UDP5R/jWronxp8IarN9nvbmfR7wNtaDUI9gB7/OMqP+BEH2r0c9Kxtb8L6H4kh8vWNKtbwYIDSx/Ov+633l/CgDStrmC8hS4tp454XGVkicMpHsQcVPXlFx8FYtNuHvPBniPUtBnY58oSGSI+gIyDj/eLfSojrPxZ8JjOqaNZeJ7NBkz2BKzH04ABJ+kZ+tAHrlFeb6N8avCeoTm0v5bjRb1W2PBqEZQKf94ZA/4FivQLS8tr+3S4tLiK4gflZInDKfxBoAsUUUUAFFFFABTJZEhheWVgsaKWZieAB1NPrz34ya4+jfDm8hgJ+16kwsYVC5Lb/vD/AL4DfjQBkfBuKTW7/wATeOLlGD6tetHbbxysKdAD6cqv/bOvWT0NYXhHQk8NeEtL0dAB9lt1Vyo6v1c/ixY/jR4k1x9GsYo7WEXOqXj+RZW/QSyEZyfRFGWY9gDQBxvxK8fNpGq6T4V0mYjVNRuoUnlQ8wRM4GB/tNnHsOe4r00DHSvBfEuiLp/xW8A6U0xubw3H2+8uW+9PMZFLOfQfusAdlUDtXvlABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRR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+54	"At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \geq 3$, the chairs are labelled $1,2,3, \ldots, n-1, n$ in order around the table. A table is considered full if no more people can be seated without having two people sit in neighbouring chairs. For example, when $n=6$, full tables occur when people are seated in chairs labelled $\{1,4\}$ or $\{2,5\}$ or $\{3,6\}$ or $\{1,3,5\}$ or $\{2,4,6\}$. Thus, there are 5 different full tables when $n=6$.
+
+<image_1>
+A full table with $6 k+5$ chairs, for some positive integer $k$, has $t$ people seated in its chairs. Determine, in terms of $k$, the number of possible values of $t$."	['Suppose that $k$ is a positive integer.\n\nSuppose that $t$ people are seated at a table with $6 k+5$ chairs so that the table is full.\n\nWhen $t$ people are seated, there are $t$ gaps. Each gap consists of either 1 or 2 chairs. (A gap with 3 or more chairs can have an additional person seated in it, so the table is not full.)\n\nTherefore, there are between $t$ and $2 t$ empty chairs.\n\nThis means that the total number of chairs is between $t+t$ and $t+2 t$.\n\nIn other words, $2 t \\leq 6 k+5 \\leq 3 t$.\n\nSince $2 t \\leq 6 k+5$, then $t \\leq 3 k+\\frac{5}{2}$. Since $k$ and $t$ are integers, then $t \\leq 3 k+2$.\n\nWe note that it is possible to seat $3 k+2$ people around the table in seats\n\n$$\n\\{2,4,6, \\ldots, 6 k+2,6 k+4\\}\n$$\n\nThis table is full becase $3 k+1$ of the gaps consist of 1 chair and 1 gap consists of 2 chairs. Since $3 t \\geq 6 k+5$, then $t \\geq 2 k+\\frac{5}{3}$. Since $k$ and $t$ are integers, then $t \\geq 2 k+2$.\n\nWe note that it is possible to seat $2 k+2$ people around the table in seats\n\n$$\n\\{3,6,9, \\ldots, 6 k, 6 k+3,6 k+5\\}\n$$\n\nThis table is full becase $2 k+1$ of the gaps consist of 2 chairs and 1 gap consists of 1 chair.\n\nWe now know that, if there are $t$ people seated at a full table with $6 k+5$ chairs, then $2 k+2 \\leq t \\leq 3 k+2$.\n\nTo confirm that every such value of $t$ is possible, consider a table with $t$ people, $3 t-(6 k+5)$\n\n\n\ngaps of 1 chair, and $(6 k+5)-2 t$ gaps of 2 chairs.\n\nFrom the work above, we know that $3 t \\geq 6 k+5$ and so $3 t-(6 k+5) \\geq 0$, and that $2 t \\leq 6 k+5$ and so $(6 k+5)-2 t \\geq 0$.\n\nThe total number of gaps is $3 t-(6 k+5)+(6 k+5)-2 t=t$, since there are $t$ people seated.\n\nFinally, the total number of chairs is\n\n$$\nt+1 \\cdot(3 t-(6 k+5))+2 \\cdot((6 k+5)-2 t)=t+3 t-4 t-(6 k+5)+2(6 k+5)=6 k+5\n$$\n\nas expected.\n\nThis shows that every $t$ with $2 k+2 \\leq t \\leq 3 k+2$ can produce a full table.\n\nTherefore, the possible values of $t$ are those integers that satisfy $2 k+2 \\leq t \\leq 3 k+2$.\n\nThere are $(3 k+2)-(2 k+2)+1=k+1$ possible values of 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+55	"At the Canadian Eatery with Multiple Configurations, there are round tables, around which chairs are placed. When a table has $n$ chairs around it for some integer $n \geq 3$, the chairs are labelled $1,2,3, \ldots, n-1, n$ in order around the table. A table is considered full if no more people can be seated without having two people sit in neighbouring chairs. For example, when $n=6$, full tables occur when people are seated in chairs labelled $\{1,4\}$ or $\{2,5\}$ or $\{3,6\}$ or $\{1,3,5\}$ or $\{2,4,6\}$. Thus, there are 5 different full tables when $n=6$.
+
+<image_1>
+Determine the number of different full tables when $n=19$."	"['For each integer $n \\geq 3$, we define $f(n)$ to be the number of different full tables of size $n$. We can check that\n\n- $f(3)=3$ because the full tables when $n=3$ have people in chairs $\\{1\\},\\{2\\},\\{3\\}$,\n- $f(4)=2$ because the full tables when $n=4$ have people in chairs $\\{1,3\\},\\{2,4\\}$, and\n- $f(5)=5$ because the full tables when $n=4$ have people in chairs $\\{1,3\\},\\{2,4\\},\\{3,5\\}$, $\\{4,1\\},\\{5,2\\}$.\n\nIn the problem, we are told that $f(6)=5$ and in part (a), we determined that $f(8)=10$. This gives us the following table:\n\n| $n$ | $f(n)$ |\n| :---: | :---: |\n| 3 | 3 |\n| 4 | 2 |\n| 5 | 5 |\n| 6 | 5 |\n| 7 | $?$ |\n| 8 | 10 |\n\nBased on this information, we make the guess that for every integer $n \\geq 6$, we have $f(n)=f(n-2)+f(n-3)$.\n\nFor example, this would mean that $f(7)=f(5)+f(4)=5+2=7$ which we can verify is true.\n\nBased on this recurrence relation (which we have yet to prove), we deduce the values of $f(n)$ up to and including $n=19$ :\n\n| $n$ | $f(n)$ | $n$ | $f(n)$ |\n| :---: | :---: | :---: | :---: |\n| 3 | 3 | 11 | 22 |\n| 4 | 2 | 12 | 29 |\n| 5 | 5 | 13 | 39 |\n| 6 | 5 | 14 | 51 |\n| 7 | 7 | 15 | 68 |\n| 8 | 10 | 16 | 90 |\n| 9 | 12 | 17 | 119 |\n| 10 | 17 | 18 | 158 |\n|  |  | 19 | 209 |\n\nWe now need to prove that the equation $f(n)=f(n-2)+f(n-3)$ is true for all $n \\geq 6$.\n\n\n\nWe think about each full table as a string of 0 s and 1s, with 1 representing a chair that is occupied and 0 representing an empty chair.\n\nLet $a(n)$ be the number of full tables with someone in seat 1 (and thus nobody in seat 2). Let $b(n)$ be the number of full tables with someone in seat 2 (and thus nobody in seat 1). Let $c(n)$ be the number of full tables with nobody in seat 1 or in seat 2 .\n\nSince every full table must be in one of these categories, then $f(n)=a(n)+b(n)+c(n)$. A full table with $n$ seats $n \\geq 4$ must correspond to a string that starts with 10,01 or 00 . Since there cannot be more than two consecutive 0s, we can further specify this, namely to say that a full table with $n$ seats must correspond to a string that starts with 1010 or 1001 or 0100 or 0101 or 0010 . In each case, these are the first 4 characters of the string and correspond to full (1) and empty (0) chairs.\n\nConsider the full tables starting with 1010. Note that such strings end with 0 since the table is circular. Removing the 10 from positions 1 and 2 creates strings of length $n-2$ that begin 10. These strings will still correspond to a full table, and so there are $a(n-2)$ such strings. (We note that all possible strings starting 1010 of length $n$ will lead to all possible strings starting with 1010 of length $n-2$.)\n\nConsider the full tables starting with 1001. Note that such a string ends with 0 since the table is circular. Removing the 100 from positions 1, 2 and 3 creates strings of length $n-3$ that begin 10. (There must have been a 0 in position 5 after the 1 in position 4.) These strings will still correspond to full tables, and so there are $a(n-3)$ such strings. Consider the full tables starting with 0100 . Removing the 100 from positions 2,3 and 4 creates strings of length $n-3$ that begin 01 . (There must have been a 1 in position 5 after the 0 in position 4.) These strings will still correspond to full tables, and so there are $b(n-3)$ such strings.\n\nConsider the full tables starting with 0101. Removing the 01 from positions 3 and 4 creates strings of length $n-2$ that begin 01 . (The 1 in position 4 must have been followed by one or two 0s and so these strings maintains the desired properties.) These strings will still correspond to full tables, and so there are $b(n-2)$ such strings.\n\nConsider the full tables starting with 0010. These strings must begin with either 00100 or 00101.\n\nIf strings start 00100 , then they start 001001 and so we remove the 001 in positions 4,5 and 6 and obtain strings of length $n-3$ that start 001 (and thus start 00). There are $c(n-3)$ such strings.\n\nIf strings start 00101, we remove the 01 in positions 4 and 5 and obtain strings of length $n-2$ that start 001 (and thus start 00 ). There are $c(n-2)$ such strings.\n\nThese 6 cases and subcases count all strings counted by $f(n)$.\n\nTherefore,\n\n$$\n\\begin{aligned}\nf(n) & =a(n-2)+a(n-3)+b(n-3)+b(n-2)+c(n-3)+c(n-2) \\\\\n& =a(n-2)+b(n-2)+c(n-2)+a(n-3)+b(n-3)+c(n-3) \\\\\n& =f(n-2)+f(n-3)\n\\end{aligned}\n$$\n\nas required, which means that the number of different full tables when $n=19$ is 209 .'
+ 'Extending our approach from (b), the number of people seated at a full table with 19 chairs is at least $\\frac{19}{3}=6 \\frac{1}{3}$ and at most $\\frac{19}{2}=9 \\frac{1}{2}$.\n\nSince the number of people is an integer, there must be 7,8 or 9 people at the table, which means that the number of empty chairs is 12,11 or 10 , respectively.\n\n\n\nSuppose that there are 9 people and 9 gaps with a total of 10 empty chairs.\n\nIn this case, there is 1 gap with 2 empty chairs and 8 gaps with 1 empty chair.\n\nThere are 19 pairs of chairs in which we can put 2 people with a gap of 2 in between: $\\{1,4\\},\\{2,5\\}, \\ldots,\\{19,3\\}$.\n\nOnce we choose one of these pairs, the seat choice for the remaining 8 people is completely determined by placing people in every other chair.\n\nTherefore, there are 19 different full tables with 9 people.\n\nSuppose that there are 8 people and 8 gaps with a total of 11 empty chairs.\n\nIn this case, there are 3 gaps with 2 empty chairs and 5 gaps with 1 empty chair. There are 7 different circular orderings in which these 8 gaps can be arranged:\n\n$$\n\\begin{array}{lllllll}\n22211111 & 22121111 & 22112111 & 22111211 & 22111121 & 21212111 & 21211211\n\\end{array}\n$$\n\nWe note that ""22211111"" would be the same as, for example, ""11222111"" since these gaps are arranged around a circle.\n\nIf the three gaps of length 2 are consecutive, there is only one configuration (22211111). If there are exactly 2 consecutive gaps of length 2 , there are 4 relative places in which the third gap of length 2 can be placed.\n\nIf there are no consecutive gaps of length 2 , these gaps can either be separated by 1 gap each (21212111) with 3 gaps on the far side, or can be separated by 1 gap, 2 gaps, and 2 gaps (21211211). There is only one configuration for the gaps in this last situation.\n\nThere are 7 different circular orderings for these 8 gaps.\n\nEach of these 7 different orderings can be placed around the circle of 19 chairs in 19 different ways, because each can be started in 19 different places. Because 19 is prime, none of these orderings overlap.\n\nTherefore, there are $7 \\cdot 19=133$ different full tables with 8 people.\n\nSuppose that there are 7 people and 7 gaps with a total of 12 empty chairs.\n\nIn this case, there are 2 gaps with 1 empty chair and 5 gaps with 2 empty chairs.\n\nThe 2 gaps with 1 empty chair can be separated by 0 gaps with 2 empty chairs, 1 gap with 2 empty chairs, or 2 gaps with 2 empty chairs. Because the chairs are around a circle, if there were 3 , 4 or 5 gaps with 2 empty chairs between them, there would be 2,1 or 0 gaps going the other way around the circle.\n\nThis means that there are 3 different configurations for the gaps.\n\nEach of these configurations can be placed in 19 different ways around the circle of chairs. Therefore, there are $3 \\cdot 19=57$ full tables with 7 people.\n\nIn total, there are $19+133+57=209$ full tables with 19 chairs.'
+ 'As in Solution 2, there must be 7, 8 or 9 people in chairs, and so there are 7,8 or 9 gaps. If there are 7 gaps, there are 2 gaps of 1 chair and 5 gaps of 2 chairs.\n\nIf there are 8 gaps, there are 5 gaps of 1 chair and 3 gaps of 2 chairs.\n\nIf there are 9 gaps, there are 8 gaps of 1 chair and 1 gap of 2 chairs.\n\nWe consider three mutually exclusive cases: (i) there is a person in chair 1 and not in chair 2 , (ii) there is a person in chair 2 and not in chair 1 , and (iii) there is nobody in chair 1 or in chair 2. Every full table fits into exactly one of these three cases.\n\nCase (i): there is a person in chair 1 and not in chair 2\n\nWe use the person in chair 1 to ""anchor"" the arrangement, by starting at chair 1 and\n\n\n\narranging the gaps (and thus the full chairs) clockwise around the table from chair 1.\n\nIf there are 7 gaps, we need to choose 2 of them to be of length 1 , and so there are $\\left(\\begin{array}{l}7 \\\\ 2\\end{array}\\right)$ ways of arranging the gaps starting at chair 1.\n\nIf there are 8 gaps, we need to choose 3 of them to be of length 2 , and so there are $\\left(\\begin{array}{l}8 \\\\ 3\\end{array}\\right)$ ways of arranging the gaps starting at chair 1.\n\nIf there are 9 gaps, we need to choose 1 of them to be of length 2 , and so there are $\\left(\\begin{array}{l}9 \\\\ 1\\end{array}\\right)$ ways of arranging the gaps starting at chair 1.\n\nIn this case, there are a total of $\\left(\\begin{array}{l}7 \\\\ 2\\end{array}\\right)+\\left(\\begin{array}{l}8 \\\\ 3\\end{array}\\right)+\\left(\\begin{array}{l}9 \\\\ 1\\end{array}\\right)=21+56+9=86$ full tables.\n\nCase (ii): there is a person in chair 2 and not in chair 1\n\nWe use the same reasoning starting with the person in chair 2 as the anchor.\n\nAgain, there are 86 full tables in this case.\n\nCase (iii): there is nobody in chair 1 or chair 2\n\nSince there is nobody in chair 1 or chair 2 , there must be a person in chair 3 and also in chair 19 , which fixes one gap of 2 chairs.\n\nHere, we use the person in chair 3 as the anchor.\n\nIf there are 7 gaps, there are 2 gaps of 1 chair and 4 gaps of 2 chairs left to place. There are $\\left(\\begin{array}{l}6 \\\\ 2\\end{array}\\right)$ ways of doing this.\n\nIf there are 8 gaps, there are 5 gaps of 1 chair and 2 gaps of 2 chairs left to place. There are $\\left(\\begin{array}{l}7 \\\\ 2\\end{array}\\right)$ ways of doing this.\n\nIf there are 9 gaps, there are 8 gaps of 1 chair and 0 gaps of 2 chairs left to place. There is 1 way to do this.\n\nIn this case, there are a total of $\\left(\\begin{array}{l}6 \\\\ 2\\end{array}\\right)+\\left(\\begin{array}{l}7 \\\\ 2\\end{array}\\right)+1=15+21+1=37$ full tables.\n\nIn total, there are $86+86+37=209$ full tables with 19 chairs.']"	['209']		['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+56	"In the diagram, $\triangle A B C$ is right-angled at $B$ and $\triangle A C D$ is right-angled at $A$. Also, $A B=3, B C=4$, and $C D=13$. What is the area of quadrilateral $A B C D$ ?
+
+<image_1>"	"['The area of quadrilateral $A B C D$ is the sum of the areas of $\\triangle A B C$ and $\\triangle A C D$.\n\nSince $\\triangle A B C$ is right-angled at $B$, its area equals $\\frac{1}{2}(A B)(B C)=\\frac{1}{2}(3)(4)=6$.\n\nSince $\\triangle A B C$ is right-angled at $B$, then by the Pythagorean Theorem,\n\n$$\nA C=\\sqrt{A B^{2}+B C^{2}}=\\sqrt{3^{2}+4^{2}}=\\sqrt{25}=5\n$$\n\nbecause $A C>0$. (We could have also observed that $\\triangle A B C$ must be a ""3-4-5"" triangle.) Since $\\triangle A C D$ is right-angled at $A$, then by the Pythagorean Theorem,\n\n$$\nA D=\\sqrt{C D^{2}-A C^{2}}=\\sqrt{13^{2}-5^{2}}=\\sqrt{144}=12\n$$\n\nbecause $A D>0$. (We could have also observed that $\\triangle A C D$ must be a "" $5-12-13$ "" triangle.) Thus, the area of $\\triangle A C D$ equals $\\frac{1}{2}(A C)(A D)=\\frac{1}{2}(5)(12)=30$.\n\nFinally, the area of quadrilateral $A B C D$ is thus $6+30=36$.']"	['36']		['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+57	"Three identical rectangles $P Q R S$, WTUV and $X W V Y$ are arranged, as shown, so that $R S$ lies along $T X$. The perimeter of each of the three rectangles is $21 \mathrm{~cm}$. What is the perimeter of the whole shape?
+<image_1>"	['Let the width of each of the identical rectangles be $a$.\n\nIn other words, $Q P=R S=T W=W X=U V=V Y=a$.\n\nLet the height of each of the identical rectangles be $b$.\n\nIn other words, $Q R=P S=T U=W V=X Y=b$.\n\nThe perimeter of the whole shape equals\n\n$$\nQ P+P S+S X+X Y+V Y+U V+T U+T R+Q R\n$$\n\nSubstituting for known lengths, we obtain\n\n$$\na+b+S X+b+a+a+b+T R+b\n$$\n\nor $3 a+4 b+(S X+T R)$.\n\nBut $S X+T R=(T R+R S+S X)-R S=(T W+W X)-R S=a+a-a=a$.\n\nTherefore, the perimeter of the whole shape equals $4 a+4 b$.\n\nThe perimeter of one rectangle is $2 a+2 b$, which we are told equals $21 \\mathrm{~cm}$.\n\nFinally, the perimeter of the whole shape is thus $2(2 a+2 b)$ which equals $42 \\mathrm{~cm}$.']	['42']		['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']	Multimodal	Competition	False	cm	Numerical		Open-ended	Geometry	Math	English
+58	"The diagram shows two hills that meet at $O$. One hill makes a $30^{\circ}$ angle with the horizontal and the other hill makes a $45^{\circ}$ angle with the horizontal. Points $A$ and $B$ are on the hills so that $O A=O B=20 \mathrm{~m}$. Vertical poles $B D$ and $A C$ are connected by a straight cable $C D$. If $A C=6 \mathrm{~m}$, what is the length of $B D$ for which $C D$ is as short as possible?
+
+<image_1>"	['Extend $C A$ and $D B$ downwards until they meet the horizontal through $O$ at $P$ and $Q$, respectively.\n\n<img_3828>\n\nSince $C A$ and $D B$ are vertical, then $\\angle C P O=\\angle D Q O=90^{\\circ}$.\n\nSince $O A=20 \\mathrm{~m}$, then $A P=O A \\sin 30^{\\circ}=(20 \\mathrm{~m}) \\cdot \\frac{1}{2}=10 \\mathrm{~m}$.\n\nSince $O B=20 \\mathrm{~m}$, then $B Q=O B \\sin 45^{\\circ}=(20 \\mathrm{~m}) \\cdot \\frac{1}{\\sqrt{2}}=10 \\sqrt{2} \\mathrm{~m}$.\n\nSince $A C=6 \\mathrm{~m}$, then $C P=A C+A P=16 \\mathrm{~m}$.\n\nFor $C D$ to be as short as possible and given that $C$ is fixed, then it must be the case that $C D$ is horizontal:\n\nIf $C D$ were not horizontal, then suppose that $X$ is on $D Q$, possibly extended, so that $C X$ is horizontal.\n\n<img_3255>\n\nThen $\\angle C X D=90^{\\circ}$ and so $\\triangle C X D$ is right-angled with hypotenuse $C D$.\n\nIn this case, $C D$ is longer than $C X$ or $X D$.\n\nIn particular, $C D>C X$, which means that if $D$ were at $X$, then $C D$ would be shorter.\n\nIn other words, a horizontal $C D$ makes $C D$ as short as possible.\n\nWhen $C D$ is horizontal, $C D Q P$ is a rectangle, since it has two vertical and two horizontal sides. Thus, $D Q=C P=16 \\mathrm{~m}$.\n\nFinally, this means that $B D=D Q-B Q=(16-10 \\sqrt{2}) \\mathrm{m}$.']	['$(16-10 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+59	"In the diagram, line segments $A C$ and $D F$ are tangent to the circle at $B$ and $E$, respectively. Also, $A F$ intersects the circle at $P$ and $R$, and intersects $B E$ at $Q$, as shown. If $\angle C A F=35^{\circ}, \angle D F A=30^{\circ}$, and $\angle F P E=25^{\circ}$, determine the measure of $\angle P E Q$.
+
+<image_1>"	['Let $\\angle P E Q=\\theta$.\n\nJoin $P$ to $B$.\n\nWe use the fact that the angle between a tangent to a circle and a chord in that circle that passes through the point of tangency equals the angle inscribed by that chord. We prove this fact below.\n\nMore concretely, $\\angle D E P=\\angle P B E$ (using the chord $E P$ and the tangent through $E$ ) and $\\angle A B P=\\angle P E Q=\\theta$ (using the chord $B P$ and the tangent through $B$ ).\n\nNow $\\angle D E P$ is exterior to $\\triangle F E P$ and so $\\angle D E P=\\angle F P E+\\angle E F P=25^{\\circ}+30^{\\circ}$, and so $\\angle P B E=\\angle D E P=55^{\\circ}$.\n\nFurthermore, $\\angle A Q B$ is an exterior angle of $\\triangle P Q E$.\n\nThus, $\\angle A Q B=\\angle Q P E+\\angle P E Q=25^{\\circ}+\\theta$.\n\n<img_3991>\n\nIn $\\triangle A B Q$, we have $\\angle B A Q=35^{\\circ}, \\angle A B Q=\\theta+55^{\\circ}$, and $\\angle A Q B=25^{\\circ}+\\theta$.\n\nThus, $35^{\\circ}+\\left(\\theta+55^{\\circ}\\right)+\\left(25^{\\circ}+\\theta\\right)=180^{\\circ}$ or $115^{\\circ}+2 \\theta=180^{\\circ}$, and so $2 \\theta=65^{\\circ}$.\n\nTherefore $\\angle P E Q=\\theta=\\frac{1}{2}\\left(65^{\\circ}\\right)=32.5^{\\circ}$.\n\nAs an addendum, we prove that the angle between a tangent to a circle and a chord in that circle that passes through the point of tangency equals the angle inscribed by that chord.\n\nConsider a circle with centre $O$ and a chord $X Y$, with tangent $Z X$ meeting the circle at $X$. We prove that if $Z X$ is tangent to the circle, then $\\angle Z X Y$ equals $\\angle X W Y$ whenever $W$ is a point on the circle on the opposite side of $X Y$ as $X Z$ (that is, the angle subtended by $X Y$ on the opposite side of the circle).\n\nWe prove this in the case that $\\angle Z X Y$ is acute. The cases where $\\angle Z X Y$ is a right angle or an obtuse angle are similar.\n\nDraw diameter $X O V$ and join $V Y$.\n\n<img_3622>\n\nSince $\\angle Z X Y$ is acute, points $V$ and $W$ are on the same arc of chord $X Y$.\n\nThis means that $\\angle X V Y=\\angle X W Y$, since they are angles subtended by the same chord.\n\nSince $O X$ is a radius and $X Z$ is a tangent, then $\\angle O X Z=90^{\\circ}$.\n\nThus, $\\angle O X Y+\\angle Z X Y=90^{\\circ}$.\n\nSince $X V$ is a diameter, then $\\angle X Y V=90^{\\circ}$.\n\nFrom $\\triangle X Y V$, we see that $\\angle X V Y+\\angle V X Y=90^{\\circ}$.\n\nBut $\\angle O X Y+\\angle Z X Y=90^{\\circ}$ and $\\angle X V Y+\\angle V X Y=90^{\\circ}$ and $\\angle O X Y=\\angle V X Y$ tells us that $\\angle Z X Y=\\angle X V Y$.\n\nThis gives us that $\\angle Z X Y=\\angle X W Y$, as required.']	['$32.5$']		['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+60	"In the diagram, $A B C D$ and $P N C D$ are squares of side length 2, and $P N C D$ is perpendicular to $A B C D$. Point $M$ is chosen on the same side of $P N C D$ as $A B$ so that $\triangle P M N$ is parallel to $A B C D$, so that $\angle P M N=90^{\circ}$, and so that $P M=M N$. Determine the volume of the convex solid $A B C D P M N$.
+
+<image_1>"	"['Draw a line segment through $M$ in the plane of $\\triangle P M N$ parallel to $P N$ and extend this line until it reaches the plane through $P, A$ and $D$ at $Q$ on one side and the plane through $N, B$ and $C$ at $R$ on the other side.\n\nJoin $Q$ to $P$ and $A$. Join $R$ to $N$ and $B$.\n\n<img_3945>\n\nSo the volume of solid $A B C D P M N$ equals the volume of solid $A B C D P Q R N$ minus the volumes of solids $P M Q A$ and $N M R B$.\n\nSolid $A B C D P Q R N$ is a trapezoidal prism. This is because $N R$ and $B C$ are parallel (since they lie in parallel planes), which makes $N R B C$ a trapezoid. Similarly, $P Q A D$ is a trapezoid. Also, $P N, Q R, D C$, and $A B$ are all perpendicular to the planes of these trapezoids and equal in length, since they equal the side lengths of the squares.\n\nSolids $P M Q A$ and $N M R B$ are triangular-based pyramids. We can think of their bases as being $\\triangle P M Q$ and $\\triangle N M R$. Their heights are each equal to 2 , the height of the original solid. (The volume of a triangular-based pyramid equals $\\frac{1}{3}$ times the area of its base times its height.)\n\nThe volume of $A B C D P Q R N$ equals the area of trapezoid $N R B C$ times the width of the prism, which is 2.\n\nThat is, this volume equals $\\frac{1}{2}(N R+B C)(N C)(N P)=\\frac{1}{2}(N R+2)(2)(2)=2 \\cdot N R+4$.\n\nSo we need to find the length of $N R$.\n\nConsider quadrilateral $P N R Q$. This quadrilateral is a rectangle since $P N$ and $Q R$ are perpendicular to the two side planes of the original solid.\n\nThus, $N R$ equals the height of $\\triangle P M N$.\n\nJoin $M$ to the midpoint $T$ of $P N$.\n\nSince $\\triangle P M N$ is isosceles, then $M T$ is perpendicular to $P N$.\n\n<img_3305>\n\nSince $N T=\\frac{1}{2} P N=1$ and $\\angle P M N=90^{\\circ}$ and $\\angle T N M=45^{\\circ}$, then $\\triangle M T N$ is also right-angled and isosceles with $M T=T N=1$.\n\nTherefore, $N R=M T=1$ and so the volume of $A B C D P Q R N$ is $2 \\cdot 1+4=6$.\n\nThe volumes of solids $P M Q A$ and $N M R B$ are equal. Each has height 2 and their bases $\\triangle P M Q$ and $\\triangle N M R$ are congruent, because each is right-angled (at $Q$ and at $R$ ) with $P Q=N R=1$ and $Q M=M R=1$.\n\nThus, using the formula above, the volume of each is $\\frac{1}{3}\\left(\\frac{1}{2}(1)(1)\\right) 2=\\frac{1}{3}$.\n\nFinally, the volume of the original solid equals $6-2 \\cdot \\frac{1}{3}=\\frac{16}{3}$.'
+ 'We determine the volume of $A B C D P M N$ by splitting it into two solids: $A B C D P N$ and $A B N P M$ by slicing along the plane of $A B N P$.\n\nSolid $A B C D P N$ is a triangular prism, since $\\triangle B C N$ and $\\triangle A D P$ are each right-angled (at $C$ and $D$ ), $B C=C N=A D=D P=2$, and segments $P N, D C$ and $A B$ are perpendicular to each of the triangular faces and equal in length.\n\nThus, the volume of $A B C D P N$ equals the area of $\\triangle B C N$ times the length of $D C$, or $\\frac{1}{2}(B C)(C N)(D C)=\\frac{1}{2}(2)(2)(2)=4$. (This solid can also be viewed as ""half"" of a cube.)\n\nSolid $A B N P M$ is a pyramid with rectangular base $A B N P$. (Note that $P N$ and $A B$ are perpendicular to the planes of both of the side triangular faces of the original solid, that $P N=A B=2$ and $B N=A P=\\sqrt{2^{2}+2^{2}}=2 \\sqrt{2}$, by the Pythagorean Theorem.)\n\nTherefore, the volume of $A B N P M$ equals $\\frac{1}{3}(A B)(B N) h=\\frac{4 \\sqrt{2}}{3} h$, where $h$ is the height of the pyramid (that is, the distance that $M$ is above plane $A B N P$ ).\n\nSo we need to calculate $h$.\n\nJoin $M$ to the midpoint, $T$, of $P N$ and to the midpoint, $S$, of $A B$. Join $S$ and $T$. By symmetry, $M$ lies directly above $S T$. Since $A B N P$ is a rectangle and $S$ and $T$ are the midpoints of opposite sides, then $S T=A P=2 \\sqrt{2}$.\n\nSince $\\triangle P M N$ is right-angled and isosceles, then $M T$ is perpendicular to $P N$. Since $N T=\\frac{1}{2} P N=1$ and $\\angle T N M=45^{\\circ}$, then $\\triangle M T N$ is also right-angled and isosceles with $M T=T N=1$.\n\n<img_3858>\n\nAlso, $M S$ is the hypotenuse of the triangle formed by dropping a perpendicular from $M$ to $U$ in the plane of $A B C D$ (a distance of 2) and joining $U$ to $S$. Since $M$ is 1 unit horizontally from $P N$, then $U S=1$.\n\nThus, $M S=\\sqrt{2^{2}+1^{2}}=\\sqrt{5}$ by the Pythagorean Theorem.\n\n<img_3846>\n\nWe can now consider $\\triangle S M T . h$ is the height of this triangle, from $M$ to base $S T$.\n\n<img_3680>\n\nNow $h=M T \\sin (\\angle M T S)=\\sin (\\angle M T S)$.\n\nBy the cosine law in $\\triangle S M T$, we have\n\n$$\nM S^{2}=S T^{2}+M T^{2}-2(S T)(M T) \\cos (\\angle M T S)\n$$\n\nTherefore, $5=8+1-4 \\sqrt{2} \\cos (\\angle M T S)$ or $4 \\sqrt{2} \\cos (\\angle M T S)=4$.\n\nThus, $\\cos (\\angle M T S)=\\frac{1}{\\sqrt{2}}$ and so $\\angle M T S=45^{\\circ}$ which gives $h=\\sin (\\angle M T S)=\\frac{1}{\\sqrt{2}}$.\n\n(Alternatively, we note that the plane of $A B C D$ is parallel to the plane of $P M N$, and so since the angle between plane $A B C D$ and plane $P N B A$ is $45^{\\circ}$, then the angle between plane $P N B A$ and plane $P M N$ is also $45^{\\circ}$, and so $\\angle M T S=45^{\\circ}$.)\n\nFinally, this means that the volume of $A B N P M$ is $\\frac{4 \\sqrt{2}}{3} \\cdot \\frac{1}{\\sqrt{2}}=\\frac{4}{3}$, and so the volume of solid $A B C D P M N$ is $4+\\frac{4}{3}=\\frac{16}{3}$.']"	['$\\frac{16}{3}$']		['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+61	"In the diagram, $\triangle A B C$ is right-angled at $B$ and $A C=20$. If $\sin C=\frac{3}{5}$, what is the length of side $B C$ ?
+
+<image_1>"	"['Since $\\sin C=\\frac{A B}{A C}$, then $A B=A C \\sin C=20\\left(\\frac{3}{5}\\right)=12$.\n\nBy Pythagoras, $B C^{2}=A C^{2}-A B^{2}=20^{2}-12^{2}=256$ or $B C=16$.'
+ 'Using the standard trigonometric ratios, $B C=A C \\cos C$.\n\nSince $\\sin C=\\frac{3}{5}$, then $\\cos ^{2} C=1-\\sin ^{2} C=1-\\frac{9}{25}=\\frac{16}{25}$ or $\\cos C=\\frac{4}{5}$. (Notice that $\\cos C$ is positive since angle $C$ is acute in triangle $A B C$.)\n\nTherefore, $B C=20\\left(\\frac{4}{5}\\right)=16$.']"	['16']		['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+62	"A helicopter is flying due west over level ground at a constant altitude of $222 \mathrm{~m}$ and at a constant speed. A lazy, stationary goat, which is due west of the helicopter, takes two measurements of the angle between the ground and the helicopter. The first measurement the goat makes is $6^{\circ}$ and the second measurement, which he makes 1 minute later, is $75^{\circ}$. If the helicopter has not yet passed over the goat, as shown, how fast is the helicopter travelling to the nearest kilometre per hour?
+
+<image_1>"	['Let $G$ be the point where the goat is standing, $H$ the position of the helicopter when the goat first measures the angle, $P$ the point directly below the helicopter at this time, $J$ the position of the helicopter one minute later, and $Q$ the point directly below the helicopter at this time.\n\n<img_3329>\n\nUsing the initial position of the helicopter, $\\tan \\left(6^{\\circ}\\right)=\\frac{H P}{P G}$ or $P G=\\frac{222}{\\tan \\left(6^{\\circ}\\right)} \\approx 2112.19 \\mathrm{~m}$.\n\nUsing the second position of the helicopter, $\\tan \\left(75^{\\circ}\\right)=\\frac{J Q}{Q G}$ or $Q G=\\frac{222}{\\tan \\left(75^{\\circ}\\right)} \\approx 59.48 \\mathrm{~m}$.\n\nSo in the one minute that has elapsed, the helicopter has travelled\n\n$2112.19 \\mathrm{~m}-59.48 \\mathrm{~m}=2052.71 \\mathrm{~m}$ or $2.0527 \\mathrm{~km}$.\n\nTherefore, in one hour, the helicopter will travel $60(2.0527)=123.162 \\mathrm{~km}$.\n\nThus, the helicopter is travelling $123 \\mathrm{~km} / \\mathrm{h}$.']	['123']		['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+63	"A regular hexagon is a six-sided figure which has all of its angles equal and all of its side lengths equal. In the diagram, $A B C D E F$ is a regular hexagon with an area of 36. The region common to the equilateral triangles $A C E$ and $B D F$ is a hexagon, which is shaded as shown. What is the area of the shaded hexagon?
+
+<image_1>"	"['We label the vertices of the shaded hexagon $U, V, W, X$, $Y$, and $Z$.\n\nBy symmetry, all of the six triangles with two vertices on the inner hexagon and one on the outer hexagon (eg. triangle $U V A$ ) are congruent equilateral triangles. In order to determine the area of the inner hexagon, we determine the ratio of the side lengths of the two hexagons.\n\n<img_3663>\n\nLet the side length of the inner hexagon be $x$. Then $A U=U F=x$. Then triangle $A U F$ has a $120^{\\circ}$ between the two sides of length $x$. If we draw a perpendicular from $U$ to point $P$ on side $A F$, then $U P$ divides\n\n$\\triangle A U F$ into two $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangles. Thus, $F P=P A=\\frac{\\sqrt{3}}{2} x$ and so $A F=\\sqrt{3} x$.\n\nSo the ratio of the side lengths of the hexagons is $\\sqrt{3}: 1$, and so the ratio of\n\n<img_3205>\ntheir areas is $(\\sqrt{3})^{2}: 1=3: 1$.\n\nSince the area of the larger hexagon is 36 , then the area of the inner hexagon is 12.'
+ 'We label the vertices of the hexagon $U, V, W, X, Y$, and $Z$. By symmetry, all of the six triangles with two vertices on the inner hexagon and one on the outer hexagon (eg. triangle $U V A$ ) are congruent equilateral triangles.\n\nWe also join the opposite vertices of the inner hexagon, ie. we join $U$ to $X, V$ to $Y$, and $W$ to $Z$. (These 3 line segments all meet at a single point, say $O$.) This divides the inner hexagon into 6 small equilateral triangles identical to the\n\n<img_3385>\nsix earlier mentioned equilateral triangles.\n\nLet the area of one of these triangles be $a$. Then we can label the 12 small equilateral triangles as all having area $a$.\n\nBut triangle $A U F$ also has area $a$, because if we consider triangle $A F V$, then $A U$ is a median (since $F U=A U=U V$ by symmetry) and so divides triangle $A F V$ into two triangles of equal area. Since the area of\n\n<img_3160>\ntriangle $A U V$ is $a$, then the area of triangle $A U F$ is also $a$.\n\nTherefore, hexagon $A B C D E F$ is divided into 18 equal areas. Thus, $a=2$ since the area of the large hexagon is 36.\n\nSince the area of $U V W X Y Z$ is $6 a$, then its area is 12 .']"	['12']		['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+64	"At the Big Top Circus, Herc the Human Cannonball is fired out of the cannon at ground level. (For the safety of the spectators, the cannon is partially buried in the sand floor.) Herc's trajectory is a parabola until he catches the vertical safety net, on his way down, at point $B$. Point $B$ is $64 \mathrm{~m}$ directly above point $C$ on the floor of the tent. If Herc reaches a maximum height of $100 \mathrm{~m}$, directly above a point $30 \mathrm{~m}$ from the cannon, determine the horizontal distance from the cannon to the net.
+
+<image_1>"	['We assign coordinates to the diagram, with the mouth of the cannon at the point $(0,0)$, with the positive $x$-axis in the horizontal direction towards the safety net from the cannon, and the positive $y$ axis upwards from $(0,0)$.\n\nSince Herc reaches his maximum height when his horizontal distance is $30 \\mathrm{~m}$, then the axis of symmetry of the parabola is the line $x=30$. Since the parabola has a root at $x=0$, then the other root must be at $x=60$.\n\nTherefore, the parabola has the form $y=\\operatorname{ax}(x-60)$.\n\nIn order to determine the value of $a$, we note that Herc passes through the point $(30,100)$, and so\n\n<img_3831>\n\n\n\n$$\n\\begin{aligned}\n100 & =30 a(-30) \\\\\na & =-\\frac{1}{9}\n\\end{aligned}\n$$\n\nThus, the equation of the parabola is $y=-\\frac{1}{9} x(x-60)$.\n\n(Alternatively, we could say that since the parabola has its maximum point at $(30,100)$, then it must be of the form $y=a(x-30)^{2}+100$.\n\nSince the parabola passes through $(0,0)$, then we have\n\n$$\n\\begin{aligned}\n& 0=a(0-30)^{2}+100 \\\\\n& 0=900 a+100 \\\\\n& a=-\\frac{1}{9}\n\\end{aligned}\n$$\n\nThus, the parabola has the equation $y=-\\frac{1}{9}(x-30)^{2}+100$.)\n\nWe would like to find the points on the parabola which have $y$-coordinate 64 , so we solve\n\n$$\n\\begin{aligned}\n64 & =-\\frac{1}{9} x(x-60) \\\\\n0 & =x^{2}-60 x+576 \\\\\n0 & =(x-12)(x-48)\n\\end{aligned}\n$$\n\nSince we want a point after Herc has passed his highest point, then $x=48$, ie. the horizontal distance from the cannon to the safety net is $48 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+65	"In the diagram, $V$ is the vertex of the parabola with equation $y=-x^{2}+4 x+1$. Also, $A$ and $B$ are the points of intersection of the parabola and the line with equation $y=-x+1$. Determine the value of $A V^{2}+B V^{2}-A B^{2}$.
+
+<image_1>"	['First, we find the coordinates of $V$.\n\nTo do this, we use the given equation for the parabola and complete the square:\n\n$y=-x^{2}+4 x+1=-\\left(x^{2}-4 x-1\\right)=-\\left(x^{2}-4 x+2^{2}-2^{2}-1\\right)=-\\left((x-2)^{2}-5\\right)=-(x-2)^{2}+5$\n\nTherefore, the coordinates of the vertex $V$ are $(2,5)$.\n\nNext, we find the coordinates of $A$ and $B$.\n\nNote that $A$ and $B$ are the points of intersection of the line with equation $y=-x+1$ and the parabola with equation $y=-x^{2}+4 x+1$.\n\nWe equate $y$-values to obtain $-x+1=-x^{2}+4 x+1$ or $x^{2}-5 x=0$ or $x(x-5)=0$.\n\nTherefore, $x=0$ or $x=5$.\n\nIf $x=0$, then $y=-x+1=1$, and so $A$ (which is on the $y$-axis) has coordinates $(0,1)$.\n\nIf $x=5$, then $y=-x+1=-4$, and so $B$ has coordinates $(5,-4)$.\n\n\n\nWe now have the points $V(2,5), A(0,1), B(5,-4)$.\n\nThis gives\n\n$$\n\\begin{aligned}\nA V^{2} & =(0-2)^{2}+(1-5)^{2}=20 \\\\\nB V^{2} & =(5-2)^{2}+(-4-5)^{2}=90 \\\\\nA B^{2} & =(0-5)^{2}+(1-(-4))^{2}=50\n\\end{aligned}\n$$\n\nand so $A V^{2}+B V^{2}-A B^{2}=20+90-50=60$.']	['60']		['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+66	"In the diagram, $A B C$ is a quarter of a circular pizza with centre $A$ and radius $20 \mathrm{~cm}$. The piece of pizza is placed on a circular pan with $A, B$ and $C$ touching the circumference of the pan, as shown. What fraction of the pan is covered by the piece of pizza?
+
+<image_1>"	['Since $A B C$ is a quarter of a circular pizza with centre $A$ and radius $20 \\mathrm{~cm}$, then $A C=A B=20 \\mathrm{~cm}$.\n\nWe are also told that $\\angle C A B=90^{\\circ}$ (one-quarter of $360^{\\circ}$ ).\n\nSince $\\angle C A B=90^{\\circ}$ and $A, B$ and $C$ are all on the circumference of the circle, then $C B$ is a diameter of the pan. (This is a property of circles: if $X, Y$ and $Z$ are three points on a circle with $\\angle Z X Y=90^{\\circ}$, then $Y Z$ must be a diameter of the circle.)\n\nSince $\\triangle C A B$ is right-angled and isosceles, then $C B=\\sqrt{2} A C=20 \\sqrt{2} \\mathrm{~cm}$.\n\nTherefore, the radius of the circular plate is $\\frac{1}{2} C B$ or $10 \\sqrt{2} \\mathrm{~cm}$.\n\nThus, the area of the circular pan is $\\pi(10 \\sqrt{2} \\mathrm{~cm})^{2}=200 \\pi \\mathrm{cm}^{2}$.\n\nThe area of the slice of pizza is one-quarter of the area of a circle with radius $20 \\mathrm{~cm}$, or $\\frac{1}{4} \\pi(20 \\mathrm{~cm})^{2}=100 \\pi \\mathrm{cm}^{2}$.\n\nFinally, the fraction of the pan that is covered is the area of the slice of pizza divided by the area of the pan, or $\\frac{100 \\pi \\mathrm{cm}^{2}}{200 \\pi \\mathrm{cm}^{2}}=\\frac{1}{2}$.']	['$\\frac{1}{2}$']		['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+67	"The deck $A B$ of a sailboat is $8 \mathrm{~m}$ long. Rope extends at an angle of $60^{\circ}$ from $A$ to the top $(M)$ of the mast of the boat. More rope extends at an angle of $\theta$ from $B$ to a point $P$ that is $2 \mathrm{~m}$ below $M$, as shown. Determine the height $M F$ of the mast, in terms of $\theta$.
+
+<image_1>"	['Suppose that the length of $A F$ is $x \\mathrm{~m}$.\n\nSince the length of $A B$ is $8 \\mathrm{~m}$, then the length of $F B$ is $(8-x) \\mathrm{m}$.\n\nSince $\\triangle M A F$ is right-angled and has an angle of $60^{\\circ}$, then it is $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle.\n\nTherefore, $M F=\\sqrt{3} A F$, since $M F$ is opposite the $60^{\\circ}$ angle and $A F$ is opposite the $30^{\\circ}$ angle.\n\nThus, $M F=\\sqrt{3} x \\mathrm{~m}$.\n\nSince $M P=2 \\mathrm{~m}$, then $P F=M F-M P=(\\sqrt{3} x-2) \\mathrm{m}$.\n\nWe can now look at $\\triangle B F P$ which is right-angled at $F$.\n\nWe have\n\n$$\n\\tan \\theta=\\frac{P F}{F B}=\\frac{(\\sqrt{3} x-2) \\mathrm{m}}{(8-x) \\mathrm{m}}=\\frac{\\sqrt{3} x-2}{8-x}\n$$\n\nTherefore, $(8-x) \\tan \\theta=\\sqrt{3} x-2$ or $8 \\tan \\theta+2=\\sqrt{3} x+(\\tan \\theta) x$.\n\nThis gives $8 \\tan \\theta+2=x(\\sqrt{3}+\\tan \\theta)$ or $x=\\frac{8 \\tan \\theta+2}{\\tan \\theta+\\sqrt{3}}$.\n\nFinally, $M F=\\sqrt{3} x=\\frac{8 \\sqrt{3} \\tan \\theta+2 \\sqrt{3}}{\\tan \\theta+\\sqrt{3}} \\mathrm{~m}$.\n\n']	['$\\frac{8 \\sqrt{3} \\tan \\theta+2 \\sqrt{3}}{\\tan 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+68	"In the diagram, triangle ABC is right-angled at B. MT is the perpendicular bisector of $B C$ with $M$ on $B C$ and $T$ on $A C$. If $A T=A B$, what is the size of $\angle A C B$ ?
+
+<image_1>"	"['Since $M T$ is the perpendicular bisector of $B C$, then\n\n$B M=M C$, and $T M$ is perpendicular to $B C$.\n\nTherefore, $\\triangle C M T$ is similar to $\\triangle C B A$, since they share a common angle and each have a right angle.\n\n<img_3335>\n\nBut $\\frac{C M}{C B}=\\frac{1}{2}$ so $\\frac{C T}{C A}=\\frac{C M}{C B}=\\frac{1}{2}$, and thus $C T=A T=A B$, ie. $\\frac{A B}{A C}=\\frac{1}{2}$ or $\\sin (\\angle A C B)=\\frac{1}{2}$.\n\nTherefore, $\\angle A C B=30^{\\circ}$.'
+ 'Since $T M \\| A B$, and $C M=M B$, then $C T=T A=A B$.\n\nJoin $T$ to $B$.\n\nSince $\\angle A B C=90^{\\circ}$, then $A C$ is the diameter of a circle passing through $A, C$ and $B$, with $T$ as its centre.\n\n<img_3228>\n\nThus, $T A=A B=B T$ (all radii), and so $\\triangle A B T$ is equilateral. Therefore, $\\angle B A C=60^{\\circ}$, and so $\\angle A C B=30^{\\circ}$.']"	['$30^{\\circ}$']		['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+69	"In the diagram, $A B C D E F$ is a regular hexagon with a side
+length of 10 . If $X, Y$ and $Z$ are the midpoints of $A B, C D$ and $E F$, respectively, what is the length of $X Z$ ?
+
+<image_1>"	"['Extend $X A$ and $Z F$ to meet at point $T$.\n\nBy symmetry, $\\angle A X Z=\\angle F Z X=60^{\\circ}$ and $\\angle T A F=\\angle T F A=60^{\\circ}$, and so $\\triangle T A F$ and $\\triangle T X Z$ are both equilateral triangles.\n\nSince $A F=10$, then $T A=10$, which means\n\n$T X=10+5=15$, and so $X Z=T X=15$.\n\n<img_3285>'
+ 'We look at the quadrilateral $A X Z F$.\n\nSince $A B C D E F$ is a regular hexagon, then $\\angle F A X=\\angle A F Z=120^{\\circ}$.\n\nNote that $A F=10$, and also $A X=F Z=5$ since $X$ and $Z$ are midpoints of their respective sides.\n\n<img_3509>\n\nBy symmetry, $\\angle A X Z=\\angle F Z X=60^{\\circ}$, and so $A X Z F$ is a trapezoid.\n\nDrop perpendiculars from $A$ and $F$ to $P$ and $Q$, respectively, on $X Z$.\n\nBy symmetry again, $P X=Q Z$. Now, $P X=A X \\cos 60^{\\circ}=5\\left(\\frac{1}{2}\\right)=\\frac{5}{2}$.\n\nSince $A P Q F$ is a rectangle, then $P Q=10$.\n\nTherefore, $X Z=X P+P Q+Q Z=\\frac{5}{2}+10+\\frac{5}{2}=15$.']"	['15']		['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+70	"In the diagram, $A C=2 x, B C=2 x+1$ and $\angle A C B=30^{\circ}$. If the area of $\triangle A B C$ is 18 , what is the value of $x$ ?
+
+<image_1>"	"['Using a known formula for the area of a triangle, $A=\\frac{1}{2} a b \\sin C$,\n\n$$\n\\begin{aligned}\n18 & =\\frac{1}{2}(2 x+1)(2 x) \\sin 30^{\\circ} \\\\\n36 & =(2 x+1)(2 x)\\left(\\frac{1}{2}\\right) \\\\\n0 & =2 x^{2}+x-36 \\\\\n0 & =(2 x+9)(x-4)\n\\end{aligned}\n$$\n\nand so $x=4$ or $x=-\\frac{9}{2}$. Since $x$ is positive, then $x=4$.'
+ 'Draw a perpendicular from $A$ to $P$ on $B C$.\n\n<img_3390>\n\nUsing $\\triangle A P C, A P=A C \\sin 30^{\\circ}=2 x\\left(\\frac{1}{2}\\right)=x$.\n\nNow $A P$ is the height of $\\triangle A B C$, so Area $=\\frac{1}{2}(B C)(A P)$.\n\nThen\n\n$$\n\\begin{aligned}\n18 & =\\frac{1}{2}(2 x+1)(x) \\\\\n0 & =2 x^{2}+x-36 \\\\\n0 & =(2 x+9)(x-4)\n\\end{aligned}\n$$\n\nand so $x=4$ or $x=-\\frac{9}{2}$.\n\nSince $x$ is positive, then $x=4$.']"	['4']		['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Kxqqf3VUevc98DPQVv0UAcV4N8CT+FlktpNXa60+Kd5rO1EewRFu7HPzHHbpkk4zjHa0UUAFc74b8Lnw/e65cfbPP8A7UvnvNvl7PL3Enb1OevXiuiooAKKKKAOH8d+BL/xvFb2za3DZ2sEplRFsi7FiCASTIAcAnsK66zjvY4AL64gnl/vwwmIfkWb+dWqKACiiigAooooAKKKKACmlc06igBu3jFKR70tFABRRRQB/9k=']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+71	"A ladder, $A B$, is positioned so that its bottom sits on horizontal ground and its top rests against a vertical wall, as shown. In this initial position, the ladder makes an angle of $70^{\circ}$ with the horizontal. The bottom of the ladder is then pushed $0.5 \mathrm{~m}$ away from the wall, moving the ladder to position $A^{\prime} B^{\prime}$. In this new position, the ladder makes an angle of $55^{\circ}$ with the horizontal. Calculate, to the nearest centimetre, the distance that the ladder slides down the wall (that is, the length of $B B^{\prime}$ ).
+
+<image_1>"	['Let the length of the ladder be $L$.\n\nThen $A C=L \\cos 70^{\\circ}$ and $B C=L \\sin 70^{\\circ}$. Also, $A^{\\prime} C=L \\cos 55^{\\circ}$ and $B^{\\prime} C=L \\sin 55^{\\circ}$.\n\nSince $A^{\\prime} A=0.5$, then\n\n$$\n0.5=L \\cos 55^{\\circ}-L \\cos 70^{\\circ}\n$$\n\n$$\nL=\\frac{0.5}{\\cos 55^{\\circ}-\\cos 70^{\\circ}}\n$$\n\nTherefore,\n\n$$\n\\begin{aligned}\nB B^{\\prime} & =B C-B^{\\prime} C \\\\\n& =L \\sin 70^{\\circ}-L \\sin 55^{\\circ} \\\\\n& =L\\left(\\sin 70^{\\circ}-\\sin 55^{\\circ}\\right) \\\\\n& =\\frac{(0.5)\\left(\\sin 70^{\\circ}-\\sin 55^{\\circ}\\right)}{\\left(\\cos 55^{\\circ}-\\cos 70^{\\circ}\\right)} \\quad(\\text { from }(*)) \\\\\n& \\approx 0.2603 \\mathrm{~m}\n\\end{aligned}\n$$\n\n<img_3214>\n\nTherefore, to the nearest centimetre, the distance that the ladder slides down the wall is $26 \\mathrm{~cm}$.']	['26']		['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+72	"In the diagram, $P Q R S$ is an isosceles trapezoid with $P Q=7, P S=Q R=8$, and $S R=15$. Determine the length of the diagonal $P R$.
+
+<image_1>"	['Draw perpendiculars from $P$ and $Q$ to $X$ and $Y$, respectively, on $S R$.\n\n<img_3755>\n\nSince $P Q$ is parallel to $S R$ (because $P Q R S$ is a trapezoid) and $P X$ and $Q Y$ are perpendicular to $S R$, then $P Q Y X$ is a rectangle.\n\nThus, $X Y=P Q=7$ and $P X=Q Y$.\n\nSince $\\triangle P X S$ and $\\triangle Q Y R$ are right-angled with $P S=Q R$ and $P X=Q Y$, then these triangles are congruent, and so $S X=Y R$.\n\nSince $X Y=7$ and $S R=15$, then $S X+7+Y R=15$ or $2 \\times S X=8$ and so $S X=4$.\n\nBy the Pythagorean Theorem in $\\triangle P X S$,\n\n$$\nP X^{2}=P S^{2}-S X^{2}=8^{2}-4^{2}=64-16=48\n$$\n\nNow $P R$ is the hypotenuse of right-angled $\\triangle P X R$.\n\nSince $P R>0$, then by the Pythagorean Theorem,\n\n$$\nP R=\\sqrt{P X^{2}+X R^{2}}=\\sqrt{48+(7+4)^{2}}=\\sqrt{48+11^{2}}=\\sqrt{48+121}=\\sqrt{169}=13\n$$\n\nTherefore, $P R=13$.']	['13']		['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+73	"In the diagram, $\triangle A B C$ has $A B=A C$ and $\angle B A C<60^{\circ}$. Point $D$ is on $A C$ with $B C=B D$. Point $E$ is on $A B$ with $B E=E D$. If $\angle B A C=\theta$, determine $\angle B E D$ in terms of $\theta$.
+
+<image_1>"	['Since $A B=A C$, then $\\triangle A B C$ is isosceles and $\\angle A B C=\\angle A C B$. Note that $\\angle B A C=\\theta$.\n\n<img_3938>\n\nThe angles in $\\triangle A B C$ add to $180^{\\circ}$, so $\\angle A B C+\\angle A C B+\\angle B A C=180^{\\circ}$.\n\nThus, $2 \\angle A C B+\\theta=180^{\\circ}$ or $\\angle A B C=\\angle A C B=\\frac{1}{2}\\left(180^{\\circ}-\\theta\\right)=90^{\\circ}-\\frac{1}{2} \\theta$.\n\nNow $\\triangle B C D$ is isosceles as well with $B C=B D$ and so $\\angle C D B=\\angle D C B=90^{\\circ}-\\frac{1}{2} \\theta$.\n\nSince the angles in $\\triangle B C D$ add to $180^{\\circ}$, then\n\n$$\n\\angle C B D=180^{\\circ}-\\angle D C B-\\angle C D B=180^{\\circ}-\\left(90^{\\circ}-\\frac{1}{2} \\theta\\right)-\\left(90^{\\circ}-\\frac{1}{2} \\theta\\right)=\\theta\n$$\n\nNow $\\angle E B D=\\angle A B C-\\angle D B C=\\left(90^{\\circ}-\\frac{1}{2} \\theta\\right)-\\theta=90^{\\circ}-\\frac{3}{2} \\theta$.\n\nSince $B E=E D$, then $\\angle E D B=\\angle E B D=90^{\\circ}-\\frac{3}{2} \\theta$.\n\nTherefore, $\\angle B E D=180^{\\circ}-\\angle E B D-\\angle E D B=180^{\\circ}-\\left(90^{\\circ}-\\frac{3}{2} \\theta\\right)-\\left(90^{\\circ}-\\frac{3}{2} \\theta\\right)=3 \\theta$.']	['$3 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d6ncgmK2iaQqOrEDhR7k4A+tXY3Z41Zl2kgErnofSgB9FFFABXLeLPA1r4xiSDUdV1KO0Rg4trcxKm8ZG7JQtnk98e1dTRQByK+B7hIVgXxh4jWFVChVniBAAxjIjzV3w94K0bw1NNc2MUj3s5/fXdxIZJZPqx/piuhooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAZJEksbRyKrIwIZWGQQeorkoPh/b6VNK/h/WNS0aKVi721uyPBuPUhJFYA/SuwooA5CT4f2l/PFLrer6tqwjcOsNxOEhDA5B2RhQfxzXXYpaKACiiigD/2Q==']	Multimodal	Competition	False		Expression		Open-ended	Geometry	Math	English
+74	"In the diagram, the ferris wheel has a diameter of $18 \mathrm{~m}$ and rotates at a constant rate. When Kolapo rides the ferris wheel and is at its lowest point, he is $1 \mathrm{~m}$ above the ground. When Kolapo is at point $P$ that is $16 \mathrm{~m}$ above the ground and is rising, it takes him 4 seconds to reach the highest point, $T$. He continues to travel for another 8 seconds reaching point $Q$. Determine Kolapo's height above the ground when he reaches point $Q$.
+
+<image_1>"	"[""Let $O$ be the centre of the ferris wheel and $B$ the lowest point on the wheel.\n\nSince the radius of the ferris wheel is $9 \\mathrm{~m}$ (half of the diameter of $18 \\mathrm{~m}$ ) and $B$ is $1 \\mathrm{~m}$ above the ground, then $O$ is $9+1=10 \\mathrm{~m}$ above the ground.\n\nLet $\\angle T O P=\\theta$.\n\n<img_3230>\n\nSince the ferris wheel rotates at a constant speed, then in 8 seconds, the angle through which the wheel rotates is twice the angle through which it rotates in 4 seconds. In other words, $\\angle T O Q=2 \\theta$.\n\nDraw a perpendicular from $P$ to $R$ on $T B$ and from $Q$ to $G$ on $T B$.\n\nSince $P$ is $16 \\mathrm{~m}$ above the ground and $O$ is $10 \\mathrm{~m}$ above the ground, then $O R=6 \\mathrm{~m}$.\n\nSince $O P$ is a radius of the circle, then $O P=9 \\mathrm{~m}$.\n\nLooking at right-angled $\\triangle O R P$, we see that $\\cos \\theta=\\frac{O R}{O P}=\\frac{6}{9}=\\frac{2}{3}$.\n\nSince $\\cos \\theta=\\frac{2}{3}<\\frac{1}{\\sqrt{2}}=\\cos \\left(45^{\\circ}\\right)$, then $\\theta>45^{\\circ}$.\n\nThis means that $2 \\theta>90^{\\circ}$, which means that $Q$ is below the horizontal diameter through $O$ and so $G$ is below $O$.\n\nSince $\\angle T O Q=2 \\theta$, then $\\angle Q O G=180^{\\circ}-2 \\theta$.\n\nKolapo's height above the ground at $Q$ equals $1 \\mathrm{~m}$ plus the length of $B G$.\n\nNow $B G=O B-O G$. We know that $O B=9 \\mathrm{~m}$.\n\nAlso, considering right-angled $\\triangle Q O G$, we have\n\n$$\nO G=O Q \\cos (\\angle Q O G)=9 \\cos \\left(180^{\\circ}-2 \\theta\\right)=-9 \\cos (2 \\theta)=-9\\left(2 \\cos ^{2} \\theta-1\\right)\n$$\n\nSince $\\cos \\theta=\\frac{2}{3}$, then $O G=-9\\left(2\\left(\\frac{2}{3}\\right)^{2}-1\\right)=-9\\left(\\frac{8}{9}-1\\right)=1 \\mathrm{~m}$.\n\nTherefore, $B G=9-1=8 \\mathrm{~m}$ and so $Q$ is $1+8=9 \\mathrm{~m}$ above the ground.""]"	['9']		['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']	Multimodal	Competition	False	m	Numerical		Open-ended	Geometry	Math	English
+75	"On Saturday, Jimmy started painting his toy helicopter between 9:00 a.m. and 10:00 a.m. When he finished between 10:00 a.m. and 11:00 a.m. on the same morning, the hour hand was exactly where the minute hand had been when he started, and the minute hand was exactly where the hour hand had been when he started. Jimmy spent $t$ hours painting. Determine the value of $t$.
+
+<image_1>"	"['The hour hand and minute hand both turn at constant rates. Since the hour hand moves $\\frac{1}{12}$ of the way around the clock in 1 hour and the minute hand moves all of the way around the clock in 1 hour, then the minute hand turns 12 times as quickly as the hour hand.\n<img_3418>\n\nSuppose also that the hour hand moves through an angle of $x^{\\circ}$ between Before and After. Therefore, the minute hand moves through an angle of $\\left(360^{\\circ}-x^{\\circ}\\right)$ between Before and After, since these two angles add to $360^{\\circ}$.\n\n\n\nSince the minute hand moves 12 times as quickly as the hour hand, then $\\frac{360^{\\circ}-x^{\\circ}}{x^{\\circ}}=12$ or $360-x=12 x$ and so $13 x=360$, or $x=\\frac{360}{13}$.\n\nIn one hour, the hour hand moves through $\\frac{1}{12} \\times 360^{\\circ}=30^{\\circ}$.\n\nSince the hour hand is moving for $t$ hours, then we have $30^{\\circ} t=\\left(\\frac{360}{13}\\right)^{\\circ}$ and so $t=\\frac{360}{30(13)}=\\frac{12}{13}$.'
+ ""Suppose that Jimmy starts painting $x$ hours after 9:00 a.m. and finishes painting $y$ hours after 10:00 a.m., where $0<x<1$ and $0<y<1$.\n\nSince $t$ is the amount of time in hours that he spends painting, then $t=(1-x)+y$, because he paints for $(1-x)$ hours until 10:00 a.m., and then for $y$ hours until his finishing time. The hour hand and minute hand both turn at constant rates.\n\nThe minute hand turns $360^{\\circ}$ in one hour and the hour hand turns $\\frac{1}{12} \\times 360^{\\circ}=30^{\\circ}$ in one hour.\n\nThus, in $x$ hours, where $0<x<1$, the minute hand turns $(360 x)^{\\circ}$ and the hour hand turns $(30 x)^{\\circ}$.\n\nIn the Before picture, the minute hand is $(360x)^{\\circ}$ clockwise from the 12 o'clock position.\n\nIn the After picture, the minute hand is $(360 y)^{\\circ}$ clockwise from the 12 o'clock position. The 9 is $9 \\times 30^{\\circ}=270^{\\circ}$ clockwise from the 12 o'clock position and the 10 is $10 \\times 30^{\\circ}=300^{\\circ}$ clockwise from the 12 o'clock position.\n\nTherefore, in the Before picture, the hour hand is $270^{\\circ}+(30 x)^{\\circ}$ clockwise from the 12 o'clock position, and in the After picture, the hour hand is $300^{\\circ}+(30y)^{\\circ}$ clockwirse from the 12 o'clock position.\n\nBecause the hour and minute hands have switched places from the Before to the After positions, then we can equate the corresponding positions to obtain $(360 x)^{\\circ}=300^{\\circ}+(30 y)^{\\circ}$ $($ or $360 x=300+30 y)$ and $(360 y)^{\\circ}=270^{\\circ}+(30 x)^{\\circ}($ or $360 y=270+30 x)$.\n\nDividing both equations by 30 , we obtain $12 x=10+y$ and $12 y=9+x$.\n\nSubtracting the second equation from the first, we obtain $12 x-12 y=10+y-9-x$ or $-1=13 y-13 x$.\n\nTherefore, $y-x=-\\frac{1}{13}$ and so $t=(1-x)+y=1+y-x=1-\\frac{1}{13}=\\frac{12}{13}$.""]"	['$\\frac{12}{13}$']		['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MYFOpD1oA5ZPDl3/wseXxHNJCbVdOFnbxqTvU7txJGMfrXU4zmkpV7jsKADbQFxS0UAJjIwTRt5z3paKAEAxS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABSEZNLRQA3b7n3pSuaWigDAm8MifxxbeJHuyRbWTWsdt5fAJYkvuz6HGMfjW8VzjnkUtFADdgzn2xS7QOnGetLRQAgGKWiigAooooAKKKKACiiigAooooAKKKKACiiigD//Z']	Multimodal	Competition	False		Numerical		Open-ended	Algebra	Math	English
+76	"In the diagram, the circle with centre $C(1,1)$ passes through the point $O(0,0)$, intersects the $y$-axis at $A$, and intersects the $x$-axis at $B(2,0)$. Determine, with justification, the coordinates of $A$ and the area of the part of the circle that lies in the first quadrant.
+
+<image_1>"	"['Since $\\angle A O B=90^{\\circ}, A B$ is a diameter of the circle.\n\nJoin $A B$.\n\n<img_3988>\n\nSince $C$ is the centre of the circle and $A B$ is a diameter, then $C$ is the midpoint of $A B$, so $A$ has coordinates $(0,2)$.\n\nTherefore, the area of the part of the circle inside the first quadrant is equal to the area of $\\triangle A O B$ plus the area of the semi-circle above $A B$.\n\nThe radius of the circle is equal to the distance from $C$ to $B$, or $\\sqrt{(1-2)^{2}+(1-0)^{2}}=\\sqrt{2}$, so the area of the semi-circle is $\\frac{1}{2} \\pi(\\sqrt{2})^{2}=\\pi$.\n\nThe area of $\\triangle A O B$ is $\\frac{1}{2}(O B)(A O)=\\frac{1}{2}(2)(2)=2$.\n\nThus, the area of the part of the circle inside the first quadrant is $\\pi+2$.'
+ 'Since $\\angle A O B=90^{\\circ}, A B$ is a diameter of the circle.\n\nJoin $A B$.\n\n<img_3957>\n\nSince $C$ is the centre of the circle and $A B$ is a diameter, then $C$ is the midpoint of $A B$, so $A$ has coordinates $(0,2)$.\n\nThus, $A O=B O$.\n\nWe ""complete the square"" by adding point $D(2,2)$, which is on the circle, by symmetry.\n\n\n\n<img_3605>\n\nThe area of the square is 4 .\n\nThe radius of the circle is equal to the distance from $C$ to $B$, or $\\sqrt{(1-2)^{2}+(1-0)^{2}}=\\sqrt{2}$, so the area of the circle is $\\pi(\\sqrt{2})^{2}=2 \\pi$.\n\nThe area of the portion of the circle outside the square is thus $2 \\pi-4$. This area is divided into four equal sections (each of area $\\frac{1}{4}(2 \\pi-4)=\\frac{1}{2} \\pi-1$ ), two of which are the only portions of the circle outside the first quadrant.\n\nTherefore, the area of the part of the circle inside the the first quadrant is $2 \\pi-2\\left(\\frac{1}{2} \\pi-1\\right)=$ $\\pi+2$.\n\nTwo additional ways to find the coordinates of $A$ :\n\n$*$ The length of $O C$ is $\\sqrt{1^{2}+1^{2}}=\\sqrt{2}$.\n\nSince $C$ is the centre of the circle and $O$ lies on the circle, then the circle has radius $\\sqrt{2}$.\n\nSince the circle has centre $(1,1)$ and radius $\\sqrt{2}$, its equation is $(x-1)^{2}+(y-1)^{2}=2$. To find the coordinates of $A$, we substitute $x=0$ to obtain $(0-1)^{2}+(y-1)^{2}=2$ or $(y-1)^{2}=1$, and so $y=0$ or $y=2$.\n\nSince $y=0$ gives us the point $O$, then $y=2$ gives us $A$, ie. $A$ has coordinates $(0,2)$.\n\n* Since $O$ and $A$ are both on the circle and each has a horizontal distance of 1 from $C$, then their vertical distances from $C$ must be same, ie. must each be 1.\n\nThus, $A$ has coordinates $(0,2)$.']"	['$(0,2),\\pi+2$']		['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+77	"Survivors on a desert island find a piece of plywood $(A B C)$ in the shape of an equilateral triangle with sides of length $2 \mathrm{~m}$. To shelter their goat from the sun, they place edge $B C$ on the ground, lift corner $A$, and put in a vertical post $P A$ which is $h \mathrm{~m}$ long above ground. When the sun is directly overhead, the shaded region $(\triangle P B C)$ on the ground directly underneath the plywood is an isosceles triangle with largest angle $(\angle B P C)$ equal to $120^{\circ}$. Determine the value of $h$, to the nearest centimetre.
+
+<image_1>"	['From the given information, $P C=P B$.\n\nIf we can calculate the length of $P C$, we can calculate the value of $h$, since we already know the length of $A C$.\n\nNow $\\triangle C P B$ is isosceles with $P C=P B, B C=2$ and $\\angle B P C=120^{\\circ}$.\n\nSince $\\triangle C P B$ is isosceles, $\\angle P C B=\\angle P B C=30^{\\circ}$.\n\n<img_3969>\n\nJoin $P$ to the midpoint, $M$, of $B C$.\n\nThen $P M$ is perpendicular to $B C$, since $\\triangle P C B$ is isosceles.\n\n\n\nTherefore, $\\triangle P M C$ is right-angled, has $\\angle P C M=30^{\\circ}$ and has $C M=1$.\n\nThus, $P C=\\frac{2}{\\sqrt{3}}$.\n\n(There are many other techniques that we can use to calculate the length of $P C$.)\n\nReturning to $\\triangle A P C$, we see $A P^{2}=A C^{2}-P C^{2}$ or $h^{2}=2^{2}-\\left(\\frac{2}{\\sqrt{3}}\\right)^{2}=4-\\frac{4}{3}=\\frac{8}{3}$, and so $h=\\sqrt{\\frac{8}{3}}=2 \\sqrt{\\frac{2}{3}}=\\frac{2 \\sqrt{6}}{3} \\approx 1.630$.\n\nTherefore, the height is approximately $1.63 \\mathrm{~m}$ or $163 \\mathrm{~cm}$.']	['163']		['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+78	"Points $A_{1}, A_{2}, \ldots, A_{N}$ are equally spaced around the circumference of a circle and $N \geq 3$. Three of these points are selected at random and a triangle is formed using these points as its vertices.
+
+Through this solution, we will use the following facts:
+
+When an acute triangle is inscribed in a circle:
+
+- each of the three angles of the triangle is the angle inscribed in the major arc defined by the side of the triangle by which it is subtended,
+- each of the three arcs into which the circle is divided by the vertices of the triangles is less than half of the circumference of the circle, and
+- it contains the centre of the circle.
+
+Why are these facts true?
+
+- Consider a chord of a circle which is not a diameter.
+
+Then the angle subtended in the major arc of this circle is an acute angle and the angle subtended in the minor arc is an obtuse angle.
+
+Now consider an acute triangle inscribed in a circle.
+
+Since each angle of the triangle is acute, then each of the three angles is inscribed in the major arc defined by the side of the triangle by which it is subtended.
+
+- It follows that each arc of the circle that is outside the triangle must be a minor arc, thus less than the circumference of the circle.
+- Lastly, if the centre was outside the triangle, then we would be able to draw a diameter of the circle with the triangle entirely on one side of the diameter.
+
+<image_1>
+
+In this case, one of the arcs of the circle cut off by one of the sides of the triangle would have to be a major arc, which cannot happen, because of the above.
+
+Therefore, the centre is contained inside the triangle.
+If $N=7$, what is the probability that the triangle is acute? (A triangle is acute if each of its three interior angles is less than $90^{\circ}$.)"	"[""Since there are $N=7$ points from which the triangle's vertices can be chosen, there are $\\left(\\begin{array}{l}7 \\\\ 3\\end{array}\\right)=35$ triangles in total.\n\nWe compute the number of acute triangles.\n\nFix one of the vertices of such a triangle at $A_{1}$.\n\nWe construct the triangle by choosing the other two vertices in ascending subscript order. We choose the vertices by considering the arc length from the previous vertex - each of\n\n\n\nthese arc lengths must be smaller than half the total circumference of the circle.\n\nSince there are 7 equally spaced points on the circle, we assume the circumference is 7 , so the arc length formed by each side must be at most 3 .\n\nSince the first arc length is at most 3 , the second point can be only $A_{2}, A_{3}$ or $A_{4}$.\n\nIf the second point is $A_{2}$, then since the second and third arc lengths are each at most 3 , then the third point must be $A_{5}$. (Since the second arc length is at most 3, then the third point cannot be any further along than $A_{5}$. However, the arc length from $A_{5}$ around to $A_{1}$ is 3 , so it cannot be any closer than $A_{5}$.)\n\n<img_4024>\n\nIf the second point is $A_{3}$, the third point must be $A_{5}$ or $A_{6}$. If the second point is $A_{4}$, the third point must be $A_{5}$ or $A_{6}$ or $A_{7}$. Therefore, there are 6 acute triangles which include $A_{1}$ as one of its vertices.\n\nHow many acute triangles are there in total?\n\nWe can repeat the above process for each of the 6 other points, giving $7 \\times 6=42$ acute triangles.\n\nBut each triangle is counted three times here, as it has been counted once for each of its vertices.\n\nThus, there are $\\frac{7 \\times 6}{3}=14$ acute triangles.\n\nTherefore, the probability that a randomly chosen triangle is acute if $\\frac{14}{35}=\\frac{2}{5}$.""]"	['$\\frac{2}{5}$']		['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+79	"In the diagram, $\triangle P Q S$ is right-angled at $P$ and $\triangle Q R S$ is right-angled at $Q$. Also, $P Q=x, Q R=8, R S=x+8$, and $S P=x+3$ for some real number $x$. Determine all possible values of the perimeter of quadrilateral $P Q R S$.
+
+<image_1>"	['Since $\\triangle P Q S$ is right-angled at $P$, then by the Pythagorean Theorem,\n\n$$\nS Q^{2}=S P^{2}+P Q^{2}=(x+3)^{2}+x^{2}\n$$\n\nSince $\\triangle Q R S$ is right-angled at $Q$, then by the Pythagorean Theorem, we obtain\n\n$$\n\\begin{aligned}\nR S^{2} & =S Q^{2}+Q R^{2} \\\\\n(x+8)^{2} & =\\left((x+3)^{2}+x^{2}\\right)+8^{2} \\\\\nx^{2}+16 x+64 & =x^{2}+6 x+9+x^{2}+64 \\\\\n0 & =x^{2}-10 x+9 \\\\\n0 & =(x-1)(x-9)\n\\end{aligned}\n$$\n\nand so $x=1$ or $x=9$.\n\n(We can check that if $x=1, \\triangle P Q S$ has sides of lengths 4,1 and $\\sqrt{17}$ and $\\triangle Q R S$ has sides of lengths $\\sqrt{17}, 8$ and 9 , both of which are right-angled, and if $x=9, \\triangle P Q S$ has sides of lengths 12,9 and 15 and $\\triangle Q R S$ has sides of lengths 15,8 and 17 , both of which are right-angled.)\n\nIn terms of $x$, the perimeter of $P Q R S$ is $x+8+(x+8)+(x+3)=3 x+19$.\n\nThus, the possible perimeters of $P Q R S$ are 22 (when $x=1$ ) and 46 (when $x=9$ ).']	['22,46']		['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+80	"In the diagram, $\triangle A B D$ has $C$ on $B D$. Also, $B C=2, C D=1, \frac{A C}{A D}=\frac{3}{4}$, and $\cos (\angle A C D)=-\frac{3}{5}$. Determine the length of $A B$.
+
+<image_1>"	['Since $\\frac{A C}{A D}=\\frac{3}{4}$, then we let $A C=3 t$ and $A D=4 t$ for some real number $t>0$.\n\n<img_3309>\n\nUsing the cosine law in $\\triangle A C D$, the following equations are equivalent:\n\n$$\n\\begin{aligned}\nA D^{2} & =A C^{2}+C D^{2}-2 \\cdot A C \\cdot C D \\cdot \\cos (\\angle A C D) \\\\\n(4 t)^{2} & =(3 t)^{2}+1^{2}-2(3 t)(1)\\left(-\\frac{3}{5}\\right) \\\\\n16 t^{2} & =9 t^{2}+1+\\frac{18}{5} t \\\\\n80 t^{2} & =45 t^{2}+5+18 t \\\\\n35 t^{2}-18 t-5 & =0 \\\\\n(7 t-5)(5 t+1) & =0\n\\end{aligned}\n$$\n\nSince $t>0$, then $t=\\frac{5}{7}$.\n\nThus, $A C=3 t=\\frac{15}{7}$.\n\nUsing the cosine law in $\\triangle A C B$ and noting that\n\n$$\n\\cos (\\angle A C B)=\\cos \\left(180^{\\circ}-\\angle A C D\\right)=-\\cos (\\angle A C D)=\\frac{3}{5}\n$$\n\n\n\nthe following equations are equivalent:\n\n$$\n\\begin{aligned}\nA B^{2} & =A C^{2}+B C^{2}-2 \\cdot A C \\cdot B C \\cdot \\cos (\\angle A C B) \\\\\n& =\\left(\\frac{15}{7}\\right)^{2}+2^{2}-2\\left(\\frac{15}{7}\\right)(2)\\left(\\frac{3}{5}\\right) \\\\\n& =\\frac{225}{49}+4-\\frac{36}{7} \\\\\n& =\\frac{225}{49}+\\frac{196}{49}-\\frac{252}{49} \\\\\n& =\\frac{169}{49}\n\\end{aligned}\n$$\n\nSince $A B>0$, then $A B=\\frac{13}{7}$.']	['$\\frac{13}{7}$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAQ8BoQMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAEJwM0bhnHekJB4PevO/DXjxvFfxS1HTdOfOjabYMN4HE8zOnz5/ugbgvrknoRQB6NRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUh6UpOBmud8V+KINAt7e3ieF9Vv5Vt7G3kbAeRiAGbvsGck+2OpFAFXXZ5de1Q+GbKVo4QiyarOhwYoW6RA9nfH1C5Pdc8X8K4opfif8QJ4UVIba4jtIlUYCqrSKAPYCMfhXX2/wAO/DaW/manZRX142ZLq8uM75pO7nnjvgdhgDgVxfwCNtJD4ourZkCTaj8qA9EAJX8PmP5UAey0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAIRkVj3vhPw7qF213e6Dpd1ct96aezjdye2WIJrZpCeDQBSvdL0/UbQWd9Y21zbdoZoVdBgYGAQR0OKq6f4X8P6Zci50/QtLs5xkCW2tI42HY8gZ6cV5J8Q9Y13xx4j1Dw74TuJY7PQ7aSa/nhcr5s20jysjr3Xbxzv8A7tJ8L/jE5is9G8Wy7DINlnqcrcSY42Skn72f4j7Z9SAe6UUmRnFLQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAVwfxR8ZyeF/DyWum5k1vVGNtYRpywY8FwP9nIx7kds47HUNQtNN064vr2YQ2sEZklkboqgZJryn4e2F1478YXXxF1eFktkJt9Ft3H3IxkFz6kZI6/eLegoA7L4d+DYvBXhaKzbEl/OfPvZhyZJT79cDoPxPUmvNYPDGlWfxG1zwDrMA/sXWx/aWllflaCXnPl/3SBvX3CKO+D7vt/LHrXmXxn0e6OhWPirTONS0C4W5Qhc5jJG4EDsCFY+wagDG0jxLrPwn1OLw94xkkvPD7nbp+rqhbyx/dccn8Oq44yOnskM8VxCk0MiyRSKHR0O5WB5BBHBFYUH9jePPB1tJcW0d1puowLIYpOcZHTI6Mp4yDkEV5lFLrnwTvxb3Bn1XwPNJiOUjMliWPOcD1J9j1GDkEA9tzS1S03U7PV9Pg1CwuI7i1nTfHLGchh2/H1HUd6uZoAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApD0pegrkPiL40i8GeF5btNsmoz/ALmxhIyXlPQ46kDqfwHUigDjfiBe3Xj3xlafDzSZnjtIyLnWbiM/cQYIT68j23Mvoa9XsLC20yxgsrOFIbaCMRxxoOFUdB+Fcd8LfBknhPw80+o7n1vUn+038rnLbjkhM+2Tn3Zua7ugAqG7tob2zmtbmMSQTIY5EPRlIwR+VTUh6UAeTfCK6m8P6t4g+H1/KzvpUxnsi+Mvbsck4HQcq31kPpXqV1awXtrLa3UMc1vKpWSKRQyup4IIPUV5X8VIpfCnizw98QbVGMdtL9j1EJnLQt0OOh6sOe+3049YiljnhSWF1eN1DKynII7UAeM3+ja38HNSm1jw6s2o+Epn33umsxZrX1dT6D19OGzjcPU/DviTS/FGkRappNys9vJwcfeRu6sOxHp+PQ1qsoZSGAKkcj1ryDxD4M1jwDrMvivwDFvtn+bUNFCkxyAc5RR6ZJAHIOccHbQB7BkHvS1zXgvxrpPjfSFv9NmxImBPbOw8yBj2YDscHB7/AJiukyM4oAWiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKQ9KAI554oLeWaZxHFGpd3Y4CgDJJP05rx7wnBJ8UPiHP4xvY2/sDSJDBpMDj5ZJByZMfkx9yoydtXPihrF54h1mz+HGgyEXd8Q+pyx8/Z4OuGx0yOSDjI2j+KvSdD0ez0HR7PS9PiEdraxiOMd/cn3JyT6k0AaAGDmloooAKKKKAMjxNoUHiXw1qGjXW3y7uExhiM7G/hf6hsH8K4z4Ma/Nf+FZtB1EkapoM5s5kY5OwEhPywy/8A969JPSvINe/4oP42adrowml+I4/sl36LMMAN2A5CHPu9AHr/tRtoB5xS0AeW+Mvh3qFnrB8YeBHW01tPmuLJeIr0cZBGQAT3HRuvB5O74C+Idh4ztpIGjNlrNtlbvT5OHQggEgHnbk49QeD2rtGGRjpXnnjv4bHW7yPxF4euP7N8T2pDR3CcLPjor9s44zg8cHI6AHoYYGlrzvwD8R/7cvJPD3iG3/szxRa5WW2cYWfHVkz374z05GR09DBzQAtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFB6UAFc5438W2ng3wrdavcYZ1Gy3hJx5sp+6v09fQA10LsFRixCgDJJ7e9eM2aH4t/Ew6g+X8KeHpNtuP4bqfrn3HGf90L/AHjQB0Hwo8J3Wm6dceJNczJ4g1lvtEzyD5oo2+ZU9vUgYxwP4a9HA5oC4bNLQAUUUUAFFFFABXG/FDwsfFngS/s4lzeQL9ptcdfNTJAHuQWX/gVdlSHpQByPw08UDxb4HsNQkcNdxg293zyJUwCT7kYb/gVdfXkHhvPgT40ap4ebKaT4gU3llnhVlGSyjsP4x9Agr17NAC0UUUAcV49+HVh40tY7hZDZaza82l9DkOhHIViOSuefUHkdwcLwb8Qr6z1oeEPHaCz1xCFt7o4EV4D0II4BPbHBPHDcH1I1zXjLwTpXjfSDY6nEVkT5re5j4kgb1U+hwMjofqAQAdICDS1494d8aav4B1mDwl4+k3QSfLp2tHOyVegDk+nAJPIyM8HdXr6urgFSCCMgjoaAHUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABQelFY/ibxFY+F/Dt3rN++ILdMhQeZG7KvuTQBw/xY8SXjiz8EaA+7Wtb+RyDjyoDncSe2QGHHQBj1xntfCnhuz8J+HbPRrFR5dumGkxgyufvOfcnP06dq4j4T+H727kvPHniBc6xrJ3QoR/qLfjAGeRkAf8AAQvqa9RAxQAtFFFABRRRQAUUUUAFIRkYpaKAPNPjPoVxdeF7bxHp3y6poE63kLAZIQEbx+GFbnsh9a7bw3rlv4k8O2GsWn+pu4RIBnO0/wASn3ByPwrQuYIrq2lt541khlQo6MMhlIwQa8p+FE0vhfxL4h+H947n7HMbuwZs/PA2P8UP1ZqAPW6KTNLQAUhGQR60tFAGR4i8N6Z4o0eXTNWt1mtpOR2aNucMp7MMnn6joTXlmna3rnwe1GDR/ETS6h4Rlby7LUlUl7XJOEf2AHT05UkDbXtJGRiqeo6Zaatp81hqEEdxazqUkjkGQwP8j9KAJ7a7t721iurWZJreVQ0csbBlYEZBBHapQc14lLHrnwT1Az24n1bwRNJ80ROZbJm9PYnv0PQ4JyfX9I1ew13TLfUtNukubSYbo5U7/wCBHII6g8GgC/RSbhnHeloAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoooPSgBCRXjGplviz8Sk0WJt3hbw+/mXbg/Lcz8gKPyI+gc9xXR/FXxddaLpFvoei5k1/WW+z2qIfnRTwz+x5wD2PPY1v+BPCNt4L8L2ulQFXmA8y5mA/1spHzH6dAO+AKAOkRAgCqAFUY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+81	"Suppose that $a>\frac{1}{2}$ and that the parabola with equation $y=a x^{2}+2$ has vertex $V$. The parabola intersects the line with equation $y=-x+4 a$ at points $B$ and $C$, as shown. If the area of $\triangle V B C$ is $\frac{72}{5}$, determine the value of $a$.
+
+<image_1>"	['The parabola with equation $y=a x^{2}+2$ is symmetric about the $y$-axis.\n\nThus, its vertex occurs when $x=0$ (which gives $y=a \\cdot 0^{2}+2=2$ ) and so $V$ has coordinates $(0,2)$.\n\nTo find the coordinates of $B$ and $C$, we use the equations of the parabola and line to obtain\n\n$$\n\\begin{aligned}\na x^{2}+2 & =-x+4 a \\\\\na x^{2}+x+(2-4 a) & =0\n\\end{aligned}\n$$\n\nUsing the quadratic formula,\n\n$$\nx=\\frac{-1 \\pm \\sqrt{1^{2}-4 a(2-4 a)}}{2 a}=\\frac{-1 \\pm \\sqrt{1-8 a+16 a^{2}}}{2 a}\n$$\n\nSince $1-8 a+16 a^{2}=(4 a-1)^{2}$ and $4 a-1>0\\left(\\right.$ since $\\left.a>\\frac{1}{2}\\right)$, then $\\sqrt{1-8 a+16 a^{2}}=4 a-1$ and so\n\n$$\nx=\\frac{-1 \\pm(4 a-1)}{2 a}\n$$\n\nwhich means that $x=\\frac{4 a-2}{2 a}=\\frac{2 a-1}{a}=2-\\frac{1}{a}$ or $x=\\frac{-4 a}{2 a}=-2$.\n\nWe can use the equation of the line to obtain the $y$-coordinates of $B$ and $C$.\n\nWhen $x=-2$ (corresponding to point $B$ ), we obtain $y=-(-2)+4 a=4 a+2$.\n\nWhen $x=2-\\frac{1}{a}$ (corresponding to point $C$ ), we obtain $y=-\\left(2-\\frac{1}{a}\\right)+4 a=4 a-2+\\frac{1}{a}$.\n\nLet $P$ and $Q$ be the points on the horizontal line through $V$ so that $B P$ and $C Q$ are perpendicular to $P Q$.\n\n<img_3382>\n\n\n\nThen the area of $\\triangle V B C$ is equal to the area of trapezoid $P B C Q$ minus the areas of right-angled $\\triangle B P V$ and right-angled $\\triangle C Q V$.\n\nSince $B$ has coordinates $(-2,4 a+2), P$ has coordinates $(-2,2), V$ has coordiantes $(0,2)$, $Q$ has coordinates $\\left(2-\\frac{1}{a}, 2\\right)$, and $C$ has coordinates $\\left(2-\\frac{1}{a}, 4 a-2+\\frac{1}{a}\\right)$, then\n\n$$\n\\begin{aligned}\nB P & =(4 a+2)-2=4 a \\\\\nC Q & =\\left(4 a-2+\\frac{1}{a}\\right)-2=4 a-4+\\frac{1}{a} \\\\\nP V & =0-(-2)=2 \\\\\nQ V & =2-\\frac{1}{a}-0=2-\\frac{1}{a} \\\\\nP Q & =P V+Q V=2+2-\\frac{1}{a}=4-\\frac{1}{a}\n\\end{aligned}\n$$\n\nTherefore, the area of trapezoid $P B C Q$ is\n\n$$\n\\frac{1}{2}(B P+C Q)(P Q)=\\frac{1}{2}\\left(4 a+4 a-4+\\frac{1}{a}\\right)\\left(4-\\frac{1}{a}\\right)=\\left(4 a-2+\\frac{1}{2 a}\\right)\\left(4-\\frac{1}{a}\\right)\n$$\n\nAlso, the area of $\\triangle B P V$ is $\\frac{1}{2} \\cdot B P \\cdot P V=\\frac{1}{2}(4 a)(2)=4 a$.\n\nFurthermore, the area of $\\triangle C Q V$ is\n\n$$\n\\frac{1}{2} \\cdot C Q \\cdot Q V=\\frac{1}{2}\\left(4 a-4+\\frac{1}{a}\\right)\\left(2-\\frac{1}{a}\\right)=\\left(2 a-2+\\frac{1}{2 a}\\right)\\left(2-\\frac{1}{a}\\right)\n$$\n\nFrom the given information,\n\n$$\n\\left(4 a-2+\\frac{1}{2 a}\\right)\\left(4-\\frac{1}{a}\\right)-4 a-\\left(2 a-2+\\frac{1}{2 a}\\right)\\left(2-\\frac{1}{a}\\right)=\\frac{72}{5}\n$$\n\nMultiplying both sides by $2 a^{2}$, which we distribute through the factors on the left side as $2 a \\cdot a$, we obtain\n\n$$\n\\left(8 a^{2}-4 a+1\\right)(4 a-1)-8 a^{3}-\\left(4 a^{2}-4 a+1\\right)(2 a-1)=\\frac{144}{5} a^{2}\n$$\n\nMultiplying both sides by 5 , we obtain\n\n$$\n5\\left(8 a^{2}-4 a+1\\right)(4 a-1)-40 a^{3}-5\\left(4 a^{2}-4 a+1\\right)(2 a-1)=144 a^{2}\n$$\n\nExpanding and simplifying, we obtain\n\n$$\n\\begin{aligned}\n\\left(160 a^{3}-120 a^{2}+40 a-5\\right)-40 a^{3}-\\left(40 a^{3}-60 a^{2}+30 a-5\\right) & =144 a^{2} \\\\\n80 a^{3}-204 a^{2}+10 a & =0 \\\\\n2 a\\left(40 a^{2}-102 a+5\\right) & =0 \\\\\n2 a(20 a-1)(2 a-5) & =0\n\\end{aligned}\n$$\n\nand so $a=0$ or $a=\\frac{1}{20}$ or $a=\\frac{5}{2}$. Since $a>\\frac{1}{2}$, then $a=\\frac{5}{2}$.']	['$\\frac{5}{2}$']		['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+82	"Suppose that $m$ and $n$ are positive integers with $m \geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is
+
+<image_1>
+
+The $(3,4)$-sawtooth sequence includes 17 terms and the average of these terms is $\frac{33}{17}$.
+Determine the sum of the terms in the $(4,2)$-sawtooth sequence."	['The $(4,2)$-sawtooth sequence consists of the terms\n\n$$\n1, \\quad 2,3,4,3,2,1, \\quad 2,3,4,3,2,1\n$$\n\nwhose sum is 31 .']	['31']		['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KACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAP//Z']	Multimodal	Competition	False		Numerical		Open-ended	Number Theory	Math	English
+83	"Suppose that $m$ and $n$ are positive integers with $m \geq 2$. The $(m, n)$-sawtooth sequence is a sequence of consecutive integers that starts with 1 and has $n$ teeth, where each tooth starts with 2, goes up to $m$ and back down to 1 . For example, the $(3,4)$-sawtooth sequence is
+
+<image_1>
+
+The $(3,4)$-sawtooth sequence includes 17 terms and the average of these terms is $\frac{33}{17}$.
+For each positive integer $m \geq 2$, determine a simplified expression for the sum of the terms in the $(m, 3)$-sawtooth sequence."	"['Suppose that $m \\geq 2$.\n\nThe $(m, 3)$-sawtooth sequence consists of an initial 1 followed by 3 teeth, each of which goes from 2 to $m$ to 1 .\n\nConsider one of these teeth whose terms are\n\n$$\n2,3,4, \\ldots, m-1, m, m-1, m-2, m-3, \\ldots, 2,1\n$$\n\nWhen we write the ascending portion directly above the descending portion, we obtain\n\n$$\n\\begin{aligned}\n& 2, \\quad 3, \\quad 4, \\quad \\ldots, \\quad m-1, \\quad m \\\\\n& m-1, \\quad m-2, \\quad m-3, \\quad \\ldots, \\quad 2, \\quad 1\n\\end{aligned}\n$$\n\nFrom this presentation, we can see $m-1$ pairs of terms, the sum of each of which is $m+1$. $($ Note that $2+(m-1)=3+(m-2)=4+(m-3)=\\cdots=(m-1)+2=m+1$ and as we move from left to right, the terms on the top increase by 1 at each step and the terms on the bottom decrease by 1 at each step, so their sum is indeed constant.) Therefore, the sum of the numbers in one of the teeth is $(m-1)(m+1)=m^{2}-1$. This means that the sum of the terms in the $(m, 3)$-sawtooth sequence is $1+3\\left(m^{2}-1\\right)$, which equals $3 m^{2}-2$.'
+ 'Suppose that $m \\geq 2$.\n\nThe $(m, 3)$-sawtooth sequence consists of an initial 1 followed by 3 teeth, each of which goes from 2 to $m$ to 1 .\n\nConsider one of these teeth whose terms are\n\n$$\n2,3,4, \\ldots, m-1, m, m-1, m-2, m-3, \\ldots, 2,1\n$$\n\nThis tooth includes one 1 , two $2 \\mathrm{~s}$, two $3 \\mathrm{~s}$, and so on, until we reach two $(m-1) \\mathrm{s}$, and one $m$.\n\nThe sum of these numbers is\n\n$$\n1(1)+2(2)+2(3)+\\cdots+2(m-1)+m\n$$\n\nwhich can be rewritten as\n\n$$\n2(1+2+3+\\cdots+(m-1)+m)-1-m=2 \\cdot \\frac{1}{2} m(m+1)-m-1=m^{2}+m-m-1=m^{2}-1\n$$\n\nTherefore, the sum of the numbers in one of the teeth is $(m-1)(m+1)=m^{2}-1$.\n\nThis means that the sum of the terms in the $(m, 3)$-sawtooth sequence is $1+3\\left(m^{2}-1\\right)$, which equals $3 m^{2}-2$.']"	['$3 m^{2}-2$']		['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Theory	Math	English
+84	"At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected
+
+<image_1>
+uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping.
+For a 2-topping pizza, determine the probability that at least $\frac{1}{4}$ of the pizza is covered by both toppings."	['Assume that the first topping is placed on the top half of the pizza. (We can rotate the pizza so that this is the case.)\n\nAssume that the second topping is placed on the half of the pizza that is above the horizontal diameter that makes an angle of $\\theta$ clockwise with the horizontal as shown. In other words, the topping covers the pizza from $\\theta$ to $\\theta+180^{\\circ}$.\n<img_3295>\n\nWe may assume that $0^{\\circ} \\leq \\theta \\leq 360^{\\circ}$.\n\nWhen $0^{\\circ} \\leq \\theta \\leq 90^{\\circ}$, the angle of the sector covered by both toppings is at least $90^{\\circ}$ (and so is at least a quarter of the circle).\n\nWhen $90^{\\circ}<\\theta \\leq 180^{\\circ}$, the angle of the sector covered by both toppings is less than $90^{\\circ}$ (and so is less than a quarter of the circle).\n\nWhen $\\theta$ moves past $180^{\\circ}$, the left-hand portion of the upper half circle starts to be covered with both toppings again. When $180^{\\circ} \\leq \\theta<270^{\\circ}$, the angle of the sector covered by both toppings is less than $90^{\\circ}$ (and so is less than a quarter of the circle).\n\nWhen $270^{\\circ} \\leq \\theta \\leq 360^{\\circ}$, the angle of the sector covered by both toppings at least $90^{\\circ}$ (and so is at least a quarter of the circle).\n\nTherefore, if $\\theta$ is chosen randomly between $0^{\\circ}$ and $360^{\\circ}$, the combined length of the intervals in which at least $\\frac{1}{4}$ of the pizza is covered with both toppings is $180^{\\circ}$.\n\nTherefore, the probability is $\\frac{180^{\\circ}}{360^{\\circ}}$, or $\\frac{1}{2}$.']	['$\\frac{1}{2}$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIASACDQMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAoopjSBEZ3Kqq5JJOAKAH0VwmvfGDwVoBeOTVkvZ1/5Y2C+cT7bh8gPsWrBPxM8aa8GHhX4f3ZiYZjutSfy0YeuDtB/BzQB6xmjNeUf2P8AGPVyrXXiHR9FhYfNFaw+Y6/mp/RqVPhJrt25bWPiPr1znqkDNEB+BZh+lAHq2fXFNMsYODIoPpmvK2+AXhu4dXvdY1+7IOSJrpCCf++M/rT/APhn3wT/ANRP/wACR/8AE0Aepk0Z4zXlLfs++EFZZLe91m1dc/NFcpn9UNEnwZuraM/2R498SWRxxvuC4/JStAHq2fajNeUt4R+K2lFTpXju01CNTkpqFttz+OHP603/AITD4p6DuOs+CLfVIVOBLpkpDN77QXJ/75FAHrNFeaaX8cPCl1P9l1QXuiXYIUx30BA3fVc4H+9ivQLDU7LVbYXOn3dvd25OBLbyiRT+INAFuim7s9MU6gAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACignAJpnmDaWzwOpPGPr6UAPJwCaz9W1rTdCsXvdUvYLS2XjzJnCgn0HqeOg5rzzxB8WnutQfQfAVh/b2r4O6dObeIdMlsgN9cheRyelcz8OfDFj8STN4i8YandatqNrOYX02X91HbEHIBUYyD1wABkMCGwaAN+4+LOreJbmWw+H3hufUip2vqF2vlwJ+BI6jpuZT7GmRfCnXvFEi3Hj7xVc3i7t39n2LbIV9BnAH5KD7mvVrOytdPtI7Szt4re2jG2OKJAiqPQAVNgUAc9oHgXwx4ZC/2To1rBIox5xXfL/wB9tlvwzXQ4FLRQAYpMClooAKK4HS9S+IWtJPdwnw5aWRnkW2We3meR41cqGbEgAzjNYHgvxv488Zz6xDbnw5bNpk4hcyWs5DklhkYl4+6aAPXaTvXK+FNV8TXGqatpviW305JrTyngksFcJNG+/wCb5mJ6oRj2PWsLxP4717R/Hfh3Ql0uC3sdTuhGbmSTe7qGUMAoOEPzDqT16UAej4ox168+9LRQBnaroWk63bmHVNNtb2PH3Z4lfH0JHFeeah8E9Ntro6h4Q1a/8PXw6eTK0kZ9iCd3P+9j2r1Q8jFIVBoA8g/4S/4h+AwB4u0aPXNKQAtqWnffQDqWGAOPcKP9o13vhfx14f8AGEG/SL9JJlGXtpBslT6qece4yPeuiCgHNefeJ/hHoGuzfb9O3aHq6ndHe2HyYf1ZRgE+4wfegD0AE06vHrfxv4t+Hdwll49sG1DS2YJFrdmu7HoHGACfrtPBxur1LStYsNcsIr/S7uG7tJOVlibIPt7HsQcEUAX6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKDwDXOeMPGmleCtHbUNTk+ZsrBAv35nxnaP0ye38wC/r2v6d4b0qXUtVuo7e1j/iY8scH5VA5J4PA9K8nVfFXxmnJY3GgeCt/wAo4E96vv6/+gjP8ZGateH/AAXq/wAQNXh8U+P0KWyEmw0TaQiL6yKfXgkHk4GcAba9fWNUVVQBVXgADgUAZfh7wzpHhbTFsNHs47aEcuV5aRv7zN1Y+5/lXmXieOT4ZfE628WwKRoOtt9n1RFztilJz5mOnbd3PEg/iFeyYzWR4l8PWfifw7e6PfD9zdRld2MlG6qw9wcH8KANRJBIodGDIwBBHII65z6U+vMPhJ4gu4o73wRrrbdZ0NvLTd/y1g/hIzyQOOfQrXp9ABRRRQAUUUHgUANVFUAKMAdAOleM/AX/AJCXjX/r/X+clez5PpXi3wFYHU/Gm1lYG+Q5H+9LQB7P5abi20biACcckDOM/ma8o+J/HxN+Gv8A1/S/+hw160eleSfFAj/hZ3w2/wCv6T/0KGgD1uikzzS0AFFFFABSYpaD0oAgubS3u7aS3uYI54JFKvFIgZWB6gg8EV4t4v8ADdz8JjL4s8H6otpZGVFudJumLRTEnGE5yfp1AyQQOK9pubqG0tZbm5lSKCJC8jucBFAySa8b0iGf4xeMhr1/EyeENJlK2Ns4/wCPqUYyzD06EjpjC8/MaAO58CfEfSfHVm32c/ZdSiXNxYyt88fYkHjcueMgfXGcV2OTxXnvjr4bJrtzFr3h+f8AsnxNa4aK6jO1ZccAPjrxxn0ODkcUeAPiK+u3E3h/xDbf2d4msyVlt34EwHG5OevqPxGQaAPQ6KaGyR7ntTqACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiqeo6paaRplxqOoTJBa28ZklkboFH8/YdyRQBmeLfFmn+DvD8+rak3yL8sUSnDTOeiL79/YAntXklnbzrf2fxE8eaff391e3S22j6PaxB2gBDMhKMR2U4HvuOSeNLwrpd18VfFf/Caa9C66DZSGPSNPkA2vg8uw6HBHOM5IxnC4Pr9xYwXNxazyqWktXMkRz0YqyZ/Jm/OgDhNb+LNj4dtIrnWPDHiOzgkfYjTW8OC3XH+tznirGqfE+PRbJrzUfCPie3tkwHme1iKp7kiXgVyf7R/Hg3SiP8AoID/ANFvXr17YW+o6fcWNym+3uY2ilXP3lYYP86AGajqlrpVm93eS+XEmc4Usx74CjJJxngA1i+DfG+n+OLO9u9MhnW3trjyA0wClztDZwM4HPfmumCgAAdhivJfgMqro3iNQMKusSgD/gK0ASfFXSrvQdS074iaLGTeaYwS+jUf6+AnHzfTJB9Ac/w16No2s2mvaRaapYSCS1uoxJG3cZ6g+hByCPUEdqt3NtFd2stvPGssMqNHJG4yrqRggj0NeSeAbmbwB48vvh9fyu2n3TG70aV+6nO5M/gfTlWP8QoA9hopM9KWgAooooAq31hDqNs1vO1wsbdTBcPC3/fSEH9awND+HXhjw3eNdaPZ3NpK5DSGO/uMSEHI3Avhup6g9TXU0UABGRiuY1T4feHNa1OHUdRtrue8gfzIZTqFwPJbOcoA+F59MV09FAEVvbpbQJDGZCqjAMkjO34sxJP51LRRQAUUUUAFNycc4px6V5l8TfGN7bzW3g7wzuk8R6p+7zG2Ps0Z6uT2JGeeoALemQDI8W6lefE3xb/wg2gzMmjWbh9ZvUPBwf8AVqfYjHfLeynPq+l6XZ6Npltp1hAsFrbII4o06Af1Pck9TyaxvA3g6x8E+GoNLtPnl4e5uCMNNJ3PsPQdh+JPS4oAK4f4hfD+HxfaRXtk/wBj1+x+eyvV4OQchGPUjPIPY89yD3NJtFAHn3w78eT+IGn0HX4DZ+JtO+W4hYYEyj+Nf0zjjkEcHj0DOa85+Jfga51dYPEvhxmtvE2mnzIXiABuFH8DepxnGeOSpznjZ+Hvja28b6At2qiC/tz5V7bYwYpPUDrtPb8R1FAHX0UUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRQelABXjPiqef4pePF8F2E0ieH9KkEur3EZ4kcHAjB9Qcge4Y4OwV1nxR8YzeFfC/l6fl9Z1J/stjGoy29urge2Rj/aK1b+HPg6LwX4Tt7F9r38p8+8lzndKeuD6D7o9cZ70AdRa2dvY2sNrawpDBCgjjjQYVFAwAB6VOelFFAHi/7SGf8AhDdKz/0EP/ab17ODXn3xD+HeoeP4re1l1+CxsreUypGlgXcsRj5mMoBxz0A612mmQahbWSR6heW93MoA82GAwg49QXfn8aAL1eS/AjP9keJOP+YzL/6Cteo3ovHtmFlPDBN2knhMqj6qGU/rXGeAfAeo+BvtcJ12C/tryc3EqtYmN95GDtbzSMcDtQB3lef/ABW8Hy+JPDkd9pgddb0h/tVi8X3yRglAO5O0Y91H0PoFJgYxQBzHgHxdD408J2mqpsWf/VXUa9I5l+8Pocgj2YV1FeM3GPhZ8Vxdr+78MeJX2y9kt7jOc+wyc+m1m/uV7IGzQA6iiigAooooAKKKKACiiigAo7UHgZrO1rW7HQNGutV1GYQ2luhd37+gA9SSQAO5I9aAMPx944tvBPh1rxgk19PmOytsnM0n0HO0ZyT9B1IrH+GXgm50eGfxJ4hY3HiXVT5lxJIOYEOCIx6ds44HCjhaw/Ami33jvxR/wsTxHb7LcNs0ayfkRoDxIR35yQe5y3Za9gCgHNAABjuaWiigAooooAQr3rxzx1p118OvGEPxC0SAtYXDiHWbRBgMGP8ArPQEn8mAPO417JVW/wBPtdS0+4sbyFZra4jMcsb8hlIwRQAmn6jb6pYW99ZyrNa3EYlikX+JTyKt15D8N7y58FeML/4c6pK7QZa60eaT/lpEckr9cAngcFZPavXQSaAFooooAKKKD0oAKKTP056Um7jNADqKaWwD04pd1AC0UxJUkzsdW2nBwc4PpT6ACiiigAooo6DNABRTd3OOKCTmgB1FIDnnjGcUtABRRRQAUUUUAFFFU7HVLHU0kewvLe7SOQxSNBKHCMOqnGcEehoAuUUUUAFFFFABRRSbsdaAFoqNZkaQxhgXUAkA8gHODjsODUlABSZpa4D4v+KT4Y+H948Emy9vf9EtyCQQWB3MCOmFDEHscUAc54WX/hY3xWvvFcpMmiaGfsmmA8rJJjlx+e7pn5k9K9hCgdK8/wDhLd+G4fBOnaXomp2tzPHEJLlUbbJ5rcuSpw2M8AkdAK7/ACc9vwoAdRRRQAYpPelooA5C4+JfhS31O7059Qna7tJDHcRw2M8pjYdiVQjt24qvD8WPBlxNLDBqVzJLEcSRppt0Shz0OI8jmun0rSbfSILiO3UA3FzLdStgZZ5HLEn6AgfQCvJ/hJ/yVP4jjt9vf/0dLQB6R4e8aaD4pmu4dGvTcS2m3z0aCSIpuzjh1H901iat8YPBmi6rJp13qbNNE5SVoIWkSJu4LAdR3Aziumk0K1bU73UU3xT3dsttM0R2sQpYq2f7w3HB+npWZ4isfD+l+BL6zvba2i0iK2ZfI24XODgKOu4npjnJz1oAj8T6LpnxF8ETWkM8U1vdx+baXSHcEkH3WHtnII64yOprF+Enim51XRJtA1jKa7ojm2uEc5Z0U7Vf36FSec4z/EK0PhX4dvvC/gCw07UyRdsXmeIn/VbyTs/Lr75rlfiTbXPgvxdp3xH0qJjFlbXV4FGPNiPAY++ABz0Ij96APYKKrWF9BqVhb3trKstvcRrJHIvRlIyDVmgAooooAKKKKACiig9DQBG8qxoWkZUUAksxwAOv8q8XlMnxq8arDGXHgnRZsuwyovJh2HrwfqFJ6FxV74g69feLvECfDnwxNh5RnV7xeVgizymfX+9jqSq55IHpPh/QrHw3olppOnR7La3TaCfvOe7N6knJJ9TQBoxQxQxpHEipGgCqijAUAYAA7cVJRRQAUUUUAFFFJmgBaD0NN3HOOM0BsnFAHm3xi8P3F3oNv4m0r5NX0CUXUTqOTGOXHvjAb6AjvXYeFPEEHinwxp+tW3CXUQZkBzsfo6/gwIrYeJJI2R1DKwwynkEehryX4ak+D/H/AIi8BTMfspf+0NML90bG5Qe/G3p3RjQB67RRRQAUHoaKKAOT8U+C7HxdrOkNq0X2jTrJZneAsQJJDsCZxyQAHOPp+PlHxa8H+HtF8UeCLbTdJt7aG9vXS5RAcSrvhGD+DH86+gsV4x8bv+Rx+Hn/AF/v/wCjIKAOo1T4T+GW+yXekaVDY6hZ3UNzDJESoOx1Yqwzg5ANdL4m01tS0C7hF9eWhELkPazeW2dhx8wGRz2GPetrHNVdTH/EpvP+uD/+gmgDzv4CsX+GULsSWN3MSSc5Oa9Pry/4BD/i18H/AF9TfzFeoHpQAgalrhfF9r4Wn1eNta8J32rXXkKFng02S4CpubC7lBAOcnHvWAdO+H2P+Sdav/4Ip/8ACgD1YNnrgVm+JdOm1fwxqumwMFlu7SW3Vj2LoVz+tcz4QtfC8GtM2jeFL/SrryWBuLjTZIFZcrkbmGMnj8jXd0AeU+L/AIZ+D9D+HWsS2WiwfabWwkaO4clpAwUkMST1qh8K/h/4U1/4W6Xe6polvc3U4mEkp3BmxM4HIIxwAPwrvPiOP+La+I/+wfL/AOgmsj4Jf8kh0P8A7b/+j5KANzwrpMPg/wAJx6dNcIttZyTlZZZBxEZXZNzeysAT7Vp6drulavu/szU7K9KffFtcJLt+u0nFedXV8vin48Dw9eKsum6HZfahbOMpJO2zDEdDgSDGehBNQ/GjTf7E0ux8ZaKFs9W065RWliG3zI2yNrY+8M7Rz2JFAHqdzf2tlGJLq5hgjZgivLIFBYnAGSepqu+u6VFqKadLqdkl++Atq06CVvoucmvG/izDa6raeB/ENuhgvNQu4CJsklVdVYDByOMA9P5mtj4ifCjw/wD8ITqeoafbPFq1nE939reVnkmKjc+8k/NkA/Q4xgcUAerzXUVrC81zLHDEi7nkkYKqj1JPSqVz4h0exMAvNWsLf7QoaHzblE8wHuuTyD7V4j4muR4o/Zvste1LdNqVvsjWZnOcrP5Rbrgkr1PU/hXYw/Cnw94h8EWz3sTy6rc2MbLfySHfG3ljYBjgKvA2gYwPXmgD00OT6f41z3hTQvDegw3sHhtYFRrkvcrFP5m2UgcHk7cDHy9s1w3wTuZPEPw9vdD1pDcRWF01qFcnmLAIUnqcHI+mB0qL4D2UEnhPxDZOm63/ALVkj2EnldiDGetAHqo1awaS5jW+tS9rzcqJlJhznG8Z+XOD19KLDVrDVYPtGnX1reQ5x5lvMsi5+oNeMfDDwppF94u8eWF1a+dp1lqYSKyc5hOHl27l/j2gYAbIGc4zzWj4P0+38O/H7xHoulp9m02XTUuPsyH5A+YuQO333/76+lAHrN7qNppsBuL+6gtbcEDzZ5Ai5PbJOKbYarYarCZtPvba7iB2l7eVZFz6ZBrzJbp7v9o1rTVube20otpSP03ttLMnq2BKM9cJ7VleLPDem3/7Qei2z2EVxDfWLS38IyASokAkOMYPCjPfHvQB7bmoLu1+1WzwefNDvABkhba4Hse31HNSxQpBEkUahY0AVVHQAdKfQB498HLdrTxx8RLVp5p/IvooxLPIXkYK04G5jyTgV7DXkvwp/wCSj/Ez/sJR/wDoc9etUAB6V5DrSDxr8edN0hl8zTfDlubu4Vh8pmbBA9DyYuP9lq9annjt4JJpXCRxoXZj2AGSa8s+CcEmoWviDxbcoyz61qLsu45xGpJAH4uw/wCAigDW8Q/BvwnrcourW2k0i+U7luNOYRYI6Ep93rzwAfesD7J8VvA//Htc2/i/S052TfLcgDryTkn8X+lewUmBQB5xonxo8OX119g1lLjw/qSnbJb6gm1VbrjfwAPdgtehQXMNzCs0EiSxMMq8bBlYeoIrO1zwvofiS1+z6zplteRgEKZE+ZB/ssMMv4EV53P8ItS8PTvd+AfE93pblixs7o+ZbufyP/jysfegD1rNLXkK/E/xX4SIi8e+FJhbg4/tPTfniPoSMlRn/eB/2a77w9438OeKYwdH1e2uXxkw7tsoHqUOGA98YoA39uB1NeMfCP8A5Kp8R/8Ar/f/ANHy17Bdy3KW7G0hhln/AIUmlMan6sFYj8jXmfgfwV4t8LeMNd1m8GizxazM0sqQ3UoMRMjP8uYufvHrjtyKAPUyOK5Xxh8P9E8bWpj1RbjzVH7qWO4dfKPqFyV/St/UxfPpd0umyww3xhYQSTKWRXwcEj0ziuR0eb4k/ZDa6nZeH/tH3RfC4kwB6mIL8x9ty/hQBh/BOXVLXT9e8OajKbhNF1BrWGQ9McgqP9kFcj03Yr0vUdNtdV0250+8iEtrcxtHKh/iUjBrP8MeHIPDWmvbRyvcXFxM9zd3UgAeeZzlnIHHoMegFbZGQRQB5H8L9RufCviLUPhxrEjM1szXGlTP/wAtoTyQP1bHrvH8NetgnvXmvxc8NXN3plt4q0XKa3oT/aI2UcyRDll98AZx6bh3rrfCPiW18WeGrLWLQjE6/vYwcmKQcMp+h/McjrQBvUUUUAFFFFAAelcB8TPHUvhyzt9H0VftHiTVD5VlCnzFMnb5hH1yBnqfUA1veMfFtj4O8NXGrX2DtGyGEHmaQg7UH5Ek9gCe1cf8M/CV9Lf3HjrxSm7XtT+aGNh/x6wkcAD+EkYHqBweSwoA3fh34Hg8F6J5cjifVrtvNvrsncXkPOATzgZP15PfFdpikwM5xzS0AFFFFABWV4lnvrbwtqs2m5/tBLSU221QxEuw7Dg5B+bHBrVpMUAeaa7YeNtA8Gahq8/ju4mu7O0ecomnWwjZgM45TOKq+CB4y8YeBbPXG8cXNtc3Ql/djTrV0UrIyD+DJyFFdb8RuPht4jPf+zpf/QTWR8Ehn4RaH/23/wDR8lAG74Rk1xvCsQ1uRbjWIpJ4pZCojWUpK6ocKMAEBeQK5HRfEniaX423nh3Vru3NlBYGaKG2iKocmMgkt8xIBPfHXivT9vua8ns+P2mr4f8AUGB/9AoA9aryT4vxN4d1rwx48tVO/Trpbe62Ly0D5OPpjzF/7aCvW653xzoY8R+CNY0naHknt2MQP/PRfmT/AMeC0Ab8cqTRpJGwZHAZWB4IPQ0+uF+EOtnXPhppMjsGmtUNpKAehj+Vc+5XafxruqACiiigAPQ14l8XTfav4s8Jy6boms3cWk3Ty3MkOmzlQN8Z4O3n/Vt0yOle29RSYoArWd7HfW6zxJMiH+GeF4m/75cA/pVTxBfJZaJduYLqZnhdUjtraSZ2YqcDaik/j0rUwKMCgDzD4IJdaX4KTSNR03UrK8juJXKXVlLGCpwQdzKF/DOeK9QpNo4zzS0AJtHv+dGBS0UAJj8vSl7UUUAcd8Sbp/8AhB9X0+Cyv7u6vLV4YY7WzkmyW45KqQOvc1lfBySex+H+naLfadqNnfWvneYl1ZSxDmVmBDMoU8MO+a9F2j0owD15+tAHkvi3RNU8NfFG08e6VY3F/aTRCDU7e2TfKq427go5YABT9Uq342mPxK0a18OaDDdtDc3Ecl5eT2skMdvEpyeXUZfOAFGffHWvTyoPXmk2igDxv42xjTdL8HxWduXFtqMawwqcFtoG1fxxiun8U+JxrfhW+0jQrHULjVtQha1W3ks5Yvs+8FWaRmUBQoJ78nGM1j/Fux1fWrvQIdK0W+vBp9+t1O8agKFG3gEnk9fyr060nF1bpN5UsW4Z8uVdrjqMEUAeT/EHw9H4W/Z8fQ43En2VbdWdR95zMpZh7FixH1rodD8cW9t4K05Dp2ovqkdjEi2C2crPK4QAbWC7ShxkPnGOuKZ8XrTUdY8D3WiaVpl1e3dy8TL5SDaoVwxySf8AZrp/Cckz+GNOhuLO4tZoLaKKWO4TadyoAe/IoA574d+HJvAngqQ6okj39zM13eLbxtMwdsDaqoCWwAM4B5yelc98F2u9F0vXbfVNJ1ayklvnu087T5huQqOny8n5enX2r1wqCADnH1pcCgDyP4XNdWXjXxrPe6Vq1pDqt/8AaLSSfT5kV03ynJJXC8MOuOtM0ya5T4/6nrTaPrC6Zc6eLSO6bTZwhfEZ5ynA+Vhk8e9ev7RnNIygg0AeO/EDVpYfiHapqPha+1XT7OzE9o+mMwnSRmwZCyYcD5SAMgZGeeMLofxK8K6NqLyXHhjXdKlumVJtQ1CJpG7YDyuxfaPTJAq7G3jzwd4q128/4R9fEWn6ndefHNbXAjlhXokZB6hVwMYx1Oea22m8SeM7CXTLzw9/YmmXSGO5lurlZZXjPDKiLnBIyNzEYzkA0AdyrBgCCCCOCKiuLpLaBpnWVlUZIjiaRv8AvlQSfyqVECIqgYAGAB6UpUHrQB5B8MpLqx8f+NLi90jV7WDV75ZbOWbTplR1Dy8klcLw69cda9gpMc570tAHGfFbVTo/wx124U4eSD7Ov/bQhP5MT+FXPh5pI0T4faFY7CjraI8inqHcb3/8eY1yHx0D32h6BoSOVbU9XihIU8sMEfzZa9UVQgCj7vYelADqKKKACk2jNLRQA0orAggEHgg964PxD8HvCOuym6ism0y+B3Lcae3lEMOQdv3evfGfeu+ooA8h/s/4q+B+bC+g8WaWn/LG6ytyqjrgk5z2+83+7Wnovxo8P3l5/Z+uwXXh3UhgPDqCFVDem/jH1YLXpWKy9a8N6L4itPs2r6ZbXkQGFEqDK/7rdV+oIoAu291b3kCXFtNHPC4ykkbhlI9iKm6/jXk9x8H7zQrh73wD4mu9HmJLG1nYywOcYAPB4/3g/tTB8SfF/hBvL8deFZZLRTj+09N+dMdAWGSAT7lT/s0AeuYpa5zw5478NeK1X+x9Wt55iM/Zydko9fkbB/EcV0O7mgAIABrxvSM/Cv4py6K2U8NeIn8yzJOEt5842+gGSF+hTng17NXI/ETwfH418JXOnLtW9j/fWcpP3JgDjnsD0PsfYUAdaCaWuE+FvjCTxV4Z8q/+XWdOf7Lfoww29cgMR23YP4hq7s9KACql9qNrpmnzX97OkFrAhkkkc8Ko71aJ7V4vr97P8XfGP/CK6TO6eF9NkEmqXkXSdwfuKeh5yB2JBbkKKADwzY3PxZ8YDxfq9s8fhvTnMek2co4mYHl2HTgjntnC5O059oI71BZWNtp1nBZ2cKQ20CCOONBwqjgAVYoAwPFPiu38KWVpcXFjfXjXdytrDBZRh5WcqzD5SRx8p71zetfFuz8OWsV1rHhjxHZwyvsR5beEAtjOP9b6A13VxY29zcWs8qEyWshkiOfusVZSfyY15B+0fx4N0oj/AKCA/wDRb0AdZqfxPj0aza81Hwh4ot7ZSA8rWsRVc9yRKcCtbxZ4407wjFbrcw3N5eXRIt7Ozj8yWQAcsB2Xpz79+lbl7YW+padc2N0m+3uYmilQ/wASsMEfkaih0e0h1e41Xaz3k8SQGRsErGuSFX0GSSfUn2GADjPDHxd0fxBro0O6sr3R9UbiOC+j27zjOAexx2IHtmvQScD3ryT4s6RHqfjDwOlnEBqz6gdroPmEKFXdj7L97n1NetDPc5oA5X4kyonw28Q73VQbGVASe5GAPrnHFZHwRkRvhJoyqwJRpwwz0Jmcj9CPzrs9S0LSNYCjVNLsr4L937VbpLt+m4HFM07w5oejymTTNH0+ycjBa2tkiJH1UCgDT7V5FaSp/wANN343ru/sgKBnqcIcflzXrjosiMjqGVhgqRkEVh/8IX4W8/z/APhG9IEuc+YLGLdn1ztzQBuZ+nWjGBSBFUAKMAcACnUAeT/CVW0Xxf448MFRHFbX4urdB0CPu6f8BEdesV5OgTSP2lnXcR/bGjBsZ4ZlOP5QmvWKACiiigBM0ZPpXKeKfD+oeINZ0eGHVdS07T4Vmku3sLloXkb92EQkHvlz+BrzX4l2N74W8Q+ELLTPEniNYtTumhud+rTMWUNEODu4OHNAHuuaTdx/9auB1PwJeWv2S+0bxJ4kae2uoZHt5tUkkjnjDrvVgTzlc98U7xD4c1vxpq9zayaxe6PodrtjRLP5JbuQgMzlj0QZ2gc5Ib8QDvNxzginV4fpreIfhn8TdI8P3GsXeraDrLbIRdMWeI5xxk8EErnHBB6V7hQAUUmT3xiloAKKaGyM1meJYr648L6rBpjul9LaSpbvGxVlkKEKQR0IODmgDUzxRk+leXeJfB0mgeAtU1GLxV4nnv7SyeZZ31aUBnC5ztBxjPaqfw58Ny+Kvhxp+r6h4m8TrfXIl3yQ6vKuNsjoMAkjoB170Aeu5OaWue8G2Go6T4Zt7DVbye8vIJp0NzO5Lyp5r7GJJPVNv8q39xNAC4oxSbvb3xQG+n4UALtFGKTJ/wAijJz04oAceBTFbPcfUUb+SOPWub8HeDLLwZaXkFldXlyLq5a4drmTeQxAHHH6nk9zQB09FN3c/wD1qC3tz6UALtGc0YFJu5/DNG7jPSgB1N3HPagvgZ/n/jXJ+K7HXdfu4NF03UJ9JsmjM15fxKRIQThY4z2Y4Yk9gB60Adbu57UteB+KtL8Q/B+5sPEGleItR1LSnuBFdWd9KXyTk/Q5AIyACDj1r3hJRJGkinCsoYZHrQB5d8RIxf8AxX+HVj1EdzNc4AyPl2MD/wCOGvVAMV5P4lkR/wBo3wZb5O5LCeQjtyk2P/QTXrNABRUF3eW9jayXV3PFBbxjdJLKwVUHqSeBWKPHfhEn/kadE9f+QhF/8VQB0NFc/wD8J14R/wChp0T8NQi5/wDHq1o9Rs5bAX6Xdu1mU8wXCyAx7cZ3bs4xjnOaALVFYOl+NPDetXxstN1ywurkZ/dRTAse/A78dxW6Cc+1AC0UUUAJgUFFPUAjpS0UAcH4i+EPhDxA7XAsDp17nctzYERMGznO37pPvjPvXOjR/in4HA/snUoPFOmR/wDLtefJOq+xJ9P9o+y9q9epCAetAHmmj/GrQ7i7/s7xDaXfh3UlwGhvkITJ/wBrAx9WAFeh2t5bX9slzaXEVzbyDcksLhkceoI4qrrHh7SNftPs2r6fb3sXOBMgJUnup6qfcYNec3XwduNFupL7wH4jvNEnY5+yyOZIH9Ac84/3t1AFTxpHJ8OfiJZ+ObRW/sjUmFrrESZIVj0kx9Bn6qf71evwzpcQpLDIjxuoZHQ5BBGQR+deK674p8Uafol3onxH8KPd6VPGUfVNKG4KOzkH5QwPIyV6dK57wX8YH0HwFdaGqPqGr2sgttHCxsRMrkhcjr8pHTgncqjuaAO9+Jniq+ur6LwF4WYSa5qAxcyKeLWAjJJPYkc+y+5Wuz8H+FbHwd4ettI09cpH88spXDTSHq7e57DnAAHasD4Z+B5fDNhPqesP9o8R6m3nXs7MGKEnPlg9OO5HGfYCu+2gGgBaKKKAA9K8X/aQz/whulZ/6CH/ALTevaK89+Ifw71Dx/Fb2suvwWNlbymVI0sC7liMfMxlAOOegHWgD0AHAqlqerWukWEt7eSbIY/RSzOTwFUDlmJIAA5JIFGmwahb2Kx6heW93MoA82GAwg/UF35/GuL8XeAfEHifXodRt/GT6bDanNpbxWO7ymIwXLGQZbqM44BwO+QDZ8P6Jcy6tP4m1qILqlxGIYLcncLK3ByIwe7k8s3QngcDJ6dHEih1Ksrcgg5BHavLZvhd4vukMN38TtSltm4liS1MZde4yJe447/jXqEESQQxwxqFjjUKqjsoGBQBLRRRQAUUUUAFFFFAHlXjgLZfGrwBf4Aab7RbE46/LjH/AJEP516rXk/xaQJ40+HFyHZWTWQnB42tJFn+Ver0ALRRRQAhUde9eMfG7/kcfh5/1/v/AOjIK9orwb4167pcvjHwWIr+3kawvHkuvLcN5S+ZF97HT7jflQB7vtAz1pMfTnioLO/tNRt1ubG6guYGOFlhcOp/EGvLvHXxY0Gy1g+G/t80US5XUbq0Xe6DvDGQeHPRm/hAI+90ANm3sR4x+JkWvAbtH8Po9vaP2uLpuJGU91QYXP8AeHBODXoRxjnpXk1h8Z/CXl2ejeHLG8luHZLWztvIEcYJIVcknhQcZr1cFjjI570AcN4vufC0OrxrrXiq/wBJujACsFvqUtsrJub5tqHk9Rn2rB+3/D7/AKKHrP8A4Prj/GvWNvPU0tAHCeELrwtLrTLo3irUNVuhC2YLjU5Z1VcrztY4z05ruscd6No49qWgDlviN/yTbxGfXT5v/QTWR8Ev+SQ6H/28f+j5KtfFLV9PsPh9rVvdXsEU1zZSRwRPIA8hIwNozk1kfA7V9Pm+Gel6dFewNe25nEtv5g8xczOwJXOcYYc+9ACarq0nir4uJ4LM0iaTp9obu+ijYp9pchdqMRzsG9Tt785zWX8SNM/4V3BZeLvCcYsDBcpFe2kRIhuI2zyydM5wM9fmqPxRDP4F+M9v41mjkbQtShFtfTohIt22hAWxnC/Kh56/MO1aXxL1Cz8beHYPC/hu8t9SvtRuYmb7LIJFgiU7mkcg4UZAHJBJPFAGN8W557q28Haxp2o3aR6jeQlbZ5T5GGCspZBjPv8AU1L8Qfhk9roN34r0/XNVfxDYIbp7mSfG9V5baBjZgdADxjHPWj4zww6HovguGJHeCx1CJVRV3MVRRjA7kgV1XjTxho1z4HvrfSb+31K91K2e1s7W1kEksjyKV+6DnAzk56Y5oA4nxZrd74l/Z/tvEzX93bXiqscyQS+XHKwm8pywHUH72O3Fa1z8Kx4r8IWV9d6xfHWfsUbWjJJsggOwFUVAOB0y2ck5PtVTxl4dfwr+zh/Y0xBmtxCZiOgkacM/1ALEA966/QvHXh218BafeTalbq1vZRpJb7wJhIqAGPy+u4kYA79uKAMf4R67feNPAFzaatd3QvbS4NtJcRylJmXAIJYc7uoz14ql8DkmvPDHiOOe7uXlbVZEa4MpMp+RBnce/vWv8LNGfwh4Jnv9daOxn1C5a9nE7hBAr4CqxOMHvz0LY61zvwI1rTI9H8QxSXtvE51F7nbJIFIiKj5ue3B5oAq/DvQJNZ8UePNMvNW1F7GDUBDLi4bzbhVaVVVpc7gAByBgnjnHB0fAVofCnxl8Q+EbCaU6MLJbyGCR9wjb9309B+8Ye+BnNQ/CPWdMfx749UX1tm91LzbYeao85fMmO5P73UHjPUUmk+INIb9pDVbgajaGCfTFtopRMu15P3R2g9CeCMD0oA1bi9fxH8crjw1q2ZNI03T/ALRFaP8A6u4lOz53Xo+A5wDkDbnHWsfW9I/sz406PoGl3upadpOrWrSy2um3TQKjqH+ZQpwo+QdB61L8QPEWh6b8UNPbWRqGlPY2e+31SyjzJMzHGz5gVKAbuCpOc9MHNzw343+HEnif+0f7fubzW7lRAt3qMLKQvZFwiog57AZzyaAPWYohFCkYZm2qF3Ocscdye5pdoAP60uc15/8AET4n6d4OWLTknH9qXAHRPMFshOPMYA8+y9z1wKAGeM7AeN/EOmeFohvsbK4S/wBWlH3VUA+XD/vNkkjsAD6V6Htx0OB7V43pPxn8AaBpn2e1OqTvlpJZGtwZbiQ8s7kkZYn8uAMAAV6xpN3cX2lWt3dW/wBmmniWRoM5Me4Z2k9yBgH3oA8u8Qow/ab8JSFfkbTJAD6kJc/4ivXu1eV+MFFt8dfAl4SwEsVxCMD/AGWH/s9eqUAZHiLRY/EGhXGkztiC5KLLz95A4LD2JUEZ96474qaHpVh8KNa+yabZweVEmwxwKu394vTA4r0fFcR8YP8AklGv/wDXJP8A0YlAFX4caBo+pfC3Qxe6XZXHm2mH8yBWLcnqSK2IPBNj/wAI5p2gXjGfS7JmP2Yk7JlBJjV/7wXIJB4JAqD4Vf8AJL/D3/XqP5muvK/KRk89+9AHjnxM8Daemp+GZfC2n29hrkmoqqfZIljHlAFmkYKMYQhefevYxjI5rxP4neFdc8NSHx5oviPU5bqzKieK5cEeWWA+UKFULkjKYx37V67oWof2xoGm6ntKfa7WK42+m9A39aANGiiigAooooAKKKKACkxjmlqKa4it4ZJp5EiijUs7u2Aqjkkn0xzQBmeIvEGn+FtDudX1KTZbQJnAxuduyL6kngf4V82a74H1nXfC998RZbSOyWa4E6adbRCPba85fIHXoc45GWPpXf2scvxo8afa5xIngnRpdsMRBX7bMO5Hp6+i4HBY17O8ETwtE0atEyFChHylfTHpQB4poA+I2maFY6z4Y1mHxVo1xGJFtb84nT+8u4tyV5H3uoPy8V0Wk/GvRnu/7P8AE1heeHNSGMxXkbGPnp82AR9WUD3NZPg6Rvht8RbvwTdOw0bVXN1pErtkIx4MeT342/VV/v16lq2iaXrtr9m1XT7a9hBJCTxhgD6gnofpQBZs7+11C2S6srmG5t3GVlhcMpH1FWM5+leUXnwcfSLh77wL4gvdCuTz9naRpLdzjgEHnGf7276VEfH/AI58FNs8aeGGv7FM/wDEz0r5htA6svT89lAHrtGK5fw38RPC3ivaml6tC1w3S2lPly/grYLfUZFdPnnFABigKAeBS0UAFJilooAKKKKACiiigAoooPTigDyj4vLv8T/D1fXW4xn6vHXqwryn4nR/bPib8N7UZO2+knKg4+60TA/+OmvV8UAFFFFAB2pNopaKAE2jGKNvqSfxpaTNABgUYoBpaACiiigAooooAQqDRilooAa0asu0jI9KjgtLe0QpbQRwr1xGgX+VTUUAePfGm6We88NW9vHPPLZ6ktxcLBCz+WgwcnA9D0r1Ky+wXAF9ZRQnzsnzUTazeuTjPar20D1o2g9eaAPNvjZMkvw7vdLgSWa9uHi8qCGNnYgSBicAcDCmul8HnTr7w3pN1FChuYrSGJ5Gh2yIwQAjJGRXSbfc0YGc0AG0HrzRj3paTPNABijFLRQB5GviXVvDHjLXZPFXhrUr22u5gljeWUH2iNIBnZHt/h9T3JJJHSr97NYeP9Mn0vTfCVxHHcqUfUdQsFgjgUjG9M/Mzj+HaOoGSK9MwMYoCgUAMjURosY6KMU/aOKAMUtACYoCgUA5paAPKPi8ps/EvgHWS2yO11dYpGwPuuyE/oh/WvVsmvNPjrp5vPhlcXKgmSwuIrlSO3zbCfwDk/hXfaNqKatolhqUX+ru7eOdfYMoYfzoAvHp0zXD/FCz1nWvBeoaJo2jz3txdogEiyxIiYcMc73Bzhew7iu4pMfWgDjvhtb6rpXg7TtG1bSJ7K4s4jGztNFIj4Y42lHJ6EdQKteLta8S6PHZTaB4eTWEZyLqMTiN0HGNueueeeenTmunwKNooA4PVodb8d6P/Y0ui3OiWFyV+2z3ckZkaMEEpEqM3JwBubGATgHpXcW0EVrbRW0CBIYUWNFXooAwAPwqTaKXFABRRRQAUUUUAFFFB4BPSgBCa8e8cavf/EDxSPh94dm8uziIfWb1TkIgP+rH6ZHdiBwA1bXxM8bXmlpb+GPDqtN4l1U+XCsZ5gQ8GQnseuPTBJ6YO54B8FWngjw9HYxES3kp8y8uSOZZP8BnAH1PUkkA29G0ey0HSrXTNPhENpapsjRT29T6kkkn1JJrRPSkx068UtAHC/FHwc/irwsXsQy6zpzfabCRThtwwSgPuB+YU9qt/DrxgnjPwnb38mEvoj5F7F02Sr147A8EfXHY112K8a1kf8Kt+KkWvxAJ4a8RMIb0fwwT8nf+eW+hk9qAPZcCjApqvuGQQQehHesfxB4t0XwtDbS6xdm3W5k8uELDJIXbGcAIpNAGL4l+FHhHxOHkuNNW0u2/5erI+U+fU4G1j7sDXKjw58T/AAQ27QNai8S6anSy1DiUD0View9GH+7XUXHxa8GWgj+06ncQ+YcJ5mnXKbj7Zj96lb4o+EFuILeTUp45Z3EcYl0+5jDMTgcmMDqRzQBg6X8bNJW6Gn+KtNvfDmod1uomMZ9CGwGA9yoHvXo1lqNpqNql3Y3UFzbP92WGQOp+hFc5461rwhpelKni77JJbykhIJofNZz/ALKgE8evHbkV5bonhvwtr9/Lc/C/xfe6JqiDe9i7PscD/ZbkrnGTlwPTmgD34HOPelryEeOfH3grKeMfDX9p2KddT0rnAHVmXp+eyuy8NfEjwt4r2JpmqxfaWOPs0/7uXPXAU/e/DNAHWUUmecUtABRRRQAUUUHpQB5RrofUf2i/DNsuGj0/TZLlx6FvMX+ez869XrybweRrXx48Zasrl4tPgjsEHZScBv8Ax6J/zr1mgAooooA5TxTqviSDVtI03w3FpzS3azS3Et+jskUabBkbWHJLgYrjvF/jfx34Q1TQbGdvDdw+sTGFGjtZwIyGQEn97z98flXrRRdwbA3AYzXjPxt48YfDzr/x/v3/AOmkFAHU6nqXxD0Z7S5nHh28sDdRR3Yt7eZJEjd1UsuZCDjPXt1wa0fEeo+Kri7bTvCltYo8IBub7US4iQkZCIqglmxgk9ACOueOraNWXayhgezDvVbUL1NOs5LpoLmcJ0jtoWkdvYKtAHn3gvxz4hfxrdeDfF9nbR6jHF59vcWoYJKvBPXrwcg8dCCM16bXj3gXxZoviP4q6tdX1rdWGvvGLWytrtAhWBRuYdc+YTkkdgOCea9hPSgAorB1dfFUl6p0O60aK1CAMt7bSyPuyeQVkUYxjjGfzqh5PxD/AOf/AML/APgDcf8Ax2gDray/Eeo3Gk+GdU1G1iWW5tbSWaGNlLBnVSVBAIJBIHQ1V0dPFSXhOt3WjS2pQhVsraWNw2RgkvIwxjPbuK3GRXUqwDKeCCM5oA861TV/iPo3ha71q9bwurWtu08lsltOx4GSu7zMZ/CoPCviL4heLPCVrr1pL4YiFz5myGW1nyNrsnJEvqvp3rpfiMB/wrbxEf8AqHzf+gmsj4J/8ki0M8/8t/8A0fJQB0XhDVdU1fw1b3es28FvqQlmiuIoQQitHK8fGSTj5fWt3NcTrfia7k8Z2fg7QGhgvZYWu727ePeLaLP8K93YnvwMg4Oax/FfiPxD8N73T9R1HUG1rw9czeRcedbxpPbsQSCpjChhgE4I7YzzQB6aSaXJrzD4l+NPEXhybRJdJe0bTdQuUj3Im+dwdpwuflAIJHTPTnms/wAZar8UfDlm/ihbnS/7OhYPNpUcW/yoycDc5GWPI3FSACeMjmgD1/NJu4yRXl/i74g6uPhZb+MPDi2kMMyIZRcKXkjZm2EKOhKt3OR7VBqd58UdR8Nwa9o9zY2iR2yzfYWhDz3I2gsWyuFJ5IUY4Iyc8UAerknpxXM+D7bxVa2t6virULS8na6ZrY20YUCLHAPA9/U47ms7wJ40u/HXgYalZpaw6rG5gmSTcYlkABzgEHBBBxn2zxWd8LPFXiDxZ4e1q61CW2a9gvnghUR7I0AVTjA5IyT1JPvQB6Ru/Klya8h8JeKfHPibWvFWkG406K4sLlbZbryMxWwDSKxSPO52YqMBmwMHJ7HT8D+JvES+Pda8F+JruG/uLOBbmC7iiWLch25BA4/5aLj6HrQB6Vu9vypc1wd74pv9W+JreC9LuPsSWln9rvroRq8hztwkYYFRw6kkg9e2KxNS8UeMdD+I9p4Rh1DTryLUIfPtLvUbYhlwGyjeUUB+4cHb3FAHrFYHiPU9ZgSKx8PWUFzqdwCyyXTFYIEGAXkI5PJGFHJ59DW3EZPKTzdhk2jeU6Z74z2olZYYpJSrkKpYhFLE49AOSfpQB5RB438Z+FvHGlaH4yttOubPV5RFbXdgGG1yQoHPXBZQQRxuzk4xXrdeLar440PVfi1o1tr9hqOnQaduNgbu3MYlncgCRweQg2jbxyTk4xXs+T6gUAZniXSRrvhjVNKyA13ayQqx6KzKQD+Bwfwrj/glq51T4aWUMhbz9PkezlDdQVO5R+Csor0U9K8k8Fk+FvjP4p8NyHbbaqBqVmTwCSSWCj/gTj6R0Aet0UUUAFFFFABRRRQAUUUUAFFFFABXL+OvGln4J8Ny6lcASXD/ALu0t+80hHA+ncnsPwB2tV1az0XS7nUtQnWG0t0LyO3Yf49sdzxXk/hDTrv4ieKG8f8AiKJotJs2K6PZOMgKp/1jDvgjOecsPRQCAbnwy8GXljJc+LvExM3iTVfnfeP+PeM9EA7HAGfQAKOhz6TiuKt/ir4Mng8+31K5miUYMiadcsOPfy6IPiz4MuoDPb6ncTQgkF49NuWUEe4j9KAO2orF0fxXouvaLJrGnXyS6fEXEk7o0QTaMtneAQAO/SuXj+NfgSTURZjWCMkKJmgcRk/XHT3xj3oA9CrD8WeGbXxZ4YvdGuztWePCSEZ8txyrAexxx3HFbEcqyxrIjKyOAVZTkMOuQe4xUh6UAeZ/CPxLd3Nhd+Etb3JrmhN5Lq55khBwrD1A4XPptPO6u8utItrzVrG/mXe9kJPKBGQrOFG764BH4mvNPijp114X8Qad8R9HjZpbRlh1OFB/roDwCf8A0HJ6ZU/w16dpepWusaba6jZSiW2uY1ljcdwR/wDX6UAeSfHrjUfBZHX7e3P4x16vrWj2ut6XJY3S5RsMrbQSjqQVYZ7ggGvKPj1/yEPBf/YQb+cdeznOOOtAGJF4asW8R3et3UaXV5IEigeWPP2aJVHyJnPVizEjrux2rzTxz4et7D4x+B7zRIFgvr25f7WIQBvjjKFmI9djSAnuB7V61f6jaaVYT319MkFvCu6RmPT2HqTxgDk5Hc1zvh/Rrm81648XaxAYr6eIW9nauMmztwcgHH/LRictjpnHY0AddtFcb4m+FnhHxTvkvNMWC7fJ+1Wf7qTPqccMf94GuxBJp1AHkJ8MfEzwSd3hvXY/EOnJ0sdT4kA7BWJ7ezD6Grum/GvTobtdP8W6TfeHb/OMXEbNE3OMhsA498Y969P2Dn3qpqWk6drFo1rqVjb3lu3WOeMOv5HvQA6x1Oy1W0W7067t7u3fpLBIHU/iOKtBsmvK7/4MW1jdtqHgvW77w9ecZRJGkhbHOCCc8+hJHtVYeM/iF4KGzxb4dGr2CHB1LTOWAAyWZQMfmE+poA9eqtf3sWnadc31ywWC3ieaRvRVBJP5CuY8M/E3wr4s2R6fqkcd0+MWtz+6lyewB4Y/7pasf42azLp/gGTTrXcb3V5ksYUT7zA8t+YG3/gQoArfAy0mfwjfa/dBTc61qE1yzAdVB24P/A95/GvUayvDejp4f8N6bpEZBFpbpESBjcwHLficn8a1aACiiigArxb43Ef8Jh8PizAbb9yfYeZBXtJGQQehrlNZ+G/hfxBfJeatZXN3cISY3k1C4/d5OflHmYXn0xQB1Bb1wD9aiS6gkuXt0miM8QBeMMCyZHGR2os7KKxtxBE87IDnM07yt/305J/WuY1T4Z+GtV8QNrskN1Bqb433FrdSQs2Bj+EjHAHTFAHOeMdGi1P4w+C5bBR9vtxJcXrJ1W3TBQv7EllB75NepHkVm6VoGm6IsosLfY8zBppXdpJJSOhd2JZiPcmtKgDD1jwho2vXi3eoQ3LzKgjBjvZohtBJ+6jgdzzjNZ//AArbwv8A8+19/wCDS6/+OV1lFAGFo/hDRtCvGu9PiuUmZDGTLezTDaSDjDuR2FbtFFAHKfEZh/wrXxFk4/4l8o/8dNZPwU/5JHoq+n2gH/v/ACV1Wu+GdL8SWhtNVjuJrYjDQpdSxIwzn5lRgG6d81D4f8H6L4Xh8jR4Li3g5IhN5NJGCTkkI7lQffFAHmbSN4d/aXkuL3K22uWSw20hPyk7EG3Pruix/wACX1rc+OOJfhybNF8y6u7yCG3jUfNJJuzx+ANdr4h8LaN4psVtNZs1uY0O5GJKvG3qrDBU/TrVbT/BmlWF9DfO17fXduCIJtQu3uDDnrsDkhT7gZ96APNPirZ/2L4c+H9ncSA/Yr23ikkJ4+RFBOT9DXo3xAuLe1+HviGW5YLGdPmTLcAsyFVH1LEAfhXDfHNIZYvCUMwVopNWQOrdGXAB/DH867dfAuiloBcNfXdtauJILW6vZJYY2HQ7WODjtuzigDy7WdKuNH/ZbitLpCkxEUrIwwVD3O8D8Awr13w1cQL4N0i5MiCAWELl8gKB5YOfTH0rk/jewHwo1QA4PmQAD/tqp4/z2q5oHgfRrvwlpkMkl+bCe0id7Fb2UQPuUE/JnoTklRxz0oAxPgRpstr4Qv8AUGjeKC/v5JbZSDzEAAGHfrkfgKi+Ax/4kniHHfWJDj/gK16g+mWraeLFEeC2CCNUtpGh2KOgUoQV6AcduKxdD8AeHPDS3C6PaXFmtwMSrHfT4b3wX4PuOfegDivhOQ3jv4kd8at0z/00m7fhS6WR/wANMa4M8/2Mo/H9z/T+ddro3gLw74fv5r7S7W5t7mdt0z/bp281uTlwzkMeTyR3NNh+H3hy21465FbXg1Nhhrk6jcFmHocycjgcHjigDj9cggtPjRbajod/Zxaz9gJ1G1vZTFDLDwqgOASH4XgKRhQTj+LbsfDE2sePYPFur3NkZrK38iys7GYypGDu3OzkKWJ3EY2jHFcfP/wgnjrx7r0fipLa0u7CUWdvFNMbd5UXrIWyN5J4A6hQPWo9W+HvgXT4w3hbUriPxGwxp8dle+a5m6gsOcIDjcTwBmgD24HjNQtdwJcRwPPEssoJSMuAzAdwO4qSMMY1EhBbHzY9e9c54m+H/h7xbdW13qtrK11bDEU8MzRuoznGQeefyoA5X42aRDrfhfT7GKMPqs2oRx2Cj75Y5D9OdoXkn2GelemxKUjVSxbAAye571kaT4T0jRp/tNvFNLdbPL+03Vw88oXrtDOSQPYYFbRUHrQAHoa8n+MVtPod74f8eWKEzaPciK5Cjl4HPQn05K/9tK9ZrP1zRrXX9CvdJvATb3cLRNjqMjgj3BwR9BQBYtbyK9tYLq2dZIJ0WSN16MrDII+oINWK8u+Der3UOnX/AIN1VsapoMzQ4J+/DuO0jPUA5H0Keteo0AFFFFABRRRQAUUUUAFITgc0teTfEfxHf+ItYT4deF3ze3Q/4mVyvK20BHzA/UEZ+oHVhQBm6jNN8ZfGn9j2crJ4N0mUNdTxni8lHZT3HUA+mW6lRXsJ0+BdNNhAiwQCHyY1iG0RrjACgdMCs/wx4a07wnoFro+mxbYYRlmb70r93Y9yf8AMACtqgDEvNOttL8F3On2sYjt7aweKNQMAAIQK4L9nr/km0v8A2EJf/QUr0jXv+Re1P/r0l/8AQDXm37PJP/Ct5h/1EJf/AEFKAOyvPBmn3lpdWLDbYXuoC9urZVAWTCrlOOzOisfXkd6pePfC+h33gDVYbixto47WzllgdIlXyWVSylcdMEDjv0rsTjrwTXHeIkfxneP4ZtHI0yJx/a9yh9MEWy+rNwWP8K+7CgCD4Ppdr8LtEF6X3lHMYfr5ZdinPcbcY9sV3dQ28UVvDHBDGscUahUReAqgYAHoO1TUAV72xttQsLiyu4lltp42iljboykYIP4V5R8OLy48EeMdQ+HOqSOYMtd6PM5+/GSSUH5E8Dqslev15z8WvCtzrOhQa1o+5dd0V/tVq8f3mA5Zfc8Bh6lcd6AKXxL8E+KvG2paU9iNHtoNMmaWMzXUpaUkrjIEWF+70yevXivSLGS9ktQ1/BBDcdCsExlX65Kr/KsbwT4rtvGXhez1iAqryLsniU58qYfeX88EZ7EHvXR7RQB5n4t8O/EbWPEtve6Xe6BBYWTbrSC4eV/n/wCejjyyC47dcdueagOgfF2+eO21DxDoMVjI6rctZq6y+UThghMXB25xyPqK9T2jOe9LigBiLtVRk4Axz/8AXp9GKKACiiigBMA0YpaKAOP8TfDHwn4qEkl/pUcV02T9qtR5UufUkDDH/eBryzwv4WnuvjMmjya1e6vo/hcmeN7piRFIQu1AckZDAen+rbivZPGXiaDwl4Tv9Zn2s0Ef7pCceZIeEX6FiM+gye1c18HvD0+keEf7T1AFtV1qQ311IygMQx+QHHsd2PVzQB6GABS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBhat4N8Pa7dC51XS4LyYdGmy23p0546DpWta2cNlbpbwKVij4UFi2PxPNT0UAZOseGdG19VXVrCO8RfupMSyj8M471Y0vSLHRrRbTToBb26/diVjtXp0z06VeooAKKKKACiiigDG1jwl4f8QOsmraPZ3cijAkliBcf8C6/rUmkeGdE0BXXSNKs7LeMOYIVUt9T1NatFACYpaKKACiiigAoPSiigDyH4lW9z4N8YaZ8RtOidoUZbTVoU/wCWkRO1WPqeg56EJXqtnewahZwXlrKk1tOiyRypyGUjIIpmp6Zaavpl1p97EJbW5iaKVD3Uj9D715b8N9UuvBvie6+G+uS7hGxn0i4bjzYmydv8zjsQ47CgD16ikzS0AFFFFABQelFYXizxVYeEPD9xq+oviKMbY0B+aaQ/dRfc4P0AJ6A0AYPxK8dn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+85	"At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected
+
+<image_1>
+uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping.
+For a 3-topping pizza, determine the probability that some region of the pizza with non-zero area is covered by all 3 toppings. (The diagram above shows an example where no region is covered by all 3 toppings.)"	['Suppose that the first topping is placed on the top half of the pizza. (Again, we can rotate the pizza so that this is the case.)\n\nAssume that the second topping is placed on the half of the pizza that is above the diameter that makes an angle of $\\theta$ clockwise with the horizontal as shown. In other words, the topping covers the pizza from $\\theta$ to $\\theta+180^{\\circ}$.\n\nWe may assume that $0^{\\circ} \\leq \\theta \\leq 180^{\\circ}$. If $180^{\\circ} \\leq \\theta \\leq 360^{\\circ}$, the resulting pizza can be seen as a reflection of the one shown.\n<img_4026>\n\nConsider the third diameter added, shown dotted in the diagram above. Suppose that its angle with the horizontal is $\\alpha$. (In the diagram, $\\alpha<90^{\\circ}$.) We assume that the topping is added on the half pizza clockwise beginning at the angle of $\\alpha$, and that this topping stays in the same relative position as the diameter sweeps around the circle.\n\nFor what angles $\\alpha$ will there be a portion of the pizza covered with all three toppings?\n\nIf $0^{\\circ} \\leq \\alpha<180^{\\circ}$, there will be a portion covered with three toppings; this portion is above the right half of the horizontal diameter.\n\nIf $180^{\\circ} \\leq \\alpha<180^{\\circ}+\\theta$, the third diameter will pass through the two regions with angle $\\theta$ and the third topping will be below this diameter, so there will not be a region covered\n\n\n\nwith three toppings.\n\nIf $180^{\\circ}+\\theta \\leq \\alpha \\leq 360^{\\circ}$, the third topping starts to cover the leftmost part of the region currently covered with two toppings, and so a region is covered with three toppings.\n\nTherefore, for an angle $\\theta$ with $0^{\\circ} \\leq \\theta \\leq 180^{\\circ}$, a region of the pizza is covered with three toppings when $0^{\\circ} \\leq \\alpha<180^{\\circ}$ and when $180^{\\circ}+\\theta \\leq \\alpha \\leq 360^{\\circ}$.\n\nTo determine the desired probability, we graph points $(\\theta, \\alpha)$. A particular choice of diameters corresponds to a choice of angles $\\theta$ and $\\alpha$ with $0^{\\circ} \\leq \\theta \\leq 180^{\\circ}$ and $0^{\\circ} \\leq \\alpha \\leq 360^{\\circ}$, which corresponds to a point on the graph below.\n\nThe probability that we are looking for then equals the area of the region of this graph where three toppings are in a portion of the pizza divided by the total allowable area of the graph.\n\nThe shaded region of the graph corresponds to instances where a portion of the pizza will be covered by three toppings.\n\n<img_3744>\n\nThis shaded region consists of the entire portion of the graph where $0^{\\circ} \\leq \\alpha \\leq 180^{\\circ}$ (regardless of $\\theta$ ) as well as the region above the line with equation $\\alpha=\\theta+180^{\\circ}$ (that is, the region with $\\theta+180^{\\circ} \\leq \\alpha \\leq 360^{\\circ}$ ).\n\nSince the slope of the line is 1 , it divides the upper half of the region, which is a square, into two pieces of equal area.\n\nTherefore, $\\frac{3}{4}$ of the graph is shaded, which means that the probability that a region of the pizza is covered by all three toppings is $\\frac{3}{4}$.']	['$\\frac{3}{4}$']		['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+86	"At Pizza by Alex, toppings are put on circular pizzas in a random way. Every topping is placed on a randomly chosen semicircular half of the pizza and each topping's semi-circle is chosen independently. For each topping, Alex starts by drawing a diameter whose angle with the horizonal is selected
+
+<image_1>
+uniformly at random. This divides the pizza into two semi-circles. One of the two halves is then chosen at random to be covered by the topping.
+Suppose that $N$ is a positive integer. For an $N$-topping pizza, determine the probability, in terms of $N$, that some region of the pizza with non-zero area is covered by all $N$ toppings."	"['The main idea of this solution is that the toppings all overlap exactly when there is one topping with the property that all other toppings ""begin"" somewhere in that toppings semi-circle. In the rest of this solution, we determine the probability using this fact and then justify this fact.\n\nSuppose that, for $1 \\leq j \\leq N$, topping $j$ is put on the semi-circle that starts at an angle of $\\theta_{j}$ clockwise from the horizontal left-hand radius and continues to an angle of $\\theta_{j}+180^{\\circ}$, where $0^{\\circ} \\leq \\theta_{j}<360^{\\circ}$. By establishing these variables and this convention, we are fixing both the angle of the diameter and the semi-circle defined by this diameter on which the topping is placed.\n\nSuppose that there is some region of the pizza with non-zero area that is covered by all $N$ toppings.\n\nThis region will be a sector with two bounding radii, each of which must be half of a diameter that defines one of the toppings.\n\nSuppose that the radius at the clockwise ""end"" of the sector is the end of the semi-circle where topping $X$ is placed, and that the radius at the counter-clockwise ""beginning"" of the sector is the start of the semi-circle where topping $Y$ is placed.\n\n\n\n<img_3683>\n\nThis means that each of the other $N-2$ toppings begins between (in the clockwise sense) the points where topping $X$ begins and where topping $Y$ begins.\n\nConsider the beginning angle for topping $X, \\theta_{X}$.\n\nTo say that the other $N-1$ toppings begin at some point before topping $X$ ends is the same as saying that each $\\theta_{j}$ with $j \\neq X$ is between $\\theta_{X}$ and $\\theta_{X}+180^{\\circ}$.\n\nHere, we can allow for the possibility that $\\theta_{X}+180^{\\circ}$ is greater than $360^{\\circ}$ by saying that an angle equivalent to $\\theta_{j}$ (which is either $\\theta_{j}$ or $\\theta_{j}+360^{\\circ}$ ) is between $\\theta_{X}$ and $\\theta_{X}+180^{\\circ}$. For each $j \\neq X$, the angle $\\theta_{j}$ is randomly, uniformly and independently chosen on the circle, so there is a probability of $\\frac{1}{2}$ that this angle (or its equivalent) will be in the semicircle between $\\theta_{X}$ and $\\theta_{X}+180^{\\circ}$.\n\nSince there are $N-1$ such angles, the probability that all are between $\\theta_{X}$ and $\\theta_{X}+180^{\\circ}$ is $\\frac{1}{2^{N-1}}$.\n\nSince there are $N$ possible selections for the first topping that can end the common sector, then the desired probability will be $\\frac{N}{2^{N-1}}$ as long as we can show that no set of angles can give two different sectors that are both covered with all toppings.\n\nTo show this last fact, we suppose without loss of generality that\n\n$$\n0^{\\circ}=\\theta_{1}<\\theta_{2}<\\theta_{3}<\\cdots<\\theta_{N-1}<\\theta_{N}<180^{\\circ}\n$$\n\n(We can relabel the toppings if necessary to obtain this order and rotate the pizza so that topping 1 begins at $0^{\\circ}$.)\n\nWe need to show that it is not possible to have a $Z$ for which $\\theta_{Z}, \\theta_{Z+1}, \\ldots, \\theta_{N}, \\theta_{1}, \\theta_{2}, \\ldots, \\theta_{Z-1}$ all lie in a semi-circle starting with $\\theta_{Z}$.\n\nSince $\\theta_{Z}<180^{\\circ}$ and $\\theta_{1}$ can be thought of as $360^{\\circ}$, then this is not possible as $\\theta_{1}$ and the angles after it are all not within $180^{\\circ}$ of $\\theta_{Z}$.\n\nTherefore, it is not possible to have two such regions with the same set of angles, and so the desired probability is $\\frac{N}{2^{N-1}}$.']"	['$\\frac{N}{2^{N-1}}$']		['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+87	"In rectangle $A B C D, F$ is on diagonal $B D$ so that $A F$ is perpendicular to $B D$. Also, $B C=30, C D=40$ and $A F=x$. Determine the value of $x$.
+
+<image_1>"	"['Since $A B C D$ is a rectangle, then $A B=C D=40$ and $A D=B C=30$.\n\nBy the Pythagorean Theorem, $B D^{2}=A D^{2}+A B^{2}$ and since $B D>0$, then\n\n$$\nB D=\\sqrt{30^{2}+40^{2}}=\\sqrt{900+1600}=\\sqrt{2500}=50\n$$\n\nWe calculate the area of $\\triangle A D B$ is two different ways.\n\nFirst, using $A B$ as base and $A D$ as height, we obtain an area of $\\frac{1}{2}(40)(30)=600$.\n\nNext, using $D B$ as base and $A F$ as height, we obtain an area of $\\frac{1}{2}(50) x=25 x$.\n\nWe must have $25 x=600$ and so $x=\\frac{600}{25}=24$.'
+ 'Since $A B C D$ is a rectangle, then $A B=C D=40$ and $A D=B C=30$.\n\nBy the Pythagorean Theorem, $B D^{2}=A D^{2}+A B^{2}$ and since $B D>0$, then\n\n$$\nB D=\\sqrt{30^{2}+40^{2}}=\\sqrt{900+1600}=\\sqrt{2500}=50\n$$\n\nSince $\\triangle D A B$ is right-angled at $A$, then $\\sin (\\angle A D B)=\\frac{A B}{B D}=\\frac{40}{50}=\\frac{4}{5}$.\n\nBut $\\triangle A D F$ is right-angled at $F$ and $\\angle A D F=\\angle A D B$.\n\nTherefore, $\\sin (\\angle A D F)=\\frac{A F}{A D}=\\frac{x}{30}$.\n\nThus, $\\frac{x}{30}=\\frac{4}{5}$ and so $x=\\frac{4}{5}(30)=24$.\n\nSolution 3\n\nSince $A B C D$ is a rectangle, then $A B=C D=40$ and $A D=B C=30$.\n\nBy the Pythagorean Theorem, $B D^{2}=A D^{2}+A B^{2}$ and since $B D>0$, then\n\n$$\nB D=\\sqrt{30^{2}+40^{2}}=\\sqrt{900+1600}=\\sqrt{2500}=50\n$$\n\nNote that $\\triangle B F A$ is similar to $\\triangle B A D$, since each is right-angled and they share a common angle at $B$.\n\nThus, $\\frac{A F}{A B}=\\frac{A D}{B D}$ and so $\\frac{x}{30}=\\frac{40}{50}$ which gives $x=\\frac{30(40)}{50}=24$.']"	['24']		['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+88	"In the diagram, $\triangle A B C$ is right-angled at $C$. Also, $2 \sin B=3 \tan A$. Determine the measure of angle $A$.
+
+<image_1>"	"['Since $\\triangle A B C$ is right-angled at $C$, then $\\sin B=\\cos A$.\n\nTherefore, $2 \\cos A=3 \\tan A=\\frac{3 \\sin A}{\\cos A}$ or $2 \\cos ^{2} A=3 \\sin A$.\n\nUsing the fact that $\\cos ^{2} A=1-\\sin ^{2} A$, this becomes $2-2 \\sin ^{2} A=3 \\sin A$\n\nor $2 \\sin ^{2} A+3 \\sin A-2=0$ or $(2 \\sin A-1)(\\sin A+2)=0$.\n\nSince $\\sin A$ is between -1 and 1 , then $\\sin A=\\frac{1}{2}$.\n\nSince $A$ is an acute angle, then $A=30^{\\circ}$.'
+ 'Since $\\triangle A B C$ is right-angled at $C$, then $\\sin B=\\frac{b}{c}$ and $\\tan A=\\frac{a}{b}$.\n\nThus, the given equation is $\\frac{2 b}{c}=\\frac{3 a}{b}$ or $2 b^{2}=3 a c$.\n\nUsing the Pythagorean Theorem, $b^{2}=c^{2}-a^{2}$ and so we obtain $2 c^{2}-2 a^{2}=3 a c$ or $2 c^{2}-3 a c-2 a^{2}=0$.\n\nFactoring, we obtain $(c-2 a)(2 c+a)=0$.\n\nSince $a$ and $c$ must both be positive, then $c=2 a$.\n\nSince $\\triangle A B C$ is right-angled, the relation $c=2 a$ means that $\\triangle A B C$ is a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle, with $A=30^{\\circ}$.']"	['$30^{\\circ}$']		['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+89	"Alice drove from town $E$ to town $F$ at a constant speed of $60 \mathrm{~km} / \mathrm{h}$. Bob drove from $F$ to $E$ along the same road also at a constant speed. They started their journeys at the same time and passed each other at point $G$.
+
+<image_1>
+
+Alice drove from $G$ to $F$ in 45 minutes. Bob drove from $G$ to $E$ in 20 minutes. Determine Bob's constant speed."	"[""Since Alice drives at $60 \\mathrm{~km} / \\mathrm{h}$, then she drives $1 \\mathrm{~km}$ every minute.\n\nSince Alice drove from $G$ to $F$ in 45 minutes, then the distance from $G$ to $F$ is $45 \\mathrm{~km}$.\n\nLet the distance from $E$ to $G$ be $d \\mathrm{~km}$ and let Bob's speed be $B \\mathrm{~km} / \\mathrm{h}$.\n\nSince Bob drove from $G$ to $E$ in 20 minutes (or $\\frac{1}{3}$ of an hour), then $\\frac{d}{B}=\\frac{1}{3}$. Thus, $d=\\frac{1}{3} B$.\n\nThe time that it took Bob to drive from $F$ to $G$ was $\\frac{45}{B}$ hours.\n\nThe time that it took Alice to drive from $E$ to $G$ was $\\frac{d}{60}$ hours.\n\nSince the time that it took each of Alice and Bob to reach $G$ was the same, then $\\frac{d}{60}=\\frac{45}{B}$\n\nand so $B d=45(60)=2700$.\n\nThus, $B\\left(\\frac{1}{3} B\\right)=2700$ so $B^{2}=8100$ or $B=90$ since $B>0$.\n\nTherefore, Bob's speed was $90 \\mathrm{~km} / \\mathrm{h}$.""]"	['90']		['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+90	"In the diagram, $D$ is the vertex of a parabola. The parabola cuts the $x$-axis at $A$ and at $C(4,0)$. The parabola cuts the $y$-axis at $B(0,-4)$. The area of $\triangle A B C$ is 4. Determine the area of $\triangle D B C$.
+
+<image_1>"	"['First, we determine the coordinates of $A$.\n\nThe area of $\\triangle A B C$ is 4 . We can think of $A C$ as its base, and its height being the distance from $B$ to the $x$-axis.\n\nIf the coordinates of $A$ are $(a, 0)$, then the base has length $4-a$ and the height is 4 .\n\nThus, $\\frac{1}{2}(4-a)(4)=4$, so $4-a=2$ and so $a=2$.\n\nTherefore, the coordinates of $A$ are $(2,0)$.\n\nNext, we determine the equation of the parabola.\n\nThe parabola has $x$-intercepts 2 and 4 , so has equation $y=k(x-2)(x-4)$.\n\nSince the parabola passes through $(0,-4)$ as well, then $-4=k(-2)(-4)$ so $k=-\\frac{1}{2}$.\n\nTherefore, the parabola has equation $y=-\\frac{1}{2}(x-2)(x-4)$.\n\nNext, we determine the coordinates of $D$, the vertex of the parabola.\n\nSince the $x$-intercepts are 2 and 4 , then the $x$-coordinate of the vertex is the average of these, or 3.\n\n\n\nThe $y$-coordinate of $D$ can be obtained from the equation of the parabola; we obtain $y=-\\frac{1}{2}(3-2)(3-4)=-\\frac{1}{2}(1)(-1)=\\frac{1}{2}$.\n\nThus, the coordinates of $D$ are $\\left(3, \\frac{1}{2}\\right)$.\n\nLastly, we determine the area of $\\triangle B D C$, whose vertices have coordinates $B(0,-4)$, $D\\left(3, \\frac{1}{2}\\right)$, and $C(4,0)$.\n\nMethod 1 \n\nWe proceed be ""completing the rectangle"". That is, we draw the rectangle with horizontal sides along the lines $y=\\frac{1}{2}$ and $y=-4$ and vertical sides along the lines $x=0$ and $x=4$. We label this rectangle as $B P Q R$.\n\n<img_3586>\n\nThe area of $\\triangle B D C$ equals the area of the rectangle minus the areas of $\\triangle B P D, \\triangle D Q C$ and $\\triangle C R B$.\n\nRectangle $B P Q R$ has height $4+\\frac{1}{2}=\\frac{9}{2}$ and width 4 .\n\n$\\triangle B P D$ has height $\\frac{9}{2}$ and base 3 .\n\n$\\triangle D Q C$ has height $\\frac{1}{2}$ and base 1.\n\n$\\triangle C R B$ has height 4 and base 4.\n\nTherefore, the area of $\\triangle B D C$ is $4\\left(\\frac{9}{2}\\right)-\\frac{1}{2}\\left(\\frac{9}{2}\\right)(3)-\\frac{1}{2}\\left(\\frac{1}{2}\\right)(1)-\\frac{1}{2}(4)(4)=18-\\frac{27}{4}-\\frac{1}{4}-8=3$.\n\nMethod 2\n\nWe determine the coordinates of $E$, the point where $B D$ crosses the $x$-axis.\n\n<img_3686>\n\nOnce we have done this, then the area of $\\triangle B D C$ equals the sum of the areas of $\\triangle E C B$ and $\\triangle E C D$.\n\nSince $B$ has coordinates $(0,-4)$ and $D$ has coordinates $\\left(3, \\frac{1}{2}\\right)$, then the slope of $B D$ is $\\frac{\\frac{1}{2}-(-4)}{3-0}=\\frac{\\frac{9}{2}}{3}=\\frac{3}{2}$.\n\nSince $B$ is on the $y$-axis, then the equation of the line through $B$ and $D$ is $y=\\frac{3}{2} x-4$. To find the $x$-coordinate of $E$, we set $y=0$ to obtain $0=\\frac{3}{2} x-4$ or $\\frac{3}{2} x=4$ or $x=\\frac{8}{3}$.\n\nWe think of $E C$ as the base of each of the two smaller triangles. Note that $E C=4-\\frac{8}{3}=\\frac{4}{3}$. Thus, the area of $\\triangle E C D$ is $\\frac{1}{2}\\left(\\frac{4}{3}\\right)\\left(\\frac{1}{2}\\right)=\\frac{1}{3}$.\n\nAlso, the area of $\\triangle E C B$ is $\\frac{1}{2}\\left(\\frac{4}{3}\\right)(4)=\\frac{8}{3}$.\n\nTherefore, the area of $\\triangle B D C$ is $\\frac{1}{3}+\\frac{8}{3}=3$.']"	['3']		['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+91	"In the diagram, $P Q R S$ is a square with sides of length 4. Points $T$ and $U$ are on sides $Q R$ and $R S$ respectively such that $\angle U P T=45^{\circ}$. Determine the maximum possible perimeter of $\triangle R U T$.
+
+<image_1>"	"['Rotate a copy of $\\triangle P S U$ by $90^{\\circ}$ counterclockwise around $P$, forming a new triangle $P Q V$. Note that $V$ lies on the extension of $R Q$.\n\n<img_3428>\n\nThen $P V=P U$ by rotation.\n\nAlso, $\\angle V P T=\\angle V P Q+\\angle Q P T=\\angle U P S+\\angle Q P T=90^{\\circ}-\\angle U P T=90^{\\circ}-45^{\\circ}$.\n\nThis tells us that $\\triangle P T U$ is congruent to $\\triangle P T V$, by ""side-angle-side"".\n\nThus, the perimeter of $\\triangle R U T$ equals\n\n$$\n\\begin{aligned}\nU R+R T+U T & =U R+R T+T V \\\\\n& =U R+R T+T Q+Q V \\\\\n& =U R+R Q+S U \\\\\n& =S U+U R+R Q \\\\\n& =S R+R Q \\\\\n& =8\n\\end{aligned}\n$$\n\nThat is, the perimeter of $\\triangle R U T$ always equals 8 , so the maximum possible perimeter is 8 .'
+ 'Let $\\angle S P U=\\theta$. Note that $0^{\\circ} \\leq \\theta \\leq 45^{\\circ}$.\n\nThen $\\tan \\theta=\\frac{S U}{P S}$, so $S U=4 \\tan \\theta$.\n\nSince $S R=4$, then $U R=S R-S U=4-4 \\tan \\theta$.\n\nSince $\\angle U P T=45^{\\circ}$, then $\\angle Q P T=90^{\\circ}-45^{\\circ}-\\theta=45^{\\circ}-\\theta$.\n\nThus, $\\tan \\left(45^{\\circ}-\\theta\\right)=\\frac{Q T}{P Q}$ and so $Q T=4 \\tan \\left(45^{\\circ}-\\theta\\right)$.\n\nSince $Q R=4$, then $R T=4-4 \\tan \\left(45^{\\circ}-\\theta\\right)$.\n\nBut $\\tan (A-B)=\\frac{\\tan A-\\tan B}{1+\\tan A \\tan B}$, so $\\tan \\left(45^{\\circ}-\\theta\\right)=\\frac{\\tan \\left(45^{\\circ}\\right)-\\tan \\theta}{1+\\tan \\left(45^{\\circ}\\right) \\tan \\theta}=\\frac{1-\\tan \\theta}{1+\\tan \\theta}$, since $\\tan \\left(45^{\\circ}\\right)=1$.\n\nThis gives $R T=4-4\\left(\\frac{1-\\tan \\theta}{1+\\tan \\theta}\\right)=\\frac{4+4 \\tan \\theta}{1+\\tan \\theta}-\\frac{4-4 \\tan \\theta}{1+\\tan \\theta}=\\frac{8 \\tan \\theta}{1+\\tan \\theta}$.\n\nBy the Pythagorean Theorem in $\\triangle U R T$, we obtain\n\n$$\n\\begin{aligned}\nU T & =\\sqrt{U R^{2}+R T^{2}} \\\\\n& =\\sqrt{(4-4 \\tan \\theta)^{2}+\\left(\\frac{8 \\tan \\theta}{1+\\tan \\theta}\\right)^{2}} \\\\\n& =4 \\sqrt{(1-\\tan \\theta)^{2}+\\left(\\frac{2 \\tan \\theta}{1+\\tan \\theta}\\right)^{2}} \\\\\n& =4 \\sqrt{\\left(\\frac{1-\\tan ^{2} \\theta}{1+\\tan \\theta}\\right)^{2}+\\left(\\frac{2 \\tan \\theta}{1+\\tan \\theta}\\right)^{2}} \\\\\n& =\\sqrt[4]{\\frac{1-2 \\tan ^{2} \\theta+\\tan ^{4} \\theta+4 \\tan ^{2} \\theta}{\\left(1+\\tan ^{2}\\right)^{2}}} \\\\\n& =\\sqrt[4]{\\frac{1+2 \\tan ^{2} \\theta+\\tan ^{4} \\theta}{\\left(1+\\tan ^{2}\\right.}} \\\\\n& =4 \\sqrt{\\frac{\\left(1+\\tan ^{2} \\theta\\right)^{2}}{\\left(1+\\tan ^{2}\\right.}} \\\\\n& =4\\left(\\frac{1+\\tan ^{2} \\theta}{1+\\tan ^{2}}\\right)\n\\end{aligned}\n$$\n\nTherefore, the perimeter of $\\triangle U R T$ is\n\n$$\n\\begin{aligned}\nU R+R T+U T & =4-4 \\tan \\theta+\\frac{8 \\tan \\theta}{1+\\tan \\theta}+4\\left(\\frac{1+\\tan ^{2} \\theta}{1+\\tan \\theta}\\right) \\\\\n& =4\\left(\\frac{1-\\tan ^{2} \\theta}{1+\\tan \\theta}+\\frac{2 \\tan \\theta}{1+\\tan \\theta}+\\frac{1+\\tan ^{2} \\theta}{1+\\tan \\theta}\\right) \\\\\n& =4\\left(\\frac{2+2 \\tan \\theta}{1+\\tan \\theta}\\right) \\\\\n& =8\n\\end{aligned}\n$$\n\nThus, the perimeter is always 8 , regardless of the value of $\\theta$, so the maximum possible perimeter is 8 .']"	['8']		['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+92	"Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown below.
+<image_1>
+
+Throughout this problem, we represent the states of the $n$ plates as a string of 0's and 1's (called a binary string) of length $n$ of the form $p_{1} p_{2} \cdots p_{n}$, with the $r$ th digit from the left (namely $p_{r}$ ) equal to 1 if plate $r$ contains a gift and equal to 0 if plate $r$ does not. We call a binary string of length $n$ allowable if it satisfies the requirements - that is, if no two adjacent digits both equal 1. Note that digit $p_{n}$ is also ""adjacent"" to digit $p_{1}$, so we cannot have $p_{1}=p_{n}=1$.
+Determine the value of $f(7,3)$."	['Suppose that $p_{1}=1$.\n\nThen $p_{2}=p_{7}=0$, so the string is of the form $10 p_{3} p_{4} p_{5} p_{6} 0$.\n\nSince $k=3$, then 2 of $p_{3}, p_{4}, p_{5}, p_{6}$ equal 1 , but in such a way that no two adjacent digits are both 1 .\n\nThe possible strings in this case are 1010100, 1010010 and 1001010.\n\nSuppose that $p_{1}=0$. Then $p_{2}$ can equal 1 or 0 .\n\nIf $p_{2}=1$, then $p_{3}=0$ as well. This means that the string is of the form $010 p_{4} p_{5} p_{6} p_{7}$, which is the same as the general string in the first case, but shifted by 1 position around the circle, so there are again 3 possibilities.\n\nIf $p_{2}=0$, then the string is of the form $00 p_{3} p_{4} p_{5} p_{6} p_{7}$ and 3 of the digits $p_{3}, p_{4}, p_{5}, p_{6}, p_{7}$ equal 1 in such a way that no 2 adjacent digits equal 1.\n\nThere is only 1 way in which this can happen: 0010101.\n\nOverall, this gives 7 possible configurations, so $f(7,3)=7$.']	['7']		['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+93	"Suppose there are $n$ plates equally spaced around a circular table. Ross wishes to place an identical gift on each of $k$ plates, so that no two neighbouring plates have gifts. Let $f(n, k)$ represent the number of ways in which he can place the gifts. For example $f(6,3)=2$, as shown below.
+<image_1>
+
+Throughout this problem, we represent the states of the $n$ plates as a string of 0's and 1's (called a binary string) of length $n$ of the form $p_{1} p_{2} \cdots p_{n}$, with the $r$ th digit from the left (namely $p_{r}$ ) equal to 1 if plate $r$ contains a gift and equal to 0 if plate $r$ does not. We call a binary string of length $n$ allowable if it satisfies the requirements - that is, if no two adjacent digits both equal 1. Note that digit $p_{n}$ is also ""adjacent"" to digit $p_{1}$, so we cannot have $p_{1}=p_{n}=1$.
+Determine the smallest possible value of $n+k$ among all possible ordered pairs of integers $(n, k)$ for which $f(n, k)$ is a positive multiple of 2009 , where $n \geq 3$ and $k \geq 2$."	"[""We develop an explicit formula for $f(n, k)$ by building these strings.\n\nConsider the allowable strings of length $n$ that include $k$ 1's. Either $p_{n}=0$ or $p_{n}=1$.\n\nConsider first the case when $p_{n}=0$. (Here, $p_{1}$ can equal 0 or 1.)\n\nThese strings are all of the form $p_{1} p_{2} p_{3} \\cdots p_{n-1} 0$.\n\nIn this case, since a 1 is always followed by a 0 and the strings end with 0 , we can build these strings using blocks of the form 10 and 0 . Any combination of these blocks will be an allowable string, as each 1 will always be both preceded and followed by a 0 .\n\nThus, these strings can all be built using $k 10$ blocks and $n-2 k 0$ blocks. This gives $k$ 1 's and $k+(n-2 k)=n-k 0$ 's. Note that any string built with these blocks will be allowable and will end with a 0 , and any such allowable string can be built in this way.\n\nThe number of ways of arranging $k$ blocks of one kind and $n-2 k$ blocks of another kind is $\\left(\\begin{array}{c}k+(n-2 k) \\\\ k\\end{array}\\right)$, which simplifies to $\\left(\\begin{array}{c}n-k \\\\ k\\end{array}\\right)$.\n\nConsider next the case when $p_{n}=1$.\n\nHere, we must have $p_{n-1}=p_{1}=0$, since these are the two digits adjacent to $p_{n}$.\n\nThus, these strings are all of the form $0 p_{2} p_{3} \\cdots 01$.\n\nConsider the strings formed by removing the first and last digits.\n\nThese strings are allowable, are of length $n-2$, include $k-11$ 's, end with 0 , and can begin with 0 or 1 .\n\nAgain, since a 1 is always followed by a 0 and the strings end with 0 , we can build these strings using blocks of the form 10 and 0 . Any combination of these blocks will be an allowable string, as each 1 will always be both preceded and followed by a 0 .\n\nTranslating our method of counting from the first case, there are $\\left(\\begin{array}{c}(n-2)-(k-1) \\\\ k-1\\end{array}\\right)$ or\n\n\n\n$\\left(\\begin{array}{c}n-k-1 \\\\ k-1\\end{array}\\right)$ such strings.\n\nThus, $f(n, k)=\\left(\\begin{array}{c}n-k \\\\ k\\end{array}\\right)+\\left(\\begin{array}{c}n-k-1 \\\\ k-1\\end{array}\\right)$ such strings.\n\nIn order to look at divisibility, we need to first simplify the formula:\n\n$$\n\\begin{aligned}\nf(n, k) & =\\left(\\begin{array}{c}\nn-k \\\\\nk\n\\end{array}\\right)+\\left(\\begin{array}{c}\nn-k-1 \\\\\nk-1\n\\end{array}\\right) \\\\\n& =\\frac{(n-k) !}{k !(n-k-k) !}+\\frac{(n-k-1) !}{(k-1) !((n-k-1)-(k-1)) !} \\\\\n& =\\frac{(n-k) !}{k !(n-2 k) !}+\\frac{(n-k-1) !}{(k-1) !(n-2 k) !} \\\\\n& =\\frac{(n-k-1) !(n-k)}{k !(n-2 k) !}+\\frac{(n-k-1) ! k}{k !(n-2 k) !} \\\\\n& =\\frac{(n-k-1) !(n-k+k)}{k !(n-2 k) !} \\\\\n& =\\frac{n(n-k-1) !}{k !(n-2 k) !} \\\\\n& =\\frac{n(n-k-1)(n-k-2) \\cdots(n-2 k+2)(n-2 k+1)}{k !}\n\\end{aligned}\n$$\n\nNow that we have written $f(n, k)$ as a product, it is significantly easier to look at divisibility.\n\nNote that $2009=41 \\times 49=7^{2} \\times 41$, so we need $f(n, k)$ to be divisible by 41 and by 7 twice. For this to be the case, the numerator of $f(n, k)$ must have at least one more factor of 41 and at least two more factors of 7 than the denominator.\n\nAlso, we want to minimize $n+k$, so we work to keep $n$ and $k$ as small as possible.\n\nIf $n=49$ and $k=5$, then\n\n$$\nf(49,5)=\\frac{49(43)(42)(41)(40)}{5 !}=\\frac{49(43)(42)(41)(40)}{5(4)(3)(2)(1)}=49(43)(14)\n$$\n\nwhich is divisible by 2009 .\n\nWe show that this pair minimizes the value of $n+k$ with a value of 54 .\n\nWe consider the possible cases by looking separately at the factors of 41 and 7 that must occur. We focus on the factor of 41 first.\n\nFor the numerator to contain a factor of 41 , either $n$ is divisible by 41 or one of the terms in the product $(n-k-1)(n-k-2) \\cdots(n-2 k+1)$ is divisible by 41 .\n\nCase 1: $n$ is divisible by 41\n\nWe already know that $n=82$ is too large, so we consider $n=41$. From the original interpretation of $f(n, k)$, we see that $k \\leq 20$, as there can be no more than 20 gifts placed on 41 plates.\n\nHere, the numerator becomes 41 times the product of $k-1$ consecutive integers, the largest of which is $40-k$.\n\nNow the numerator must also contain at least two factors of 7 more than the denominator. But the denominator is the product of $k$ consecutive integers. Since the numerator contains the product of $k-1$ consecutive integers and the denominator contains the product of $k$ consecutive integers, then the denominator will always include at least as many multiples of 7 as the numerator (since there are more consecutive integers in the product in the denominator). Thus, it is impossible for the numerator to contain even one more\n\n\n\nadditional factor of 7 than the denominator.\n\nTherefore, if $n=41$, then $f(n, k)$ cannot be divisible by 2009 .\n\nCase 2: $n$ is not divisible by 41\n\nThis means that the factor of 41 in the numerator must occur in the product\n\n$$\n(n-k-1)(n-k-2) \\cdots(n-2 k+1)\n$$\n\nIn this case, the integer 41 must occur in this product, since an occurrence of 82 would make $n$ greater than 82 , which does not minimize $n+k$.\n\nSo we try to find values of $n$ and $k$ that include the integer 41 in this list.\n\nNote that $n-k-1$ is the largest factor in the product and $n-2 k+1$ is the smallest.\n\nSince 41 is contained somewhere in the product, then $n-2 k+1 \\leq 41$ (giving $n \\leq 40+2 k$ ) and $41 \\leq n-k-1$ (giving $n \\geq 42+k$ ).\n\nCombining these restrictions, we get $42+k \\leq n \\leq 40+2 k$.\n\nNow, we focus on the factors of 7 .\n\nEither $n$ is not divisible by 7 or $n$ is divisible by 7 .\n\n* If $n$ is not divisible by 7 , then at least two factors of 7 must be included in the product\n\n$$\n(n-k-1)(n-k-2) \\cdots(n-2 k+1)\n$$\n\nwhich means that either $k \\geq 8$ (to give two multiples of 7 in this list of $k-1$ consecutive integers) or one of the factors is divisible by 49 .\n\n- If $k \\geq 8$, then $n \\geq 42+k \\geq 50$ so $n+k \\geq 58$, which is not minimal.\n- If one of the factors is a multiple of 49 , then 49 must be included in the list so $n-2 k+1 \\leq 49$ (giving $n \\leq 48+2 k$ ) and $49 \\leq n-k-1$ (giving $n \\geq 50+k$ ). In this case, we already know that $42+k \\leq n \\leq 40+2 k$ and now we also have $50+k \\leq n \\leq 48+2 k$.\n\nFor these ranges to overlap, we need $50+k \\leq 40+2 k$ and so $k \\geq 10$, which means that $n \\geq 50+k \\geq 60$, and so $n+k \\geq 70$, which is not minimal.\n\n* Next, we consider the case where $n$ is a multiple of 7 .\n\nHere, $42+k \\leq n \\leq 40+2 k$ (to include 41 in the product) and $n$ is a multiple of 7 .\n\nSince $k$ is at least 2 by definition, then $n \\geq 42+k \\geq 44$, so $n$ is at least 49 .\n\nIf $n$ was 56 or more, we do not get a minimal value for $n+k$.\n\nThus, we need to have $n=49$. In this case, we do not need to look for another factor of 7 in the list.\n\nTo complete this case, we need to find the smallest value of $k$ for which 49 is in the range from $42+k$ to $40+2 k$ because we need to have $42+k \\leq n \\leq 40+2 k$.\n\nThis value of $k$ is $k=5$, which gives $n+k=49+5=54$.\n\nSince $f(49,5)$ is divisible by 2009 , as determined above, then this is the case that minimizes $n+k$, giving a value of 54 .""]"	['54']		['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+94	"A regular pentagon covers part of another regular polygon, as shown. This regular polygon has $n$ sides, five of which are completely or partially visible. In the diagram, the sum of the measures of the angles marked $a^{\circ}$ and $b^{\circ}$ is $88^{\circ}$. Determine the value of $n$.
+
+(The side lengths of a regular polygon are all equal, as are the measures of its interior angles.)
+<image_1>"	"['The angles in a polygon with $n$ sides have a sum of $(n-2) \\cdot 180^{\\circ}$.\n\nThis means that the angles in a pentagon have a sum of $3 \\cdot 180^{\\circ}$ or $540^{\\circ}$, which means that each interior angle in a regular pentagon equals $\\frac{1}{5} \\cdot 540^{\\circ}$ or $108^{\\circ}$.\n\nAlso, each interior angle in a regular polygon with $n$ sides equals $\\frac{n-2}{n} \\cdot 180^{\\circ}$. (This is the general version of the statement in the previous sentence.)\n\nConsider the portion of the regular polygon with $n$ sides that lies outside the pentagon and join the points from which the angles that measure $a^{\\circ}$ and $b^{\\circ}$ emanate to form a hexagon.\n\n<img_3516>\n\nThis polygon has 6 sides, and so the sum of its 6 angles is $4 \\cdot 180^{\\circ}$.\n\nFour of its angles are the original angles from the $n$-sided polygon, so each equals $\\frac{n-2}{n} \\cdot 180^{\\circ}$.\n\nThe remaining two angles have measures $a^{\\circ}+c^{\\circ}$ and $b^{\\circ}+d^{\\circ}$.\n\nWe are told that $a^{\\circ}+b^{\\circ}=88^{\\circ}$.\n\nAlso, the angles that measure $c^{\\circ}$ and $d^{\\circ}$ are two angles in a triangle whose third angle is $108^{\\circ}$.\n\nThus, $c^{\\circ}+d^{\\circ}=180^{\\circ}-108^{\\circ}=72^{\\circ}$.\n\nTherefore,\n\n$$\n\\begin{aligned}\n4 \\cdot \\frac{n-2}{n} \\cdot 180^{\\circ}+88^{\\circ}+72^{\\circ} & =4 \\cdot 180^{\\circ} \\\\\n160^{\\circ} & =\\left(4-\\frac{4(n-2)}{n}\\right) \\cdot 180^{\\circ} \\\\\n160^{\\circ} & =\\frac{4 n-(4 n-8)}{n} \\cdot 180^{\\circ} \\\\\n\\frac{160^{\\circ}}{180^{\\circ}} & =\\frac{8}{n} \\\\\n\\frac{8}{9} & =\\frac{8}{n}\n\\end{aligned}\n$$\n\nand so the value of $n$ is 9 .'
+ ""The angles in a polygon with $n$ sides have a sum of $(n-2) \\cdot 180^{\\circ}$.\n\nThis means that the angles in a pentagon have a sum of $3 \\cdot 180^{\\circ}$ or $540^{\\circ}$, which means that each interior angle in a regular pentagon equals $\\frac{1}{5} \\cdot 540^{\\circ}$ or $108^{\\circ}$.\n\nAlso, each interior angle in a regular polygon with $n$ sides equals $\\frac{n-2}{n} \\cdot 180^{\\circ}$. (This is the general version of the statement in the previous sentence.)\n\nConsider the portion of the regular polygon with $n$ sides that lies outside the pentagon.\n\n<img_3345>\n\nThis polygon has 7 sides, and so the sum of its 7 angles is $5 \\cdot 180^{\\circ}$.\n\nFour of its angles are the original angles from the $n$-sided polygon, so each equals $\\frac{n-2}{n} \\cdot 180^{\\circ}$.\n\nTwo of its angles are the angles equal to $a^{\\circ}$ and $b^{\\circ}$, whose sum is $88^{\\circ}$.\n\nIts seventh angle is the reflex angle corresponding to the pentagon's angle of $108^{\\circ}$, which equals $360^{\\circ}-108^{\\circ}$ or $252^{\\circ}$.\n\nTherefore,\n\n$$\n\\begin{aligned}\n4 \\cdot \\frac{n-2}{n} \\cdot 180^{\\circ}+88^{\\circ}+252^{\\circ} & =5 \\cdot 180^{\\circ} \\\\\n340^{\\circ} & =\\left(5-\\frac{4(n-2)}{n}\\right) \\cdot 180^{\\circ} \\\\\n340^{\\circ} & =\\frac{5 n-(4 n-8)}{n} \\cdot 180^{\\circ} \\\\\n\\frac{340^{\\circ}}{180^{\\circ}} & =\\frac{n+8}{n} \\\\\n\\frac{17}{9} & =\\frac{n+8}{n} \\\\\n17 n & =9(n+8) \\\\\n17 n & =9 n+72 \\\\\n8 n & =72\n\\end{aligned}\n$$\n\nand so the value of $n$ is 9 .""]"	['9']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAmwGCAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiijzAo39szhZogDImcj+8vp/n39aqqRNGMfeHT3Hf/AD61rhcVlXUP2a63x4CSkk+zYz+uD+IPrXy3EGWpxeLgv8X+H/NG9KfRkVFPkAzkDhgD9KZXwc4uMmmdIUJE9zJ5MeVH8b4+6uOce5H+NNdiqk4yegHqTWra232eHaTliSzn1J/zgegFe5w/laxlZzq/BB6+b6Izq1OVWJlRUUKoAUDAAHQelOoor9IX9eRxhRRRTA83vMn9oXTw5+VfDzMn+8ZmB/SvR81594ujOmfE7wbrp+W3maXS5295FJiH4sDXezRmWCSNZDGXUqGHVeMZFAHm3h2TQ9H8VeIYfCNpqOs3t5debqLmZVtraTLErvbHOWbhQ56+nFDwy93/AMNC+IVuoIbeSbSUkkjhmMqkhoVU5Krk49uKufDnStZ8AaRd6DqGiXN1i6eaC8sGjdJwQB825gUPH8XFQtoHifTviTeeKU0w3iavpr2hiiuEH2KTK7NxZhldsaklc8lsA8ZAPVs5FeceAuPiR8REU/uheWzD/eKNmu7063ax0u0tpJTI0EKxtKx+9tABJ/KuL+FkRvLbxB4kK/LrWqzS27f3oEOyP+TUAeg1HNPHbwyTSsEijUuznoABkn8qkrgfiHN4s1DSbvRNA8NPcpcoIpbyS8hjTy2+8FUsGyRlckDrQBD4VvPFCaZLqUXhu3m/tW4bUDJJqPlsVcDywV8s4IjCLjPauK0ye+8IftAB9RsY7C28SRkGKOfzU8xuh3YGT5i+nG+vZ9Iurq60yOW90uTTJ8YNs8iPsx6MhIxXmXxb8O+JfFl1py6D4em+06XcGWK/e7gQMCAcKpfd1VeuOlAHr2aWsXQNT1a/tyNW0KfS50RSd88UiOxznaUYnjHcDrW1QAV5T8b7Zbqx8LW7HCza5DGSvBAIYHB/GvVq8v8Aibp/iXxFd6PBpHhueeLS9RjvWne7gjWUKM7VBfcM5PUDpQB6FLo+mzXdtdyWFs9zaqVgmaJS8QPXa2Mj8K8j+G2oarqXjvxzqwsY7y6+2Lb7pZ/LaGMM4VB8p4wq/wDfNetaVqF1f2hmvNKuNOkD7RDPJG7EYB3Axswxye+eOlcX/wAI3qfhHxzf+ItCsvt+mavhtRsY3VJY5ASfNj3EK3VsrkH5jQA7xJ4f8Q6/4g8NalBY2llJpN8JXlN2WZoW4kQYTuB+tZXiEjRPjpo+uarKiaXc6c9rFPIwC27gnO4/wglwMnjLe1dk11q2tz28UNjd6XZRypLPPO6pJKFO4RoqsSASAGJxxkDOcjmvEmqa2vj1m8IWsGr3MNmLbU7SdtkcA3F4z5hIG85bK8nGOKAKniDPij4teFH0KVJ10fzZr+7gbckaNt2xsw43MARt64bOMc13/iJf+KZ1XPP+hzf+gGuHbxd4+0aAzah8PY3sYhmT+z79HdR6hBkn6V22n6hYeLvC6XlnIXstQgYK2MEAggg+45GPagDlvgnhvhHon1nP/keSs3wvj/hf3jhRgZtbU/8AkOPP86v+AYdY8GeFIfDl/oV7czWUsiwz2pjaK4VnZwwJYbfvYIbGMVi6BpvirQfizr2sXOgT31rqaRKZ7aaMLEcKSPnZSwXlc8E7c45xQBe+K/8AyM3w9/7D0X/oSV0XxFsvD174Mu4PE97JZaZuVmmibDhgcrt4OTxjGD1P1HOfEPTvEuu+IvD02meGria30XUUu3la6gQTqCpwgL5HQ/eAqb4ieG9a8ZaBo93ZWJivbC8W5k0q7mQGYAkbSyMUBxz1xhj34oAXWb291D4YapBZ+Hp4NMTSZFik1K5EcxjERAbYquc8Z+YqT3xWp8J5Hm+FmgM7FmFuVBPorMAPwwPyqbVptS8VaDdaRb6Td6c19A0M894E2wKy4bAVjuYgkDHHvWJ4F0vxFomhaXY3+kTxtolvcoUSeIreuzZTy/m6bd2d+3BI98AFPwdaJJ8dvHVyfvRRwKvp8yqSfr8v6mvS7HSrDTFlWwsra1EzmSUQRLGHc9WO0DJ96808PWvivSPiB4k8QXHhC7a11YReXHHe2xkj2DHzAyAc+xr1egAooooA8o+J8aSfEj4bo6hl+3TcEZHBhNen3NrDeWs1rcxrJBOjRyI3RlIwR+VeV/Fn7SfH3w7Nn5X2kXk/lCXO0sDDgEjoD0z2rr9Y1vxDNpklro3h67TVZV2JJcyRLDbkjG8sGO4D0UEn0FAHn3wm1y9074f+LrVJDMmhNM1mTz0Rm2j23Ln/AIEa0PgxLqq+AEuoNNiu2vLueee4lu9jSSbsHI2n09fWuw8B+CLXwb4V/sguLqWdjJdysOJXYYIx6AAD/wDXWJ4X0XV/h1Nd6XDp8+qeHpp2mtJLdlM9qTjMbqxG5eOqk/TngAt23h7XH+KkXih7a1s7N9PNpdRLcGR5GDEq33QP7o9sGsPV7SO5/aU0USAFY9GMu0jIZg8uMj2JBHoQPQV3dm+qalqkN5PbzadYwKwjtZJF8ydjgbnCkgKBnC5JycnbgCuFubPxZN8WrTxavhC8FlBYGzMJvbXzSSWO7HmY/ixjPagCf4+jb8NGkH30vYSrdween510vxIGfhp4hP8A04Sfniud+LWmeIvFnhkaHpPh6eV3linad7mBUXAOVwXySCcdMe9dRe2t74t8Fanpt7p02k3F1byWwjnkjlwSoAfMbMCMn2PB46UAQfDuWKD4Y+HpJpESMWMQLMcAEjA/U4rO+Jdv4Wlh0i48RXF4kttdh7GCyOZbiTI+RUwd2fl9MZHIzVXw7Y64PBml+FNQ0We2ns5IUuLoujQGKOQPlSGJYsqgYxwW56cr4z8M6xN8QPD3i7T7U6lb6arRzWIkVHAO750LEAn5umR91fWgDI+MN1qmpfC++nu9DSyt1khdftF0pnUmRQDsRWXvj7/Ge9d7fW95rHw+uLa1m2X13phSOQtjDvFwc9uT1rnfHmmax498JXek6fZT2SsolLXhVTM6nKxBQxI5GSxwBgdcnE1pcTWPh++1TxLHLolnHpUNk8csquVZd+91CE5yXUL3O3oOKAMv4aeKNC0j4XWNvqF9BaT2EbrcW0rgSZLFgVXqwYMCMZB3cZq78GdD1LRPACw6lE8Dz3T3EUEgw0UbbcKR1HQn8ay9H1H4l2mkWUZ8G6ddi2hEUU81ysEjxqMAlSfkJABI7HPArf8AD/jy/udfi0DxN4em0PU7hGe1JnWaKfaMsFccBgOcc8UAc/8ACq8l1rxv471e7Ba5+2JbJu6xRKzhUHpwFz7jPWtn4zwo3wt1WY/LLbNDNC44MbiVACCOhwSKIvDmoeEfGup6/otp9u0zWAGvrKJ1WaKUEnzU3EKwO5sjIOWyM9KTxZp+sfEGyi0KPTrnStIkmWS9urtkEkiIc+XGisTkkD5mwBjvQBT8T6jPrH7Pk+o3Y/0i50qKWU4xlztJP4nmul+HdpHbfDnw8Ix/rNPhmYnqWdQ7H8yazPH+manP4HuPDPh/QZblZ7ZYI3SeKOOEAjAO9gTwOwNWvCdzreleCbOxvfDN4l5ptpBbLElzbt9pKqFJQ+ZgYA3Hdj0GTQBz/wAIyf7c8fRDhE16bA9Msw/oPyr1KvLfh1pvibQPEHiCTU/DU8VvrWovdrMl3A4gVmY4cB8nqPug16lQAV5F8cLqWZ/CmgsxWy1PUgtzg4DKrIAD7fPn8BXrtch8QvBi+NPD62sU4ttQtZRcWdweiSD19iP6HtQBv6npNpq2i3WkXKEWlzA0DpGdpCkY49MV5J8WLGLwf8MdH8LaQ0sdhPfLBI0j7mZCWkIJ92OeMdMV6Bp/iDXxYpBqPha9GpouH8iWEwSNj7yvv4UnnBGR6GqfiLwPJ4s8ES6Rq17nUJJDdJcLkrBMSSAvfYAdnuOetAHYxQR28SQxIEjRQqqowAo6D8K8q8OXcmn/ALQninR4ABZXdrHdvEOAsoSL5vqS7Z9c+1ddp+v6/a6fHbar4avZ9TjUKz2TxNDOwGN4ZnXaD6MAR71U8G+Eruw1zV/FOtmI61qrANFE25LWFQNsQb+I/KuTxnaKAOfnH/CFfHOCcfJpfiqHy37AXSdD9TkfjIa7OFF1bxtPdEbodHi+zRE8jz5QrSEfSMRjP+2w+uX8U/DF14n8IMumqTq1jMl3YlSA3mKeQCe+0n8QK6Hw3p0+maDbw3bB76TM9246NM5LOR7biQPQAelAHl/h7UNT1L46+KrkWiXclhb/AGS3jlm2eVHuXlflPXBP/Aq6Px1oXibxXpljFZ2FnZ3tlfRXkNw16TsK59Ez3H4gVLq3hfU9I8e/8Jn4dt0vGuYhb6np5dY2mXjDxs2F3javBIzjrzWzcXura5D9itNMvtLjlwJ7y4KI8aZ+YRhWJLnkA8AZzk9KAOS+PaM3gK12uVf+04QrA8qcPyPevQzoOlMNPEmn20h04AWbSRKxgwABsJGV6Dp6CuD+LWmeI/E+kw6Ro3h+W4EV3HcG6a5hSNgoPygM+7OTjkCu60jUr3UIHe90e40yRSAI55YpN3HUGNm4+uKAPONTtfHXgzxPq2peGtPttc0fU7k3U1sXxLDJgKwGCOuOOG6dBjnV8F+NLDxV4mmjvtHu9G8S2lo0b2lz/FCWB4OATg46gfe4zU/h7/hIdBv9e/tDSZrnS7vVJprQ27o8saM3VkJHyHqMEnrkc1ch0q61jx3a+IrnT3sYLC0e3hWZlMszOyksQpICqBwCc/MTgY5AOyooooA4b4vqB8Ktf9PJTj/tota/giJIvAfh5I0VB/ZtucKOATGpP6n8ayfi/wD8kp1//rin/oxaZ4Q1bUtP8CaLFPot1duNPg8mSy2Msq+Wu0fMwKkDAOeOOCaAOYhmfQv2lZLGyO231qyEl1Gv3d6ozBseuU/8ePrWxdnw9pfxVutQ02LUtU8U3FqI57K1dfKhj+Ub5GbAThVHJPUfLkirXhLwfqC+LtR8aeJBEmq3ieTb2kT71tIuBt3YAZsAcjjr61n+HNC1nwX458TX1xps+pWGtXH2iK7tGVniO5z5bozA/wAeMrkcfkAZ+uTX/wDwvLwVcXllBaTSQXEe2G4MpZdjn5vkXGCxPfvXTfF0D/hVWvDAx5SfT/WL/WsjxJoviK+8b+H/ABhb6VLLFp0jRNpoljE/lMpBkyWCbssfl3dADnrtk8X6X4jvvh3eaBZ6JNe3momSd5ftUKxwF7gy7CWcE7QcDAI460AdD4W0DTLr4e+HrO9sbe7iS0gn23EQkHmlQxfnPzZZuf8AaNcv8cuPD3h//sO2/wD6BJXXeDJtWj0Kx07VdCuNPks7SKEyPPDIkjKoU7djE9s8gVmfFLwxqHirwvBDpQRr+yvI72GJ22iQoGBXJ4Bw3f0oA7NpYklSN5FDuTsUnBb1xU1cxaPfa5rOnX9xpVzp0NlFIzLdFNzSuAuFCseFG/JOM5GM846egAryb4130jSeFNBbP2LVNTVboZwHVWQbT7fPn6gV6zXJePvBy+MtCjtoZxbX9pMtzZXBGRHIvTPsR/Q9qAOpC4wBwOwryjwBeS2fxf8AHGhQnFi0ovQnOFlJXcR/vbuf90V2EPiLXYrER3XhS/fU1XayQSwmB2A6iQuMKfcA+1UvBfhO68ODWNc1Lbda7q0rT3CQH5EAJKwoWx0zjJxnj0oAxtCH/GRHigDj/iVw/wAoq9P215Tp1p4ss/ipq/iqTwfdmyvrRLdIlvbbzVK7OSPMx/Ce9eqxyM8asyFGIBKEgke3HFAD6rXYiNnOLlgsHlsJGLbQq45JParNYXi/R5/EPhLVNItbgW813A0ayHOAffHY9D7GgDj/AADPp2naNJpXgnT7/U9MjuHc6jfzrFC0hxwh27mAxj5UwCOpqn8H2lXxB49t3jWMJrDMYo3LJGxZwQCQCRwOcDOBW14Kl1jw/wCEbHQrzwzdi+so/JHlSRGGXn74ffwDnJyMj0NYmheEvEuia34jtJbdriPXLuC7OpQSqqRHfumUqW3jqwXAOeOnOAA+KFrHefEr4dW8w3RPdT717MAYjgjuDyD7Gum+K6Kfhb4h3AN/owPPqGUj+Vc94z0/xRrHjvw3q1j4XuXs9DuJWdmu7dTcKxUAoN/AwufmweeQK2vHw13X/BF1pWneG7t7rUINrB7m3UW53Dhj5nJwP4cj3oAxde1e80j9nO1vrV3W5Oj2cXmd1EixoWz64Y4PqRXW/Dy0hs/h14djgVVU6fBKQP7zoGY/iWNZ2gaNdat8O4/CviPRJ7BIdPhs3czRSCQqm3ehRjggqrcgckdcGm+GG17whokOg6npN1qUVknl2t9YFGWWIfdDIzBkYDA7jj71AHP+J7uTR/j94YmtAQdStDbXQQ48xNzYJ9cHB/4CO3FTeOBj40eAB2/0j/0GtjS/DGoav4+/4TPX7dbVraD7NpthvV3hXnc8jLld53NwCcA9eKb488Panc+J/DPifSrU3kukTP59qrqryROACU3ELkc8EjOaAO686MzeTvXzQu8pu+bHTOPTNeVT2kd1+0tCZRkQaN5qj3yQPyzn6gV3OmLdXWtXutXFjLao9tHbW8MxXzCFZ2ZjgkKCWAAJz8vOM1wws/Fg+LZ8W/8ACIXf2I2H2Pyfttt5uc7t2PMx7YzQBP8AHjMfgnT5VJDLq8DKR1B2vzUHx91C5t/Bljp8DmOLUL5IpyDjcgBbbn6gH8Kt/FrS/EfirR7fR9H8PTzeVdx3RuHuoERgqnKgF92ct3AHFbfiTw83xE8FT6dqen3GkXBffB5zxymKRRw+Y2YEYJU8g4JoA1YZtZtraO2h0C1SGNAiIL7gKOAPudMVkfDbwzqnhXQL3TdSMIja+lmtIopS4ihbBCcgdG3H8am0jXPEVppsdnrfhy8m1KFdhmsniaG4IGAwZnUrn0YDFX4pNZ0zS7i8ns5tTv7ibzPsdtOgSBcABFaQqNoCgk9SWJxzigDhPgXpttL4O1p5oklFxqMsEiuMq0SgYQg8EZdz/wACNep2Wn2um2kdpZW8VtbRDEcMKBVUdeAOOvP515v8KtP8TeFdLl0jV/DVxGtxeyXH2qO6gdI1YD7wD7uMdga9RoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiikz7UALRSZpaACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACqeorutOgyrqR+Y/pmrlZmvzPBod1Mhw8ahx+BFYYqk6tCcF1TX3pocXZ3Im/1SH3Ix/n60yvI4fiN4jMYM8MTIOQY7N0545zIVHp+dQzfFTVIxndp0eP+fiaJc9f7rse1fC1OGMdNq1rWS37JeTOpVkeyRLvuoFIJG/J9sAkfqBWzXjvgXx9f+IvF1np88+nujGR/wDRWdsgI2ASVAPJHftXsVfU5HgamCw7hVtdtv8ArRGFSXM7oKKKK9gzCiiigDJ8R6Bb+JNCudLuHaLzQGjmT70MgOVdfcEA1H4e1C+u7I2+q25g1O1Pl3GFPlyntJGehVuvqDkEAitqm7eepoANo/TFKFxzmlooA5nxab7UrX/hHtMEkc1+hW4vAp22sB4ds93IJVQOep6KTW5p2n22ladbWFnGI7a3jWKJB2UDAqwVz3P4U6gApNvvS0UAJtGc96NtLRQA3bTqKKACk28cUtFADdvv160bPenUUAIFAGK42DQdW8OeI9U1PRo4b6x1WVZ7i0ll8uSKXABaN8EMCB904we+OK7HcM4pdtAHNXl/4mvrZ4NM0VLKZwR9pv7hGWP3CRlt/wBDt+tXPCnhy38J+GLHRLaV5Y7VSPMcYLszFmPtkseK2QuAOaWgBu3P0+lGzpzTqKAEKg8Gk2c9adRQA3bxjNGwfh6U6igBMe9LRRQAUUUmfagDzHxj4f8AF3iDxjoGq2em6clvod1JLGJb4h7lWZfSP5MhPfrXo1k9zLZxveW6W9wy5eJJfMCn0DYGfyFT8dPSlxQAY/OjbS0UAJtFG3BpaKAGlc9TmjYD15p1FADdvvzS496WigBuzrWH4u8Mx+K/DVxpLXD2zOVeKZBuMbqwZTg9RkdPSt6igDm7XUfEcEKxahoKXNwoAM2n3SeW57ttkKsuT25+pqiNA1TXfFuma7rawWdvpQkNnZQSGR2dxtLSPgDgdFXPPUnpXYlc96Me9ACFc9/0oCAdDinUUAN28Yzx6UBBTqKAEC+//wBeloooAKTaKWigBpXjrS7felooAbt6ZNAXAxmlzRu60AJt560oFLRQAm2k2+hp1FADdvvS7felooAaVz3o2+9OooAKKKKAOK+I+la94j8NXehaRZWjx3kah7me6MflkODjaEO77vqOtaPg231mx8PWel6zY21u9jbRW6SQXJlEu1dpblV29Bxz1ro8e9G33oANoxSFQadRQAm3r70BcUtFACBQABSFQQQehp1GaAG7PenUUgPtQAtJto3cZpaAG7RijZnvTqTNABtpAuBjPFOooAKTb70tFADSpPegoCadRQA3YKXbS0UAN2DOc0u3jFLRQAm2gLilooAbtA6Uu0ZzS0UAN2+9G33pc8ZpaAE20mz3p1FACbeaWiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACikz7UhbH/wBegB1JuFcR4k+LXhHwyWim1EXt2OPs1j+9fPoSPlB+pFcyfEHxQ8aHGhaND4a058/6XqJ3Slc9lI4yP9j/AIFQB6fqmtabolobrVL63s7cceZPKEGfQZ6n2FecXvxmg1G6aw8FaDf6/djH7xYmjhT3JIzj6hR71NpXwV0lrsaj4q1K+8Raj3a6kKxjByAFBJ49CxHtXo9jp9nplqlrY2sFtbp92KGMIo/AcUAeb+DfiBrQ8US+F/HNjBpurTAS2ZiI8qRSB+7BBYZyDyCcnI4I59P3e30rj/iH4Hg8baH5aOINVtW82wuxkGN+uCRyFPfHoD1FZvw08cT63DcaBr0f2bxNph8u5ifjzlBx5g/TPuQRwwFAHolFJmloAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigArI8S/wDIuX/b90a16qalYjUdNnszIYxKhXcBnFAHyY8FsjBjo+0g4zcpHF+ssjfy9adE05x9mitYzjkxTo3p/wA8Ys9q1vFPgTxv4f1c29knn2smTFNaW6IMHs2xR6enf3rBuPB/juZTI51CZsZKF5Qf/HgBRYDv/hgL4fEKwa7kZgY5gFxcAfcP/PU4/IV9C1438I/hdL4dnTX9WZ/7QZCEjyflVhgg+vX/ACBlvZKACiiigAooooAKKKKACiiigAooooAKKK4rxX4+n8J6jZxXegXD2F1dR2sd/wDaEWPewycjlhgbuoGdpoA7WiuKm+Itvb+I9F0ybS7pINZZls7gnlsYwzJjhTnjnOOSBWjq3i6Oz16Hw/p1m+oaxLF55gVxHHDFnG+RyDgZ9Ax9qAOkormIvFslr4itdD12wTT7q9UmzlinM0M5X7yByikMPQgZyME1Y8QeLLTQbmwsTBLd6nqEhjtLOEjfJjqxJICqBySaAN+kzXH3nje50zWLPRtQ0UwahqBK2LfalNvKw6hpMAqRxxtJORjJp58Va5Dod7fT+E7lri0uZIWhiuUxJGgz5yM+3cp7cZoA66iuZ8I+L18XeEU161sJIvMMgS2MgLMUJGM8DnFUfDvj2XW/GGoeGrvQrjTbyxtxPJ506PkErgDbkdGB60AZnxB8T+KfCd5ptxaNpUmk3t/FZhHgfz0LDJ537SPlbtxxXo9eX/G3jRPDfPB8QW3/AKDJXoGtazZ6Bo91qmoSeXa2y7nYDJ64AA9SSAPc0AaFFcheeLtUsdE/t5tBhuNIEQnZrPUFlnEWM79u0I2BycSH2J61u2etW+q6EmraUDewTQ+bAqEKZDjhfmwFOeOcYPXFAGlRXBaV8Snv9f1HQrjw9d2uqWbRotsZkdpGcFhyPlChRuLZIx74B1PB/jWDxadUjSyltp9NumtZlY7lZhnlWwM9O4BoA6miiigAorP1e+vbCxM1hpcupT7gPIilSMkeuXIFcl4U+JTeMrCSXStBuTcxTGOSOSZVjiHGC7kdTz8qhiMc4BGQDvaK5XTvG8E3iX/hHNWsZtL1do/NgilYPHcJ6xuOvQ8EA8H0Namv+IrDw3pjX+ou6x71jjRF3PLI33URe7HsKANaqt8t2bOVbB4Euiv7tp0LID7gEE/mKxZNc8QwWZvZPDCtAF3tbw3we6A9NmwIW9hIfYmtLRdcsPEOj2+qaZOJ7ScZVh2PQgjsQeCKAOR+H/inXtc8Q+KNJ137D5mjzRRI1nG6K27fk/MxP8Ir0CvLvhxz8TPiR/1+W4/SWtXUviVLpXiS00C48Lap9vvM/ZVEsGyYDPIbfgdOhwRxQB3lFYHirxN/wi2gvq8mmXV5DEQZ0t2TdEp/iIYjIzgcetPsfEfm6BPrGrafcaNbwqZGW9ZNwjAB3EKTjuMHnjpQBuUma40eOrl/Dx8SReHrp9FCed5nnKJ2hH/LURf3cfNgsDjnFWNK8WX2r3ekzWuiNJouppI8d9HcBzCFXI81QuF3HgAMeQc4PFAHV0Vzk/ivz9cn0XRLL+0b62UNdO0vlQW2egd8E7jg4VVY8c4pqeLGs9ZtdK12wGnz3hK2k8c3nW8z9dgfapD+zKM9s0AdLRXH+OPG9z4Lsmv28P3N7YIqmS6jnjRUZm2hSCd3XHOMcjmri+Jb6+tGutG0Vr2BQf3j3KxCRh1WPIO7ByMnaCRwTQB0lFYfhXxTYeL9Cj1XT1kRCzRyRSjDxOOqsPX/ABFblABRRRQAUUVn61rNnoGj3WqahJ5drbLudgMnrgAD1JIA9zQBoUVyF54u1Sx0T+3m0GG40gRCdms9QWWcRYzv27QjYHJxIfYnrXQaZrFlq+kW+q2U6SWU8fmJJnAx3z6Y7/Q0AX6K5rT/ABHqOuWaaho2lW8unS58mW6vGheVQSNwQRthTjIyQSOwqx4b8Rt4gTUjJp01jJYXr2bxSurMWVEYn5eMfPxzyAD3oA3aKKKACis/Wtas9A0i41O+Zlt4ACdoyzEkBVUdySQB7mqE2r6/b2j3knh+FokQyGGG+3XBXHQLsCFvbfj0JoAPFlxrtlok93oUmnrLbxPLIt7G7BlVScDYwweOvNVPh3r954q8CaZrWoCJbq5EhcQqVUbZHUYBJ7KKlbVl1/4cSassJhF5pTTiMtnZujJxnjNcf8ItZnX4ZaPY6bYNeXEKS+czSiOKImaQhWbBO4gg4AOAQTjIyAeq0VzfhrxfB4gvtS02S0msdU0yQJdW0pDYz91lYcMpHfj6VNrnim20e9tNNjglvdVvSfs9lCRuIHV2J4VB3J/AGgDeorldQ8WXvh+BLzxBpEdvpxYLJdWV0bgQEnAMilEIXJxld1R+L/HieELFdSm0e9vdLKqTeWjxMgLHAGCwPPHOMc0AddRXIXvja7sPC6a/L4Y1Jrby2nljjlhaSKIAEOQH5BBJ45GDnFW9C8VT6/4abW7bQrxEePzLaB5Yt9wvbGGwv/AiKAOkorkPD3je58S2V3d2nhjU4ooA6obh4k86VG2mNfm65yMnAyCM1R0T4lnVtV1TSZdAu7XU7GSOJbQzI7yswYnkHaFULktnHI74BAO9orlP+Ezex8T2Gha3pbWM2oqxsp45hLFIygFkJwpVuR2I5611dABRRRQAUUUUAFFFFADJJBHGztwqgk/QVwHhfXfH/iPRLbVXsNBsYLlBJCkrTFyh5DEDgAjBHNd1e2zXdhcWyyeW00TRh8Z2kgjOO9JYWSafp1rZxn5LeFYlIGOFAA4/CgDzK08e+MLv4j3fgwWWhrc2sXmvcEy7CuxWGBnJPzj9a1de1/xx4cjgvbmx0W60/wA+KO4e2aUSRKzBd209smuV0j/k6jXR/wBOS/8AoiGvWtc0ldb0O901pfK+0xNGJNu7YT0YDPY4PUUAcb8UPEvijwfosmtaQ2lPZRsiSR3EMjS5Y4yGDgYzjjHeu/gYyQRu3VkBP415x8d+PhXfH/pvD/6GK6NPEd9Lpv2nR9FfULaJMCQ3CxGcqDnygQd3IwCdoPYkc0AdRXGaV4zvtQ+Iep+GZdAube1s4y6XzE7ZMbccbcAHJwcnpW14Z8TWHizQINY07zPJlypSQYdGBwVI9c1haN48m1HxxL4WvtAudNu47U3RMs6SArkAY2ZHOfXtQB21FYV14hc63Joul2iXd/DCs1x5k3lRQqxIUMwVjubBwAp4HJHGa/8Awk19beJNM0XUNHET34lZLmC58yEbF3YyVVt3sQBjkE9KAOlooooAKKK4KX4lvYeLF0DVvD15YyyQNPE5mjlMoyVUKEJyWbgDOcmgDvaK5PRPHCar4v1Lwzc6ZPZ39jCs7EuHQowXjOB83zDgZHXBOKdD4wl1fVr2w8O6al+LCTyrq7nufIgSTuisFcsw74XAyOaAOqorndC8WQ6tql9o91ayWOsWODPaO28FD92RGH3kORzgEdwKZd+L4z4kk8PaTZNqGpQxiW4HmCOK3U4273IPJzwACcUAdJmlriE8eXV3qGoaPZ6E39t6fEZp7W4ulRWTA2mJgGL7sgD5RgnkjIzH4k+Id94Z8OWut3XhS9a3kgikuP38a/ZpH48tgfmyDgZ245HfigDu6KyZ9Wu/7Bg1Kw0qW+lmjSRbaOVEbDAH7zkDjNZfgTxrH440i7v49Plsfs121q0UrhmyqqxPA/2sY9qAMLTvE/iqD4qQeFNabSZbeWwa8SWzhkQkZIGdztjlTXo9eX3o/wCMkdNH/UAP/oySu08ReKLLw6lmtwVa4vZhb20TSKgd/dm4UDufoACSBQBuUVyGqeNbjw1Nav4l0uK00+5kEIvrW6M8cLnp5oKIVB5+YAj1xW/q9/e2FgZ7DS5dSnBGLeKVEYj1y5AoA0KK880r4qxaz4autWtNGnMlpHNNcQGYfuo4weWYDqxDBQAc7T2FdX4Y8RW/inw3Za1awyww3SFgkowykEqR+YOPUc0AbFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRTdwPSuH8VfFnw14Vvl0+Rp9Q1AnD21gqyNH1+9yADx069OBQB3VFeUr8etCI+bw74kB9rRD/AO1Ki/4aH8JpP5U2m65CwOG8y3jG36jzM/pQB63RXlf/AA0F4J/6iX/gMP8A4qpovj34FkTc11eRt/de1bP6ZH60AenUV5snx28Bu4VtRuEB/ia0kwPyBNTf8Lx+H/8A0GpP/AOb/wCIoA9Dorz8fG34e4518j62Vx/8RUkPxm+H877E8QoDjOXtpkH5sgFAHeUVxX/C3PAf/Qx23/fL/wDxNWF+J3ghkDDxNp2D2MuD+VAHW0Vy8fxH8FynC+J9KH+/cqv8zT/+FheDf+hp0f8A8DY/8aAOlornv+E78If9DTof/gxh/wDiqmh8Y+GLnPkeI9Ilx18u+ibH5NQBt0VlDxPoJ/5jene3+lJz+tct4h+MPhHQn8iK9Oq3rHaltp2JiWJwBuB2jntnPtQB3u6s3WPEOkeH7Q3Wr6hb2UWCQZpAC2Oyjqx9hk15n/afxV8b4Gn6fB4T00n/AF91l7gj0AIyD/wFf96tPRvgnoFtdDUPEF1eeItROC01+5KEjvsySR7MWFAFG5+Mc+t3Elj4C8NXutXAO37TKhjgQ9ifb/eKVH/wr7xt4x+fxt4pa0spM7tN0oBRg/ws3Q/iH+tesW1pb2VultawRQQRjCRRIFVR6ADgVLt96AOa8N/D7wv4UCtpOkwRzj/l5kHmSn/gTcj6DArpcUtFABiiiigBNvvXm3xK8FXl7JB4t8MMYPE2m/Ovljm6jHVD6nGceoJU8EY9KpNvvQByvgPxrZ+NvD63sGIryEiO8tTw0Eg6/geoP9QQOrrx7x1oWoeBfER+IPha33wk/wDE5sE4WRM8yAdvUnscNgjdXp2ga/YeJdFttW02YSWtwu5fVT3Vh2YHII9qANOiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAE20BcAe1LRQAmPeloooAKKKKACiiigAooooAKKKKACiiigAryn47QJceFtEgcZSXXIEYY6gpKDXq1ec/Ebwt4n8YpaWenDR4LSyvY7yOW4uJPMdlUjBUR4AyzfxHpQB6GI1UKFAAUYGB0rxf4dPq138TfiFPHNaLdpeiIm6jZiIw8iqFwRxhVHvha9Z0mXV3tnOs21lBOGwos53lUrjqSyLg5yMc/WuZ1HwhfWHjFvFnhqW3S7uEEWoWNyxWK6UYwQwB2OMDnB/nkAZ4l8I6z4ol0d7rUbC3bTb+O9SSC2csSp+6CX4B659qyvGdncaf8AFnwv4lhiuLuKG3miuLa3jZ5I4xlTKFXkgGYA8Z6Yz26qS01jXJIYtStrfT7BHWWSKK4Mss5UhgpO1QqbgM9dwGOBnPOeJpNW1vxxFbeDbtLbVtGh2X81yM24jmwyxkAEs/yBhjAAzzmgCDxRYz+OvFvhVdMguRY6Td/brq+mheFBgqQibgCzHac46cZNeiamP+JVd/8AXB//AEGuBeP4u2KGZbnwzqQQZNvskjZ/YHgAn3OPWup8JeI4fF/hiHUfszW7yF4bi2c5MUinayn19foRQBy3wMBPwq0/PTzp/wAf3jVU0Ej/AIaM8UY/6BcIPscQ/wBK1/B2geIPBGjyaHa2lnqNkk7vZ3BujEwRjnEi7Dgg55GfpWPYeEPGGjfE3UPE1r/Zl7Ff26xXH2i4eLB+QnaArYVdu0A54HXPNAE3xtP/ABJPDf8A2H7b/wBBkrqfHd74esfB98/ihBJpbgJJEPvStnKqvIO7IBHPbPQVg/ETwv4p8XtZ2un/ANj29nY3sd7FLcXEnmSMqkAMojIAyzdzVjxt4N1Px74Mj0++ktLDU4LlbiIwSNNCWUMoBJVTghj24469KAG2f9uz+C3jttL0/QtMSxZILSffcTLFs4DgFAhI92PrVb4IyqPhTo6u4BMs6ICcZPmucD9TxW+8XiTV9KOm6hZ2Vl58ZiurmC6aT5Tw3lrtGCRnBYjbkfexXP8AhLwXrvhrTrTTJX0+e10i4nutPlErhp3dHVVkGz5APNbJG7txxQBS8O26SftDeMrhh88VlbKpI6Bo4s/yr1QKByOteZ6Z4W8b6b491jxT5Ph6VtTjSNrb7ZMvlhFVVw/k+i88d69NBO0FgAe4BzigBaKKKAE215T8B8Hwbqo9dXnzjnPyR16tuFeMfBP+1oPDOqz2UUF3bNqkytbyyGN0YKmWVsEHIwMHHTOetAEnx4ZrKDwvqVl8up2+o/6Nt+8cgE4x6sqVP8RJ5pfjB4Bspc/YhKZlB6NJuH5kbV+mfeuhbwdqHiPxhZ+IPEzW6W+mnOn6bbuZFRzjMkjkDLcDAAwNq89c6vjHwhD4qs7RkuWs9TsJhc2N4i7jFIDnkHqDgZHsKAOkxz615L8G5pF8Q+PbCPP2GDVi0AH3QWeQNj8ET9K7OS48aSWP2aPTdJhvCu03n2t3iU4++I/LDE/7OR9aXw34VHgzwvJYaUUvL52M0s125jFxMerMQGIHboeAPrQBzHw44+JnxI/6/Lf+UtWfjFo1xceFYNe07jU9BuBfQMBztGN4+mAGP+5TPCfhfxfoHi7W9Xuk0SaDWrmOSeOK6lDQKpb7uYsMQG6HGcV6HPDHcwvDMgeKVSjqeQwIwR+INAHJrq9t4xtvDsVnza3yLqNyoOdscZBCH3Mu0Y7hHHrVj4j6WNY+HeuWf2lbbNsZPNc4UbCH59jtwfrWR8JvDEfh/wAPXMqXD3EdzcyC1du1osj+Vj2OWf8A4HWz49v7G28K3FnercO2qH+z7eG2wJZJZQQoUngHqcngYoA5yx1yWL4XxaK+j6n/AG5Hpv2AWH2OTLSCPywd+Nmw8Nu3YwfXit/4e+Hbrwr4C07SLoq93DGzyKpBCszF9oI9M4z3xnvXP2ul/F+G0jV9d8OySIgB82BySR6kLyfcCtHwz4r8QL4nbwv4usLSHUXtzc2t1YsWguEBwww3KsP6HgcZAOf+AMsl54T1q/ujvvbjV5Gndh8zHZGee/Ut+Zqf4+ER/D2KdWKzw6hE8Lg4ZGw3IPsPSt228Laj4V8QahqPhwW1xYanJ5t3ptxIYtkvd4nAI57qR26im6n4W1Pxlquny+IktrXSNPmFxHp0Epla4lH3WkfaAFGT8oBzk5NAGb8XJJZvgnfzXKbZ3itWkG3oxljz+tdr4bt47fwro8Ea7Y47KFVHoAgxXOfEnw/4h8XeH7jQtKXTI7S4WNnuLm4kEgZX3YCqhGPlXnPrxxWnpaeKrHw3FayWejSX9uqQx7byQROoABYnysqePu4P1oA5L4G/8i54gA4A124x/wB8R16nmvOfhx4V8UeDhdWeof2RcWd7eSXkssFxJ5iMygYCmMBhlV7jqal1HTfFVn8VLHWI9dDaBdMtsNMLN18picLjbwVL7s54x0oA9CooooAK5jx5e+HrHwffP4oQSaW4CSRA/NKc5VV5B3ZAI57Z6CunrjfiR4Kfxz4YXTorsWtzBOlzC7ruQuoYYYemGPP/AOqgClZ/27P4LeO20vT9C0xLFkgtJ99xMsWzgOAUCEj3Y+tUPg3Jb/8ACndOjvJFWF2uUIY4BXfIxH5Bj9M10rxeJNX0o6bqFnZWXnxmK6uYLppPlPDeWu0YJGcFiNuR97Fcz4c8A61o+gx6FcyWUllp8lzLZTxyN5kzSxyIodSoCgCV+ct1HpQBj2egfErwZbLF4T1HTte0FRvtLe7IDrGeQM5Xj0w+D1wK7P4e+J4/ElrqZm0o6XrMFyF1K1bOfM2BQ4J7FVGP93vwSnhex8U+GfDlho9zDZaqbeBY1uEuWiKADhWBU5C/dDA5IA+UVp+HNBm0291bVr94W1LVpklnEBJRFRAiICcFsAdcDJJ4HSgDoqKKKAOc8a6NZeIvCl1pF/d/ZEu2SOKcH7s28GPHr8+0Y4z6iuAN18YPCEJkuLbT/EtjANzPHxOUHsNpLY/2WP1ru/HWg3/iTw0dP026S1uvtMEqTv8A8s9kisWGOpGMge1SfaPFItzb/wBm6Y1zjaLo3bCJj/e2bC2e+3Ptu70AUtP1XTta+FbX+kQ+TYSabMsUOP8AV7VZSv4EEfhWT8EIEj+E2kOi7Wled3I7nzXH8gBWkPDOp6J4Bi8OaB9iuXMEkUs97K0YzJuLOFRGzlmJxkemarfD/QPFHhDwqmhXkej3CWsUxtpYrqQF5GcsquDHwuWbLAkj+6aAMnwv8vx88bhcAG2ticD/AKZx/wCNVPDM8t5+0V4q+1j54NPWKBW/gjzEeB753fVq0NH8LeNNN+IOreKZY9BkGpxxxSW6XcwMaqFAKt5XPC+g61u654RuG8UW3ivQZ4YNYhiMFxFcZEV5Cf4HI5UggENg9BkGgC/42SOTwH4hSXBQ6dcE56cRsf6Vxnw/0+Xxb8A4NJvyT9otpreJ35wFdhGf+AkL/wB8it7XtM8TeLtMk0aeK00awucLeTpcmeZ48/MkYCqBkcbienauq0zS7XR9LttNsYxFa20YjjQdlH+efWgDy3wXrc+t/Cm18Pzl11MXJ0Kdc4ZFGdx+qwhufVal8CaxN4W8MeJfDcuZb7w7dPFaRv1mWU/uB/wJmx/wIVvaD4COkfEvXfEYlT7HeKrwQAk7JmH7xyOx68+jkU6/8DyXPxRsvE0UypZfZwLyDOPOljP7liO+Mg57eWPWgDqdF0tdI0azsFcyNBEFaQ9XbHzMfcnJP1NeceB4I2+N3jycr+8jEKBsdiAT/wCgj8q9Qu2uVtJWtI4pLkKfLSZyiFu2SASB74NedeHfC3jHRvHGs+IJ49Dlj1d4/OhjupQYlTjKnyvmIHbjNAEHxR+Xx38OD3/tRh/49FXqeeelebeM/Cni7xJ4m0TUbRNFht9GuzcQLLcylp+UI3Yjwp+X1PWun1uy1/V/Bl3Z21zBpmtTx7VmgkZ0Rtw6NtB5UYzt4z0OKAOizz0paxPCun6npXhiwsdZvxf6hChEtwCTvOSRyeTgEDPfGa26ACiiigAooooAKKKzdUk1dI4/7ItbGeQn5/tdw0QA9RtR8/pQB5NpPH7VGun/AKcVH/kGGva68js/Avji0+Jl3403eHnluo/Ke2+0TBQu1VGG8vrhBzj1r03TJdTe1J1W2tILjcQFtZ2lUrxg5ZEOevGO3WgDg/jtz8K7/tieH/0YK7zSLeO30Sxt4V2Rx28aKB2AUAfpXH/Evw14i8Z6LJoemDS4bOUxu1xczyCTcpzjYqEY6c5rbtD4rg8PpE9lozajGVjUC8l8lkC4LE+VkHP8OMe9AHIfAX/kQ70dhqk4HsNqU+PA/aQkHYeH+np+8FWfht4V8UeCtPl0y+GkXNrLcPcmaC4kDqzKoxtMeDyvXNQL4W8bD4jnxeYfD+WsvsZtftk3C53Z3+T1z7UAL4r8La/P4rufEHgfXbe11URJb6hZz4ZJQACpOQcHaR1HToRzVfQ/F3iGHxfpeiePfD8FrdTF/wCztQtWJjMm0gg/McEjI7Yz0AORvrofiLTfHGt6/ZSWdxZX6wJ9hlcozbE279+CFbOQBggg9RVqfRtR1/XdKvtVt7e0s9MkNxDAkxlkllKlQWOAFChicDOTjnjBAOsooooAK8p8RW8c/wC0N4T3qCE0+ZwCO4EuPxHX8BXq1eZah4W8bXvxD07xXHFoCfYLdrdLZruY71bdklvK4PzenagDt/Ectza+F9XuLPIuo7KZ4iOu8ISvT3rzf4KR6v8A8K3tpLCfT1je4md/Pjdn3bsHJDDsB2r1W0NzLZRG9ihjuGX96kLmRAe4DFQSPwFcXo/hXVvBGoXqeHltr3Q7uUziwnmMUlrIcA+W20hlIA+U46DnrkAsQ+ENSk+Ilt4svL+0DR2Rs2t7eBlEgySCSzHufTsK5/RVPhT4q+K7i6S4uNP1LypPtkMDyi3lALeU+0Ej5XyCeMbRkHiu2srHUbzVY9S1YQwGBWW2tLeQyKhbhndyBuYjgDGACeTnjhrc+Kdf17Vdd8D6jaWujXcoVzqSFluJowIzJEqgkKQijJwSV6UAXtF0i91j4tX3i821xa6ZDZiytjPG0b3LZyW2t8wUdsgZ4xU/xtA/4VLrJ94P/R6VQ1DWviV4RtX1bWbfQ9Y0mAbroWJeOaNO7Ddwf1/Dk113iTR7bx14FudOjn2Qalbo8M2M4OQ6Nj0yFNAGjoOP+Ed0vjn7JF/6AK8++BpB8O+IMHg67Pj3G2Oum02HxVDoVrpE1rY288UKwNfpcl1GFxvRNgJb2OBnvWB8NPCfibwS17ptxb6dJo9xeyXCyi5czoCoUcbMNkKvcYyaAGXp/wCMkdNP/Uvn/wBGSVpfEW88ORzaNa6tpLaxqz3O7TLGJiHaQEfMWyAqZxnOR7HHGfP4W8bzfEa38X+T4eDQ2RsxafbJjlcsc7/J65b07VoeKfBuqX/i7QvF2lSWp1LTUMUlpcsRFKhByFcKSCN7YJHp6cgHP/FweIJvhXqM+qf2bBGHiJtoFeV1PmrjEpKjI5z8nsK9M0WUSaNYqXDSfZImYd8FRzj35/KuW8X+GdY8eeHbjSbsW+lwlN8YWYys8o5Xf8owgPJ6knHTHM+laX4n0+0N49vpL6r9kt7IRG6kEISLeTJv8vOWL/d28Y6mgDnfgRbovg/VJQPnk1abcfYKgA/n+deqBAAAOg6e1edfDnwt4r8GWb6beLotzZz3T3Mk0NzKJE3AZwpjw33fUda9HoAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooqKW4ighkmmkSOKNSzu7ABQOpJPQCgB+72rI8Q+KdH8K6f9t1m9jtoiSEB5eQ+iqOSa4HW/itPq+ovoXw90461qIOJLxlxbw9t2TjP+9wvTBap/D3wkje/Gu+N74+INZbnbLzbxeyoeG6nqAv+zQBkHV/HHxUJh0GKTw14Zc86hN/r51/2MH/ANBOO27tXeeEPh54f8F2xGmW267dcS3s3zzSevPYcdBgfWuoCAAAcAdAO1PoATFGOMUtFADWQMMEA/UU0W8IGPKTH+6Kkqreajaafbm4vbmK3h3Bd8rhQSeg57+3egBG0uwdizWVsWPUmJST+lQTaHo9zxNpVjJj+/bo38xT7TW9L1C4NvZahaXE4jEpjimVmC5xkgHIFebzN4elstZvfEWovB4jiurhVzdNHPb4dhAIEyDgpsIwDuJ5zQB6D/wjWgdtD03/AMBI/wDCqp8CeEScnwtohPXJ0+L/AOJrQ0V72XQdOk1Jdl89tE1wvpJtG4fnmtCgDnW8A+D26+FdE/CwiH8lph+Hvg0jB8LaP/4BR/4V0maZJMkUTSysqRqCWdmwAPUmgDmP+FZ+Cf8AoWNN/CEVXl+FngaZhv8ADNmMf3AU/kRWdr/xm8LaTN9j0+WXW9QY7Ut9OXzAW7Df90/8B3H2rF8z4r+OSPLjt/B+lydS3z3RU/8AjwP/AH7NAGh4g8F/Cbw7ZGXW9P0+yjKkgNNIHfH91Q25vwFeZzWHhLxHLJa+APhtd6mQdpvru7uI4FP/AH8H6sp9q9T0L4LeGdNuDfat5+vai5DST6i29S2OTs6H/gW7616FFbxQRJFDGkUaDCqigAD0A7UAeCeHv2dmmk+0+JNRSFS2fsenZO0enmPnH0wfrXsHh7wR4c8KoBo2lW9tIBgzYLykdwXbLY9s49q38UtACbRnNLRRQAUUUUAFFFFABRRRQAUUUUAMeNZFZXAZWGCrDII9K8WvIrj4LeL/ALdbB5PBWrS/6RCFLfYpfUf55GVxkKa9sqlqelWes6Xc6bqEKz2lyhSSNuhH9D3BHIPIoAntruG8t4ri2kSWCVQ0ciNlWBGQQe4IqavGPC2p3nwt8VL4K1+aSTQbxydI1CTkJk/6tuwGcA+hOejZHsu7nAH40AOooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApMc0tFACbcHNGPelooATbXLXXhe5tfEtz4g0G9jtru8REvbe4jLw3AQYVuCCjAHGRkY/h711VJj3oA525i8WXkBgSbStOLDDXEfmXLj1KoyoAfTJI9jVzw34csvDGiw6XZGRo4yWeSVtzyuxyzse5JrVzkdOKXNABigLjpS0UAJjBo280tFACY96NtLRQAUmKWigAooooAoanHqb2ZGkzWkVzn713E0iY+isp/WuP+H3gnW/BEM1hJq1je2E9w9zIBaskodlA+Vt5AHyg4we9d9igigAx70uKKTd7UAG2jHvS0UAN2DGDXB6X4G1PSTrVjBqhex1a7eea4kldp0RuqIOgbGRv3H1xxXfUm3jFAEdvbRWltFbwII4YlCIijAVQMAD2AFZPibwzbeJ9Nitp55bea3nS6tbmHG+CZPuuM5B6ngjkGtuigDn4x4tgQRP/AGPeMOPtBaSDPuYwHH/j1QaX4XmTxG3iPWb1b3VPINvAsUflw20ZOSEUkkse7E59MDiumxRigAx70AY70tFACbc9TSbfenUUAJjJrPk0xp9bivppt0VvGVgg24Cu33nJzycfKOBgFuueNGigAooooAKTHSlooATHvQVyKM+1LQA3bnqaXHvRuozQAtFFFACbaNvGORS0UAJik285yadRQAmKMUtFACFfeloooATBz1ox70Z9qM+1ABt96NtLRQA0LgYz+lAT3p1FACbfc0tFFABRRRQAUUUUAFJj3paaWwM4oAdim7adRQA3bz17Ubec5p1FACY96MUtFACbfegLjvS0m72oAWiiigApMUZpaAE2+9GPejNGaAADHeuQ07wxqPhYTweH7u2fSpZWmSwvAy+QzHJEcq8hcknBU49a7Ckx70AclrGga54osn03VLy1sNNm+W5isd0ksyZ5TzWChVI64XJ9eorqLe2itLaK2gQJDEgjRB0VQMAfkKlxS0AJtzQFxwKWigAxTdtOooAQrkdTRilooATHvS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFJu9qAFpN3tWdrOvaZ4f057/AFa9htLZOC8jdT6AdWPsMmvLZfGHi/4lTPY+CbWTSdEDFJtaugVZ17+X6Hrjbz0yUoA7Hxl8TNC8G/6NM7XuqvgRafbfNIxPTd/dB468nsDXGxeEvGHxMlW88Z3Mmj6FuDxaPbHa79x5np2+9z1wFrsfBvwy0LwcftUatfas+TLqNz80jMeu0dFzz05Pcmuz2igDN0Xw/pfh3TksNIs4rS2TnYg+8fUk8k+5Oa0se9LRQAUUUUAFFFFABXnvhS8TxRr+t+Kb6RRaadcSWOnI5G2GOP8A1k3PRmOeeCAMV6FXGaX8OtO0q5vhFqGpPYXc7XP9nvKPIjkbksBtycHGAxI4GQSM0AZHwtSHVbzxR4sjQiLVtSZbckY3QR5CsM+pLH8K9Ceyt5LlLh4Y2mQYVygLAex6iuf8L+GNM8C+H0tY72R7e3Qj7ReOo2IWLkZAAAyxOep7k4GOe1z41eHLG5+waJFc+INSYlUg09Cys3pvxz/wENQB6Pn25rE8Q+MvD/hWHzNa1S3tTjKxFt0jD1VBlj+Veem1+LHjcf6RcweEdLcHCQ5e5K+uc5B/FPpW94f+DfhTRZzd3cEmsX7Hc9xqLeZlj1O37vXnJBPvQBiP8UfEXiqRrfwF4VnuYs4/tHUB5cI5wSBkA/8AfWf9mkh+FGu+J3W48feK7m9UEEafYny4QR+AHT0UH3r1pIljRUjUKqjACjAH5U+gDF0DwloPheDytG0u2tONrOiZkYf7Tn5j+JrZxS0UAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBz/jDwhp/jPw/PpOoDAb5oZguWhk7MP8ADuOK4v4beK9RsdSm8BeLSE1qxBFpcFuLyIdMH+Igc56lRyMq2fVK4b4j+BB4v0yO6sJPsuv6e3m2F0pwdw52MfQ449Dz6ggHcZpa4P4beO/+Er06Ww1JPs3iLTz5d9auNrHBxvA9M8Edjweoz3lABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFcN4z+Ikvgu4ha90C4fTZp0gS/FxGIyzDJ+XJYYw3UD7poA7miuYn8T6qLN7+z8MXVzZou8KZlS4kTruWI+o5CsVbtgHitDw54l0zxVo8WqaVMZLdyVIYbXRh1Vh2I4/n0oA16K5y78Vhtek0LR7M6jqUKCS5/eeXDbA/d8x8Ehj1ChScc4xzTD4tfTtXtNN8QWC6e96/l2tzFP51vK/wDzzLFVZX9AVAODgmgDM+JOu+JfC/h651vRn0s21qqmaK7hkaQkuFypDgY5HGK6jQ7qXUNA069m2iW4to5n2jA3MgJx+JrlfjFn/hVGvf8AXOP/ANGpWtpd9dWHgXRprTTJ9RkFlBmGB0RseWOfnIHbpnvQB0tFcf4K8fR+N/tMtno99a2tuxjee4aMYkGDs2hi2cNnOMcdaW98cXFr4uPhyLwxql1dGPzkliaLymizguWLjaM8YOD+YoA6+iuf13xba6LqFhpaQSXmragxFtZwkBmA5Z2JOFUAHJPocZrOufHFxY65a6Fe6KYdUvgTZk3Sm2mx94ebjIIxyNpPIwDmgDsM0tc/YeJHTQL3VfEFl/Y6WUsqS+bJuUohx5inAJDduMntnjMFp4h1vVLFdR0/w6n2KRQ8QvL3yZ5UPRggRgMjoGYH1xQB09FYugeJ7LxHZTzWaSpcW0jQ3NnMAssEi9UYZx9DnBrm7T4mFvFdx4e1Lw9e2F5FbfaEQyxyvMSyqqIFOCTu67sDBzgAkAHfUVyGp+NpvD97py69oz2dlqE4torqO4WURyN91ZBgbc4PILDjrXX0AFFFFABXnPjbxP4p8MeIdDWF9Kk0nU9Siswpgfz0DFc5bftP8XavRq8s+MkiRTeDJZXCRpr0BdmOAADkk0Aep0Vyep+ML2w0efW4vD9xNpVvGZnczKkxiAyXWMjpjnDFTjtW9pOq2ut6TaanYvvtbqNZY2PBwfUdj7UAXqK5Wz8XXGu/a5PDdlZ39vaTNBJJPfeSWdeoVVRz9N23PUcc1Z8MeL7PxOl5FHDNaX9hL5F5Zz43wPkgZIJBBwcEdcUAdDRWDN4hmm1q60jSLKO7urNUa7eecwxxFxlVyFYliOcAYAIyeahtvE15/wAJVb6BqGkfZpprSS6FxHcCSJgjIu1TtVifn5yq44xnPAB0lFFFABRRVPUtSh0rTp764DGKFclUGWY5wFA7knAA7kigC5RXGxfEK1i1y60PU7Ca11aGNJYraJxMbhW6BCMfMO46DrnAJE974yl0fUNMg1nSXtLbU7gWsE6XCy7JT91ZAB8ueRkFhxQB1dFFFABRRRQAUVgP4hnutXvNM0exju5bHaLqWe48mONmG4ICFZi2OT8uBkc54qKz8TXcniweHr7SDbT/AGJ7zz0nEkTAOqbVOASfmycgYx3zQBQfxnfL8To/Cg0C5Nm0HmHUudgOwt/dxjI25z1rs64gePZ4/H9j4VvfD91aSXoleG5edGVkRWbcApPXb0OCM10HiHxJpvhjSzf6nKyRlxHHGilpJXP3URR1Y4OBQBNrCau2nkaLNZR3mRg3sbvGR34VlOffNcx8LfFWp+MPC82o6qtulzFeSW5W3QqoChexJ9TWtHruqrbi9u/D00FnjcyrcLJcRr6tGoxwOoVmPoCa4/4DMJPAt66kFW1Wcgj6JQB6lRXMTeLjc+ILjQ9CsRqN5aANeSvN5MFvnorPtYlj6BT74p+l+LBc+IZ/D2p2RsNXji89IxJ5kU8XTfG+1c4PBBANAHSUVz0fiO51LUL210SwhulsZjb3E9xcmFBKACUXajliARngDnGTg4NM8SXN54mu9CvNJe0ntrVLgyiYSRyBmZRsOASPlPJAOc8cUAdDRRRQAUUUUAFZ2tazbaDol5qt7uFvaxNI+0ZJx2HuTgCtGvOPjJe3afD3V7aLTJZrWSBDJdrLGEiPmLwVLbj26A9aANXQ5/EviPSIdYuryPSIbmITW9nbQLLIqEZUu75ySMcALinfDzxVd+LvDst7eWUlrNBdPb/PEY/MCgENtyccNgjJwQeTVTw/r+vL4W0sW/g+8nQWcWxxe2yhxsGCMvkfjV7wN4zXxpaajN/Zktg9ldG1khlcOdwAJ6D3oA62iuBk+JcsPiq38NS+FNUTVrhPNjhM0GCmCd27fjGFbvnit3xb4qPhPS11F9KvL+3DqkhtShZCxCrwSCcsQOKAOhormr3xlBpHhpdZ1rT7vT2dxHFZPsknlc/dVVRiCx9M8d8VV1Dxdq2iaYNX1jw8INNXBmMF5509uhIG549gGBnnazY96AOvoqG1u4L20huraRZYJkEkbqchlIyCKmoAKKKKACisDWfFNvpeo2mlW1vJf6tdgvFaQkAhB1kdjwqD17ngAmquoeLLvw8sdx4j0qO0093WNr20ujPHAzHA83KIVUk43AEdM4oA6O5uYrS0mupm2xQoZHb0UDJP5V5kvinxtrng/wD4SrQLSNxNcYstMEYJaFXKl5WJyScdF24HOTXVeOb65h8Hal9j06S+imsZw8kUsaiJTGfmO5hkY9M9K5z4RanqEngTQLQaHcCy8l1+3+fFs4Z+du7f146fpzQB6RC7vDG0kexyoLJnO09xnvUlJj3paACiisbxJrs/h/THv49Iu9QhhVpJxbMgaNFGS2HYbu/AyaANmiuP0DxzceJfDj63pvhrUJLcn9wjTQB5wGZWKgvxgr/FjPbNP8G+Ol8ZrcTW2i31paQM0TT3LRj96uMptDFs/N1xjjrQB1tecW/ijxTZ/Fax8Laq+ky2d1aSXKyWkEiOAN2Adznn5f1rX0fxxdav4judF/4RfVbaW0ZRdSzNF5cIYZU5D/NkdlzXKeKNQg0z4+aFdTklE0d1CqMszFpAqqO7EkADuSKAPW6q3929lp11dR2z3DwxNIsMf3pCASFHucfrXM6143m8NR215rmiyWmmTzLCblLhJGhLdDIg6D12lsV0OqXl3Z2Dz2OnSahOMbbeKRELZ93IH60AYngHxXd+MPDzaleaPLpciztEIZCTvAAO4EgcZJHTqDXVVyvgTxjH420e61GOwlsfs929q0UrhmJVVJPH+909qW+8ZL/wkb+HtFsG1PVIYxLcjzRFDbKenmSYJBORhQpNAHTOG2HaQGxxkZxXnuieKPE//C0p/CmtnS5II9PN4klnDIhb5lA++7erVv2fi5R4mi8Oatbw2mqzwmeFILnz0kUdedqsD1PKgHBwa4u81GDTP2hZp5g7ltCWOOKNdzyOZRhVHc/oBknABNAHrVFchqfjabw/e6cmvaO9nZahOLaK6juFlEcjfdWQYG3ODyCw46119ABRRRQAUUVgaz4pt9L1ax0eJY5tVvgzQQSTiJdq9SzHJ+gAYnB44OADforkpPGj6Vr9jpHiHTlsH1Btlncw3HnQSScfuyxVWVjkYBXB9a1PEmtXehaU99aaPPqYjBeVIZUQogBJb5iM9OgyaANmivO4/itDP4IPie30aeS3hj825TzQBFl9oXdj5nIw2MYAIyRkE797440nT/CNp4juROltdxxtBDs/fSNIMqir3Y/XHfOOaAOlorkNQ8XatommDV9Y8PCDTVwZjBeedPboSBuePYBgZ52s2PeuptbuC9tIbq2kWWCZBJG6nIZSMgigCaiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKTcAMmuX8X/ABB8P+CrXfql2DcMu6O0hw00g9QvYdeTgcevFAHT7xznA4zXmfiP4tQLqJ0HwZZHX9cfKjyuYIj3LN/Fj2IHqwrFGmeOvisQ2sSSeGvC7nIs4z/pFwn+1nsf9oAdMKeteneG/CejeEtOFlo1klvGcF26vIfVm6k/y7YoA4DRvhXea3qMeu/ETUW1a+HzR2Cti3g9VIHB6DgYB5zuzXqkNvDbQpDBGkUUahUjRQqqB2AHQU/Zj6U6gAooooAKKKKACikLYHSoLu+tbC1e5vLiK2gjGXlmcIij1JPSgCxTdwrzTV/jXokd0bDw3Y3viLUOcR2UZ2cf7WCT9VBFZp0f4p+Nvm1bVIfCumn/AJdrLLzsPcg5zjj7w/3aAO88RePPDXhRSNX1WCGbGRAp3yn/AIAuSB7nj3rhT8SfF/i8mLwL4VkS1JH/ABMtTwseOhKgHBx7Mx9q6Lw58IvCPh11uBY/2jeg7jdX581s5zkL90HPcDPvXchAMY7dB6UAeT2/wgv9fljvPH3ie81eUHd9jt38uBD3Axj/AMdC16JonhnRfDdr9m0fTbazjwA3lxgM/wDvN1b8TWriloAbtOOtOoooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACkAxS0UAeXfEnwlfwahD458Jrs17TwGuIU/wCXyIdQR/EQOMdSOBkhRXXeDPGNh408Ow6rZfIxOyeBmy0MndT/ADB4yOa6HZznvXjni7SL34Z+KT458OwGTR7ptmsadGMKAT/rAOgGc89ie4YgAHstFUNI1ix1zSrbUtOnWe0uUDxuvce47HsQeR3q/QAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFeVfHbjwpoeeR/bcH/ouWvVa8o+PSmTwjoqhmQtrcADL1GY5eaAPVMZP614/8M3a1+LfjvTrQsNPE3mlQMKsm8jA9PvMB7KPQV3tzL4uSxa1trPTJbzGxL6S4ZY+mN7RhSQf9kEj/aqPwR4Mt/B2nXC/aHvNRvZDPfXsgw00h9uwGTxz1NAHF/Ai4mvbDxLfXhJv59VZpyeudoP5ZJrQ+O6p/wAKxnkJ2yRXULxHurbsce+Cela0XhS/8N+KNQ1rw19mlttTYPfadcSGMeaM/vI3AbBOTlSOcnkcYZq/hbVPG17Yp4hjtbPRbOYXBsIZjLJcuM7fMbaoVRk/KM5yeemACh8Rbie6+A95cXQP2mWxtnlz/fLRlv1zXY+Fl/4pHRf+vCDjH+wtYfxD0LX/ABR4dudC0hNMjtruNRJcXU7qykODhUVCCOBzkdela3hS31mw0O2sNaisFe1hjhR7OZ3EgVduSGRdvTpz9aAOG0z/AIov43X+lk7NM8Tx/a7f+6tyuS6/U/Mf+BIK7bw2Pt8+oa83IvZfLtif+faPKpj2JMjj/frA+KPhpPEdpoUVvcNbanHqcYtZo+HVTkyYPbCqX+qV3dtaxWdrDa26COCFBHGijhVAwAPoBQB5p4ptJ9M+MXh/xIsVxd2osZYLqG3jaV7dASPN2jJK5lAPGcAn6O8QWNx438f+FbjTra6j07RZ2u7m+mhaJWJKFY03AFidnOOBmpPED6zrfjpl8F3cdvqGlQ/Z9Rnuxm3KyYdY8AEs4xnI4AJBPNLIvxc06M3Cy+GtUCDJt1WSJ39lPAyfc4oAzfj/AHE0XhHS7YEraXOpxpcNk4ICsQp9uM/8B9q9YUYAUDAFcYi6V8Wvh0n2m3kghvF+ZM/PbzIxBwcdiDz3HpmrVg3jLTdOjsZrPTdTmiURx35vGhEoAwGkTYxDeuC2fagDjtDmmt/2kPE9rbZFvPp8bzqvQOEiw31G4/8AfZqZoI5P2lUZlGYtE8xDjoclc/kSK6rwn4PfQJtT1a9uEvte1STzLqcKVjXH3I0HJCL0z1P4DHPDwp42HxIPi/yfD242f2T7J9smxtznO/yeufagCP488eBrBgcEatAQf+AvXqWa86+JXhXxP410+HTLAaRbWsNwlyJp7iTezKrDG0R4HLdc122lSao9nnV7e0hutx+W0maRCvY5ZVIPXjH40AaFFFFABXlHxshS5j8JQSjdHJrUSOp7g16vXm3xA8KeKvFl9pv2EaPb2umXq3cLz3EpeUqMgMojwv5mgDrvFw/4orXv+wdcf+i2rnvhDIsfwl0OSRgqLFIWZjgACV+STWjrtv4p1TwrPYQ2mjR3l3BNBOXvJTHEGUqrKfKy3XJBC496y/BvhTW9L8Et4S1sae1kLWWBbqyncyHzGbOVZABgMecnpQBR8F3drdy6jJ4B0K1s9JluiZtTvHfbPL38qPrtA/2lHPA61Q8Ema3+M/jkXs8DOsEDyyRIYk+4pJwWYjAPr69M1t+C9B8S+C9CHh9LXT9RtYHc2t4LkwkqzbsSLsJByTypPb0qgfh7rlr4j1u+t7yzuIvEFmYL6SZmR4HPBaNQCGVRwFJHbJNADPEHhfxKvizUfEPgHXbWO4uGjj1GwucGNpEQYxwQDtKnHBGc55xT/DHjDWpvGdpoXjbQIbDWXglNhdW7ZjlXgyKPmIydgPB7dB319N0PxHoPibXtStDZ3Wn6hcI8dg0pjZNsSrvDYIycYK46AHdxir66LqGreKdO1vVYre1j0tJRaW8UhlYvINrO7EADC5AUZ65z0oA6qiiigAqpfWEGoRRxXKlkjmjmUA4G9GDr+TKD+FW6hn83yJPI2+dtbZu6bu2aAPMvASLr3xS8aeJZF3JbTLplq3UAJw+PTO1T/wACpfHx/t/4l+DPDMR3CCc6pdDqFSP7ufrtZfxFdB8NPCl14P8ACCaffPHJfyzvcXTxsSpdj69/lC1Uk8MavY/EbVPEtgtrdf2hZx20bXEpX7IVxk4Cncp2hsAjJyOOtAHcxzRyqWidXUErlWyMg4I/Agj8KkqjpOmR6TpkFlE7SeWCXkf70jsSzu3uzEsfc1eoAKKKKAPKtb8L+KbfxRqfiDwBrloDdyBdQ065wyGVFAODg4JGMjgj1wcVZ8J+L9Xu/GEeh+M9Ai07Xfs0hs7mBsxzR5Uuo5PPyKep+7276elaN4l8P67r19D9kvLHUr03CWbSGN4/lUbw2CCTjBUj+EHd2rQi0W+1TxXZa9q0dvb/ANnQyxWdtBKZTukADO7EAA4GAoB6k5PGADl/EgH/AAv/AMFdv9Duu3/TOSqPjeSS5+O3gixuATYxxtPGD083Ln8x5cf04rS1bwv4z1L4haV4oii0FF02OWKO3e7mzIrhgSzeVwcN0weldB4t8IN4otdNu47hbDXdNkFxaXCjesb8EoRwWQkAHp0B9qAOpPqSR36+n9K8t+AuR8PbzaASNSn2g9PupxXZCTxVd2n2aS0sLG4YbZLyO5Mqp6siFASeuAxGOM5xXMfCvwv4p8F6fJpOpQaa9g87z+dHcuZlLAADbswfu+ooA534GtrV34c1i9huLMXU+qOblrqN2kZ9iHkhhxkn8c12194Q1XVPGmg+Ibu/som0nzBtt7dgZVdcFSSxxjtx3NRw+E9T8L+Jr/V/DLW01jqbeZe6XcSGMCXn95E4BAJzypGD6jjG1BY6pqmpW93qscFpb2jmSG0glMjPJgqGkbaBgAnCgHk5zwMAHDX3hfxfpWu6lrXw+1qznsr68kmutPugCouASsm0467gQeVPYk4FafgzxZqGp+KbnR/FWhJpfiSG1Do8RJjng3c7ck9Cc8E9+mCKuaBo3ifw1c6y4FnqFpf6nc3cVt5xjkgV3LD5iCGyDkrxg9zmtPT9DvLjxYfEeqLbxTQ2n2O1toJC4RSwZmZioyxwAAAABnrnNAHT5pa89n03xTYfFe21RtcEnh6/c26acXbKEQOxIXG3rHndnJzivQqACiiigArh/i9z8Ktexz+5T/0YtdxXGfEPRNf8T+HbnQ9JXTEgu4wJZ7qeQMuGB+VVQ56dcigDZ8J5/wCEN0M85/s+3/8ARa1xPwcGG8Y/9h+4/pXS6HB4p0zwtDYzWmjSXtpBDBAUvJRHKFUKzMfKyvTIADZ9qw/AHhbxV4UvtSF8NHntNSvXvJnguZN8TMDkKpjwwzgdRQBV+MNjPp9ppHjSwTdeaDdq8gHG+BiAykjnrgfRmrpbi8g8Tatodvav5tl5S6tKw/iQf6gH/eb5x/1yrodT0631fS7vTrtd9vdRNDIv+ywwce/Ncr8NPCF34P8ADRs9RuVub1pCDIpJVYlJEagntjLY7FzQByHxPkvJfix4Fs1dFt/O82ISglDMHHJAPOML7jPvXfanpniDVNKu9OmudJEV1A8L4t5OAykE43+9J4z8Hw+LbC2C3LWepWMouLC8RctBICDnHccDI9hSQXHjF7ZbWew0mO7xta8W6doumN4i2Bj/ALpYD/aoAseDdBm8MeE9P0We7+1vZoU84IUDDcSBjJxgEDr2/CugqjpWmR6TpsNnHI8mwEtJJ96RySzMfcsST9avUAFFFFAHkPgqaW8+O3jaW7JM0UKRRBu0YKgY9Bwp/Wu1+IcUL/DjxGJACo0+dhn+8EJB+ucH64qtrPhG5j8WxeLfD8sEWqeV5F3b3JIhvIuMBmAJVhgYbB6DiofEGj+IvGVh/Y15HbaRpkxU3ckVyZp5UBzsQbQFBIHzEn6UAY/hC5ubz9noTXZJkGk3UYLDkqvmKv8A46F/IVrfCAAfCnQSf+eUhz/20etHXdG1FfCTaB4atNOjge1e0H2uZ0EKFNoKhUbcee5FUfh7oniTwtodnoWqR6VLaWiOFuLa4kMhJYsAUaMDHPXdQB29FFFABUbRK6lWAKnqCM5qSigDxzwHqn/CES+M/CtwGcaRK17YRFiWkifGxB9SYxgd3qx4Khn8EfETVfDmo3JeHVLOPVUmdvlacDE/4k7m9gorf13wEdU+JujeJo5VS3gh2XseSDKUYPDx3w3J/wBxaf4/8ETeLLjRLi1nFvPZ3LJPJnazWsg2yqPcjA/E0AbHhGNptIfVJFZZtVma9YNwQrYEYPuIljB9x2riNfhSb9o7wruG4JpkrDPOCBNXqyoFUKoCqBgADoK8y1Dwp40vPiNp3i5YtAQ2Vu1stqbuY71O/JLeVwfn9O1AEvx0GPhXfn0mh/8ARgH9a9A0/wCbTrVjkkwoTn6CuI+IvhvxP4z8Of2LZppFvDKI5JpprmQsHU5KqBH0yBhs8+lb1kfFMGgLFJZaM2oxbY0VbyQQsgGCxPlZB/2cH60Acj8D+PDWvAf9B64/9BjrP+BDtd2fie+uwf7Rn1Rjc5+90yM/Qs9bvw28K+JfB8d5Zaj/AGTPaXd3JeNNbTyGRHYKNu0xgEfKO4qwPCOpeGvFt9rvhg201tqZDX2mXMhiUyDP7yNwGw3JyCMHJ56YAOkk8O6RL4gh1+SwhOqwxeTHckfMq8jA7dGIz1wcV5+9vFN+0qjyJkxaJ5iHHQ5K5/IkfjXapZ6vqeoW93qMVvaW9oTLDaRTmQyS7SoaRtoAABOFAPODngCuTHhTxsPiQfF/k+Htxs/sn2T7ZNjbnOd/k9c+1AEfx5yPA9gwOCNWgIP/AAF69SzXnXxK8K+J/Gunw6ZYDSLa1huEuRNPcSb2ZVYY2iPA5brmu20qTVHs86vb2kN1uPy2kzSIV7HLKpB68Y/GgDQooooAK888W32hDxzo9tBon9seLYkL2iCUxrbR8/PKwyAvUjIY56AZFeh1wF/4O1Sw+JbeNNFNtdG5t/s13ZXLmM4wAGjcAgH5E4I9eeeADmPi0NZSx8Lz6nNp4Ya3BsjtYnyhwxz5jNyOP7o7V6nrTxtoOqKrAlbaUMAeh2E8/hXG+NfB+t+NbCCVpLWwurCZbizt2lMiO4PJlYDjjIAAOOeTnjTuNL8Sf2TqjW1vpY1HVpC0yS3UnlW6+SsQCsIyX+7novWgDL+CNtFH8J9LkVQDO8zyEDqRK6g/koFYPxOe8l+LXgW0R0WESmSHzQSnnbh1AIzjC+/NdR8O/D/ibwjoVnoOox6TNZW3mEXFvcyeb8zM4GwxgHlj/FWr4z8Hw+LbC2C3LWepWMouLC8RctBICDnHccDI9hQAup6Z4g1TSrvTprnSRFdQPC+LeTgMpBON/vU3g3QZvDHhPT9Fnu/tb2aFPOCFAw3EgYycYBA69vwqvBceMXtltZ7DSY7vG1rxbp2i6Y3iLYGP+6WA/wBqtrStMj0nTYbOOR5NgJaST70jklmY+5Ykn60AXqKKKACiiigAooooAKKKKACiiigAooooAKKKKAEzVW/1Kz0qykvdQuYbW1jGXlmcIo/E1x/jn4jQ+FrqDR7DT7jVdfu03QWUKngE4DMcHjg8DJ47DmucsPhtrnjG9i1f4j35kVTuh0a1fbDD7MQev0JPT5qAG3vxE8R+ObyXSvh1YMlsp2Ta3doUjT12gjjt1y3+zXQeD/hTpPh25/tXUZZNZ1523vf3ZLFWz1QEnB/2iS3XkA4rt7OwtdOtIrSyt4ra2iGEihQKqj2A4qxj3oATb78+pp1FFABRRSbvagBaTPtVW/1Ox0q1a61C7gtYF6yTyBFH4mvONS+NWnTXh07wlpN94iv+wt42SIc4JLEZx3ztx70AeoFsdelcv4l+I3hbwpvTVNVhFyv/AC7Q/vJc+hVfu/8AAsVxZ8MfE3xqN3iLXYfD2nvjdZaaMyEdwWB7j1cj2rqfDXwp8I+Fyktrpq3N2vP2q8xLJn1GeFPuoFAHK/8ACe+O/GWV8GeFzp9i/A1PVflyD/Eq9M/TfU9p8Gm1e4S/8deIb3XLkHIt0cxQJ6qAOcf7uyvV8UY96AM/SdB0rQrUW2k6fbWUP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+95	"A circle has centre $O$ and radius 1. Quadrilateral $A B C D$ has all 4 sides tangent to the circle at points $P, Q, S$, and $T$, as shown. Also, $\angle A O B=\angle B O C=\angle C O D=\angle D O A$. If $A O=3$, determine the length of $D S$.
+
+<image_1>"	['Since $\\angle A O B=\\angle B O C=\\angle C O D=\\angle D O A$ and these angles form a complete circle around $O$, then $\\angle A O B=\\angle B O C=\\angle C O D=\\angle D O A=\\frac{1}{4} \\cdot 360^{\\circ}=90^{\\circ}$.\n\nJoin point $O$ to $P, B, Q, C, S, D, T$, and $A$.\n\n<img_3604>\n\nSince $P, Q, S$, and $T$ are points of tangency, then the radii meet the sides of $A B C D$ at right angles at these points.\n\nSince $A O=3$ and $O T=1$ and $\\angle O T A=90^{\\circ}$, then by the Pythagorean Theorem, $A T=\\sqrt{A O^{2}-O T^{2}}=\\sqrt{8}=2 \\sqrt{2}$.\n\nSince $\\triangle O T A$ is right-angled at $T$, then $\\angle T A O+\\angle A O T=90^{\\circ}$.\n\nSince $\\angle D O A=90^{\\circ}$, then $\\angle A O T+\\angle D O T=90^{\\circ}$.\n\nThus, $\\angle T A O=\\angle D O T$.\n\nThis means that $\\triangle A T O$ is similar to $\\triangle O T D$.\n\nThus, $\\frac{D T}{O T}=\\frac{O T}{A T}$ and so $D T=\\frac{O T^{2}}{A T}=\\frac{1}{2 \\sqrt{2}}$.\n\nSince $D S$ and $D T$ are tangents to the circle from the same point, then $D S=D T=\\frac{1}{2 \\sqrt{2}}$.']	['$\\frac{1}{2 \\sqrt{2}}$']		['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+96	"In the diagram, $A B=21$ and $B C=16$. Also, $\angle A B C=60^{\circ}, \angle C A D=30^{\circ}$, and $\angle A C D=45^{\circ}$. Determine the length of $C D$, to the nearest tenth.
+
+<image_1>"	['By the cosine law in $\\triangle C B A$,\n\n$$\n\\begin{aligned}\nC A^{2} & =C B^{2}+B A^{2}-2(C B)(B A) \\cos (\\angle C B A) \\\\\nC A^{2} & =16^{2}+21^{2}-2(16)(21) \\cos \\left(60^{\\circ}\\right) \\\\\nC A^{2} & =256+441-2(16)(21)\\left(\\frac{1}{2}\\right) \\\\\nC A^{2} & =256+441-(16)(21) \\\\\nC A^{2} & =361 \\\\\nC A & =\\sqrt{361}=19 \\quad(\\text { since } C A>0)\n\\end{aligned}\n$$\n\nIn $\\triangle C A D, \\angle C D A=180^{\\circ}-\\angle D C A-\\angle D A C=180^{\\circ}-45^{\\circ}-30^{\\circ}=105^{\\circ}$.\n\nBy the sine law in $\\triangle C D A$,\n\n$$\n\\begin{aligned}\n\\frac{C D}{\\sin (\\angle D A C)} & =\\frac{C A}{\\sin (\\angle C D A)} \\\\\nC D & =\\frac{19 \\sin \\left(30^{\\circ}\\right)}{\\sin \\left(105^{\\circ}\\right)} \\\\\nC D & =\\frac{19\\left(\\frac{1}{2}\\right)}{\\sin \\left(105^{\\circ}\\right)} \\\\\nC D & =\\frac{19}{2 \\sin \\left(105^{\\circ}\\right)} \\\\\nC D & \\approx 9.835\n\\end{aligned}\n$$\n\nso, to the nearest tenth, $C D$ equals 9.8 .\n\n(Note that we could have used\n\n$$\n\\begin{aligned}\n\\sin \\left(105^{\\circ}\\right) & =\\sin \\left(60^{\\circ}+45^{\\circ}\\right)=\\sin \\left(60^{\\circ}\\right) \\cos \\left(45^{\\circ}\\right)+\\cos \\left(60^{\\circ}\\right) \\sin \\left(45^{\\circ}\\right) \\\\\n& =\\frac{\\sqrt{3}}{2} \\cdot \\frac{1}{\\sqrt{2}}+\\frac{1}{2} \\cdot \\frac{1}{\\sqrt{2}}=\\frac{\\sqrt{3}+1}{2 \\sqrt{2}}\n\\end{aligned}\n$$\n\nto say that $C D=\\frac{19}{2\\left(\\frac{\\sqrt{3}+1}{2 \\sqrt{2}}\\right)}=\\frac{19 \\sqrt{2}}{\\sqrt{3}+1}$ exactly, and then evaluated this expression.)']	['$\\frac{19 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+97	"In the diagram, the large circle has radius 9 and centre $C(15,0)$. The small circles have radius 4 and centres $A$ and $B$ on the horizontal line $y=12$. Each of the two small circles is tangent to the large circle. It takes a bug 5 seconds to walk at a constant speed from $A$ to $B$ along the line $y=12$. How far does the bug walk in 1 second?
+
+<image_1>"	['Consider $P$ on $A B$ with $C P$ perpendicular to $A B$. Note that $C P=12$.\n\nSince the small circle with centre $A$ is tangent to the large circle with centre $C$, then $A C$ equals the sum of the radii of these circles, or $A C=4+9=13$. Similarly, $B C=13$.\n\nThis tells us that $\\triangle A P C$ is congruent to $\\triangle B P C$ (they have equal hypotenuses and each is right-angled and has a common side), so $B P=A P$.\n\nBy the Pythagorean Theorem in $\\triangle A P C$,\n\n$$\nA P^{2}=A C^{2}-P C^{2}=13^{2}-12^{2}=169-144=25\n$$\n\nso $A P=5$ (since $A P>0)$.\n\nTherefore, $B P=A P=5$ and so $A B=10$.\n\nSince it takes the bug 5 seconds to walk this distance, then in 1 second, the bug walks a distance of $\\frac{10}{5}=2$.']	['2']		['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+98	"In the diagram, $A B C$ is a right-angled triangle with $P$ and $R$ on $A B$. Also, $Q$ is on $A C$, and $P Q$ is parallel to $B C$. If $R P=2$, $B R=3, B C=4$, and the area of $\triangle Q R C$ is 5 , determine the length of $A P$.
+
+<image_1>"	"['Let $A P=x$ and $Q P=h$.\n\nSince $Q P$ is parallel to $C B$, then $Q P$ is perpendicular to $B A$.\n\nConsider trapezoid $C B P Q$. We can think of this as having parallel bases of lengths 4 and $h$ and height 5 . Thus, its area is $\\frac{1}{2}(4+h)(5)$.\n\nHowever, we can also compute its area by adding the areas of $\\triangle C B R$ (which is $\\left.\\frac{1}{2}(4)(3)\\right)$, $\\triangle C R Q$ (which is given as 5), and $\\triangle R P Q$ (which is $\\left.\\frac{1}{2}(2)(h)\\right)$.\n\nThus,\n\n$$\n\\begin{aligned}\n\\frac{1}{2}(4+h)(5) & =\\frac{1}{2}(4)(3)+5+\\frac{1}{2}(2)(h) \\\\\n20+5 h & =12+10+2 h \\\\\n3 h & =2 \\\\\nh & =\\frac{2}{3}\n\\end{aligned}\n$$\n\nNow, $\\triangle A P Q$ is similar to $\\triangle A B C$, as each has a right angle and they share a common angle at $A$. Thus,\n\n$$\n\\begin{aligned}\n\\frac{A P}{P Q} & =\\frac{A B}{B C} \\\\\n(A P)(B C) & =(P Q)(A B) \\\\\n4 x & =\\frac{2}{3}(x+5) \\\\\n4 x & =\\frac{2}{3} x+\\frac{10}{3} \\\\\n\\frac{10}{3} x & =\\frac{10}{3} \\\\\nx & =1\n\\end{aligned}\n$$\n\n\n\nTherefore, $A P=x=1$.'
+ 'Let $A P=x$ and $Q P=h$.\n\nSince $Q P$ is parallel to $C B$, then $Q P$ is perpendicular to $B A$.\n\nSince $\\triangle A B C$ is right-angled at $B$, its area is $\\frac{1}{2}(4)(5+x)=10+2 x$.\n\nHowever, we can look at the area of the $\\triangle A B C$ in terms of its four triangular pieces:\n\n$\\triangle C B R$ (which has area $\\left.\\frac{1}{2}(4)(3)\\right), \\triangle C R Q$ (which has area 5), $\\triangle Q P R$ (which has area $\\left.\\frac{1}{2} h(2)\\right)$, and $\\triangle Q P A$ (which has area $\\frac{1}{2} x h$ ).\n\nTherefore, $10+2 x=6+5+h+\\frac{1}{2} x h$ so $x h-4 x+2 h+2=0$.\n\nNow, $\\triangle A P Q$ is similar to $\\triangle A B C$, as each has a right angle and they share a common angle at $A$. Thus,\n\n$$\n\\begin{aligned}\n\\frac{A P}{P Q} & =\\frac{A B}{B C} \\\\\n(A P)(B C) & =(P Q)(A B) \\\\\nx(4) & =h(x+5) \\\\\n4 x & =h x+5 h \\\\\n-5 h & =h x-4 x\n\\end{aligned}\n$$\n\nSubstituting this into the equation above, $x h+2 h-4 x+2=0$ becomes $-5 h+2 h+2=0$ or $3 h=2$ or $h=\\frac{2}{3}$.\n\nLastly, we solve for $x$ by subsituting our value for $h$ : $-5\\left(\\frac{2}{3}\\right)=\\frac{2}{3} x-4 x$ or $-\\frac{10}{3}=-\\frac{10}{3} x$ and so $x=1$.\n\nTherefore, $A P=x=1$.']"	['1']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+99	"In the diagram, sector $A O B$ is $\frac{1}{6}$ of an entire circle with radius $A O=B O=18$. The sector is cut into two regions with a single straight cut through $A$ and point $P$ on $O B$. The areas of the two regions are equal. Determine the length of $O P$.
+
+<image_1>"	['Since sector $A O B$ is $\\frac{1}{6}$ of a circle with radius 18 , its area is $\\frac{1}{6}\\left(\\pi \\cdot 18^{2}\\right)$ or $54 \\pi$.\n\nFor the line $A P$ to divide this sector into two pieces of equal area, each piece has area $\\frac{1}{2}(54 \\pi)$ or $27 \\pi$.\n\nWe determine the length of $O P$ so that the area of $\\triangle P O A$ is $27 \\pi$.\n\nSince sector $A O B$ is $\\frac{1}{6}$ of a circle, then $\\angle A O B=\\frac{1}{6}\\left(360^{\\circ}\\right)=60^{\\circ}$.\n\nDrop a perpendicular from $A$ to $T$ on $O B$.\n\n<img_3642>\n\nThe area of $\\triangle P O A$ is $\\frac{1}{2}(O P)(A T)$.\n\n$\\triangle A O T$ is a $30^{\\circ}-60^{\\circ}-90^{\\circ}$ triangle.\n\nSince $A O=18$, then $A T=\\frac{\\sqrt{3}}{2}(A O)=9 \\sqrt{3}$.\n\nFor the area of $\\triangle P O A$ to equal $27 \\pi$, we have $\\frac{1}{2}(O P)(9 \\sqrt{3})=27 \\pi$ which gives $O P=\\frac{54 \\pi}{9 \\sqrt{3}}=\\frac{6 \\pi}{\\sqrt{3}}=2 \\sqrt{3} \\pi$.\n\n(Alternatively, we could have used the fact that the area of $\\triangle P O A$ is $\\frac{1}{2}(O A)(O P) \\sin (\\angle P O A)$.)']	['$2 \\sqrt{3} \\pi$']		['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+100	"In the diagram, a straight, flat road joins $A$ to $B$.
+
+<image_1>
+
+Karuna runs from $A$ to $B$, turns around instantly, and runs back to $A$. Karuna runs at $6 \mathrm{~m} / \mathrm{s}$. Starting at the same time as Karuna, Jorge runs from $B$ to $A$, turns around instantly, and runs back to $B$. Jorge runs from $B$ to $A$ at $5 \mathrm{~m} / \mathrm{s}$ and from $A$ to $B$ at $7.5 \mathrm{~m} / \mathrm{s}$. The distance from $A$ to $B$ is $297 \mathrm{~m}$ and each runner takes exactly $99 \mathrm{~s}$ to run their route. Determine the two values of $t$ for which Karuna and Jorge are at the same place on the road after running for $t$ seconds."	['Suppose that Karuna and Jorge meet for the first time after $t_{1}$ seconds and for the second time after $t_{2}$ seconds.\n\nWhen they meet for the first time, Karuna has run partway from $A$ to $B$ and Jorge has run partway from $B$ to $A$.\n\n<img_3496>\n\nAt this instant, the sum of the distances that they have run equals the total distance from $A$ to $B$.\n\nSince Karuna runs at $6 \\mathrm{~m} / \\mathrm{s}$ for these $t_{1}$ seconds, she has run $6 t_{1} \\mathrm{~m}$.\n\nSince Jorge runs at $5 \\mathrm{~m} / \\mathrm{s}$ for these $t_{1}$ seconds, he has run $5 t_{1} \\mathrm{~m}$.\n\nTherefore, $6 t_{1}+5 t_{1}=297$ and so $11 t_{1}=297$ or $t_{1}=27$.\n\nWhen they meet for the second time, Karuna has run from $A$ to $B$ and is running back to $A$ and Jorge has run from $B$ to $A$ and is running back to $B$. This is because Jorge gets to $A$ halfway through his run before Karuna gets back to $A$ at the end of her run.\n\n<img_3542>\n\nSince they each finish running after 99 seconds, then each has $99-t_{2}$ seconds left to run. At this instant, the sum of the distances that they have left to run equals the total distance from $A$ to $B$.\n\nSince Karuna runs at $6 \\mathrm{~m} / \\mathrm{s}$ for these $\\left(99-t_{2}\\right)$ seconds, she has to run $6\\left(99-t_{2}\\right) \\mathrm{m}$.\n\nSince Jorge runs at $7.5 \\mathrm{~m} / \\mathrm{s}$ for these $\\left(99-t_{2}\\right)$ seconds, he has to run $7.5\\left(99-t_{2}\\right) \\mathrm{m}$.\n\nTherefore, $6\\left(99-t_{2}\\right)+7.5\\left(99-t_{2}\\right)=297$ and so $13.5\\left(99-t_{2}\\right)=297$ or $99-t_{2}=22$ and so $t_{2}=77$.\n\nAlternatively, to calculate the value of $t_{2}$, we note that when Karuna and Jorge meet for the second time, they have each run the distance from $A$ to $B$ one full time and are on their return trips.\n\nThis means that they have each run the full distance from $A$ to $B$ once and the distances that they have run on their return trip add up to another full distance from $A$ to $B$, for a total distance of $3 \\cdot 297 \\mathrm{~m}=891 \\mathrm{~m}$.\n\nKaruna has run at $6 \\mathrm{~m} / \\mathrm{s}$ for $t_{2}$ seconds, for a total distance of $6 t_{2} \\mathrm{~m}$.\n\nJorge ran the first $297 \\mathrm{~m}$ at $5 \\mathrm{~m} / \\mathrm{s}$, which took $\\frac{297}{5} \\mathrm{~s}$ and ran the remaining $\\left(t_{2}-\\frac{297}{5}\\right)$ seconds at $7.5 \\mathrm{~m} / \\mathrm{s}$, for a total distance of $\\left(297+7.5\\left(t_{2}-\\frac{297}{5}\\right)\\right) \\mathrm{m}$.\n\nTherefore,\n\n$$\n\\begin{aligned}\n6 t_{2}+297+7.5\\left(t_{2}-\\frac{297}{5}\\right) & =891 \\\\\n13.5 t_{2} & =891-297+7.5 \\cdot \\frac{297}{5} \\\\\n13.5 t_{2} & =1039.5 \\\\\nt_{2} & =77\n\\end{aligned}\n$$\n\nTherefore, Karuna and Jorge meet after 27 seconds and after 77 seconds.']	['27, 77']		['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']	Multimodal	Competition	True		Numerical		Open-ended	Algebra	Math	English
+101	"In the diagram, rectangle $P Q R S$ is placed inside rectangle $A B C D$ in two different ways: first, with $Q$ at $B$ and $R$ at $C$; second, with $P$ on $A B, Q$ on $B C, R$ on $C D$, and $S$ on $D A$.
+<image_1>
+
+If $A B=718$ and $P Q=250$, determine the length of $B C$."	['Let $B C=x, P B=b$, and $B Q=a$.\n\nSince $B C=x$, then $A D=P S=Q R=x$.\n\nSince $B C=x$ and $B Q=a$, then $Q C=x-a$.\n\nSince $A B=718$ and $P B=b$, then $A P=718-b$.\n\nNote that $P Q=S R=250$.\n\nLet $\\angle B Q P=\\theta$.\n\nSince $\\triangle P B Q$ is right-angled at $B$, then $\\angle B P Q=90^{\\circ}-\\theta$.\n\nSince $B Q C$ is a straight angle and $\\angle P Q R=90^{\\circ}$, then $\\angle R Q C=180^{\\circ}-90^{\\circ}-\\theta=90^{\\circ}-\\theta$.\n\nSince $A P B$ is a straight angle and $\\angle S P Q=90^{\\circ}$, then $\\angle A P S=180^{\\circ}-90^{\\circ}-\\left(90^{\\circ}-\\theta\\right)=\\theta$.\n\nSince $\\triangle S A P$ and $\\triangle Q C R$ are each right-angled and have another angle in common with $\\triangle P B Q$, then these\n\n<img_3870>\nthree triangles are similar.\n\nContinuing in the same way, we can show that $\\triangle R D S$ is also similar to these three triangles.\n\nSince $R S=P Q$, then $\\triangle R D S$ is actually congruent to $\\triangle P B Q$ (angle-side-angle).\n\nSimilarly, $\\triangle S A P$ is congruent to $\\triangle Q C R$.\n\nIn particular, this means that $A S=x-a, S D=a, D R=b$, and $R C=718-b$.\n\nSince $\\triangle S A P$ and $\\triangle P B Q$ are similar, then $\\frac{S A}{P B}=\\frac{A P}{B Q}=\\frac{S P}{P Q}$.\n\nThus, $\\frac{x-a}{b}=\\frac{718-b}{a}=\\frac{x}{250}$.\n\nAlso, by the Pythagorean Theorem in $\\triangle P B Q$, we obtain $a^{2}+b^{2}=250^{2}$.\n\nBy the Pythagorean Theorem in $\\triangle S A P$,\n\n$$\n\\begin{aligned}\nx^{2} & =(x-a)^{2}+(718-b)^{2} \\\\\nx^{2} & =x^{2}-2 a x+a^{2}+(718-b)^{2} \\\\\n0 & =-2 a x+a^{2}+(718-b)^{2}\n\\end{aligned}\n$$\n\nSince $a^{2}+b^{2}=250^{2}$, then $a^{2}=250^{2}-b^{2}$.\n\nSince $\\frac{718-b}{a}=\\frac{x}{250}$, then $a x=250(718-b)$.\n\nTherefore, substituting into $(*)$, we obtain\n\n$$\n\\begin{aligned}\n0 & =-2(250)(718-b)+250^{2}-b^{2}+(718-b)^{2} \\\\\nb^{2} & =250^{2}-2(250)(718-b)+(718-b)^{2} \\\\\nb^{2} & =((718-b)-250)^{2} \\quad\\left(\\text { since } y^{2}-2 y z+z^{2}=(y-z)^{2}\\right) \\\\\nb^{2} & =(468-b)^{2} \\\\\nb & =468-b \\quad(\\text { since } b \\neq b-468) \\\\\n2 b & =468 \\\\\nb & =234\n\\end{aligned}\n$$\n\nTherefore, $a^{2}=250^{2}-b^{2}=250^{2}-234^{2}=(250+234)(250-234)=484 \\cdot 16=22^{2} \\cdot 4^{2}=88^{2}$ and so $a=88$.\n\nFinally, $x=\\frac{250(718-b)}{a}=\\frac{250 \\cdot 484}{88}=1375$. Therefore, $B C=1375$.']	['1375']		['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ish
+102	"How many equilateral triangles of side $1 \mathrm{~cm}$, placed as shown in the diagram, are needed to completely cover the interior of an equilateral triangle of side $10 \mathrm{~cm}$ ?
+
+<image_1>"	"['If we proceed by pattern recognition, we find after row 1 we have a total of 1 triangle, after two rows we have $2^{2}$ or 4 triangles. After ten rows we have $10^{2}$ or 100 triangles.\n\n<img_3802>'
+ 'This solution is based on the fact that the ratio of areas for similar triangles is the square of the ratio of corresponding sides. Thus the big triangle with side length ten times that of the smaller triangle has 100 times the area.']"	['100']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+103	"A rectangle PQRS has side PQ on the x-axis and touches the graph of $y=k \cos x$ at the points $S$ and $R$ as shown. If the length of $P Q$ is $\frac{\pi}{3}$ and the area of the rectangle is $\frac{5 \pi}{3}$, what is the value of $k ?$
+
+<image_1>"	['If $P Q=\\frac{\\pi}{3}$, then by symmetry the coordinates of $R$\n\nare $\\left(\\frac{\\pi}{6}, k \\cos \\frac{\\pi}{6}\\right)$.\n\nArea of rectangle $P Q R S=\\frac{\\pi}{3}\\left(k \\cos \\frac{\\pi}{6}\\right)=\\frac{\\pi}{3}(k)\\left(\\frac{\\sqrt{3}}{2}\\right)$\n\nBut $\\frac{\\sqrt{3} k \\pi}{6}=\\frac{5 \\pi}{3} \\quad \\therefore k=\\frac{10}{\\sqrt{3}}$ or $\\frac{10}{3} \\sqrt{3}$.\n\n<img_3853>']	['$\\frac{10}{\\sqrt{3}}$,$\\frac{10}{3} 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+104	"In determining the height, $M N$, of a tower on an island, two points $A$ and $B, 100 \mathrm{~m}$ apart, are chosen on the same horizontal plane as $N$. If $\angle N A B=108^{\circ}$, $\angle A B N=47^{\circ}$ and $\angle M B N=32^{\circ}$, determine the height of the tower to the nearest metre.
+
+<image_1>"	['In $\\triangle B A N, \\angle B N A=25^{\\circ}$\n\nUsing the Sine Law in $\\triangle B A N$,\n\n$\\frac{N B}{\\sin 108^{\\circ}}=\\frac{100}{\\sin 25^{\\circ}}$\n\nTherefore $N B=\\frac{100 \\sin 108^{\\circ}}{\\sin 25^{\\circ}} \\approx 225.04$,\n\n<img_3946>\n\nNow in $\\triangle M N B, \\frac{M N}{N B}=\\tan 32^{\\circ}$\n\n$$\nM N=\\frac{100 \\sin 108^{\\circ}}{\\sin 25^{\\circ}} \\times \\tan 32^{\\circ} \\doteq 140.6\n$$\n\nThe tower is approximately $141 \\mathrm{~m}$ high.']	['141']		['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+105	"The points $A, P$ and a third point $Q$ (not shown) are the vertices of a triangle which is similar to triangle $A B C$. What are the coordinates of all possible positions for $Q$ ?
+
+<image_1>"	['$Q(4,0), Q(0,4)$\n\n$Q(2,0), Q(0,2)$\n\n$Q(-2,2), 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+106	"In triangle $A B C, B C=2$. Point $D$ is on $\overline{A C}$ such that $A D=1$ and $C D=2$. If $\mathrm{m} \angle B D C=2 \mathrm{~m} \angle A$, compute $\sin A$.
+
+<image_1>"	['Let $[A B C]=K$. Then $[B C D]=\\frac{2}{3} \\cdot K$. Let $\\overline{D E}$ be the bisector of $\\angle B D C$, as shown below.\n\n<img_3399>\n\nNotice that $\\mathrm{m} \\angle D B A=\\mathrm{m} \\angle B D C-\\mathrm{m} \\angle A=\\mathrm{m} \\angle A$, so triangle $A D B$ is isosceles, and $B D=1$. (Alternately, notice that $\\overline{D E} \\| \\overline{A B}$, and by similar triangles, $[C D E]=\\frac{4}{9} \\cdot K$, which means $[B D E]=\\frac{2}{9} \\cdot K$. Because $[C D E]:[B D E]=2$ and $\\angle B D E \\cong \\angle C D E$, conclude that $\\frac{C D}{B D}=2$, thus $B D=1$.) Because $B C D$ is isosceles, it follows that $\\cos \\angle B D C=\\frac{1}{2} B D / C D=\\frac{1}{4}$. By the half-angle formula,\n\n$$\n\\sin A=\\sqrt{\\frac{1-\\cos \\angle B D C}{2}}=\\sqrt{\\frac{3}{8}}=\\frac{\\sqrt{6}}{\\mathbf{4}}\n$$']	['$\\frac{\\sqrt{6}}{4}$']		['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+107	"Two equilateral triangles of side length 1 and six isosceles triangles with legs of length $x$ and base of length 1 are joined as shown below; the net is folded to make a solid. If the volume of the solid is 6 , compute $x$.
+
+<image_1>"	['First consider a regular octahedron of side length 1. To compute its volume, divide it into two square-based pyramids with edges of length 1 . Such a pyramid has slant height $\\frac{\\sqrt{3}}{2}$ and height $\\sqrt{\\left(\\frac{\\sqrt{3}}{2}\\right)^{2}-\\left(\\frac{1}{2}\\right)^{2}}=\\sqrt{\\frac{1}{2}}=\\frac{\\sqrt{2}}{2}$, so its volume is $\\frac{1}{3} \\cdot 1^{2} \\cdot \\frac{\\sqrt{2}}{2}=\\frac{\\sqrt{2}}{6}$. Thus the octahedron has volume twice that, or $\\frac{\\sqrt{2}}{3}$. The result of folding the net shown is actually the image of a regular octahedron after being stretched along an axis perpendicular to one face by a factor of $r$. Because the octahedron is only being stretched in one dimension, the volume changes by the same factor $r$. So the problem reduces to computing the factor $r$ and the edge length of the resulting octahedron.\n\nFor convenience, imagine that one face of the octahedron rests on a plane. Seen from above the plane, the octahedron appears as shown below.\n\n\n\n<img_3368>\n\nLet $P$ be the projection of $A$ onto the plane on which the octahedron rests, and let $Q$ be the foot of the perpendicular from $P$ to $\\overline{B C}$. Then $P Q=R_{C}-R_{I}$, where $R_{C}$ is the circumradius and $R_{I}$ the inradius of the equilateral triangle. Thus $P Q=\\frac{2}{3}\\left(\\frac{\\sqrt{3}}{2}\\right)-\\frac{1}{3}\\left(\\frac{\\sqrt{3}}{2}\\right)=\\frac{\\sqrt{3}}{6}$. Then $B P^{2}=P Q^{2}+B Q^{2}=\\frac{3}{36}+\\frac{1}{4}=\\frac{1}{3}$, so $A P^{2}=A B^{2}-B P^{2}=\\frac{2}{3}$, and $A P=\\frac{\\sqrt{6}}{3}$.\n\nNow let a vertical stretch take place along an axis parallel to $\\overleftrightarrow{A P}$. If the scale factor is $r$, then $A P=\\frac{r \\sqrt{6}}{3}$, and because the stretch occurs on an axis perpendicular to $\\overline{B P}$, the length $B P$ is unchanged, as can be seen below.\n\n<img_3791>\n\nThus $A B^{2}=\\frac{6 r^{2}}{9}+\\frac{1}{3}=\\frac{6 r^{2}+3}{9}$. It remains to compute $r$. But $r$ is simply the ratio of the new volume to the old volume:\n\n$$\nr=\\frac{6}{\\frac{\\sqrt{2}}{3}}=\\frac{18}{\\sqrt{2}}=9 \\sqrt{2}\n$$\n\nThus $A B^{2}=\\frac{6(9 \\sqrt{2})^{2}+3}{9}=\\frac{975}{9}=\\frac{325}{3}$, and $A B=\\frac{5 \\sqrt{39}}{3}$.']	['$\\frac{5 \\sqrt{39}}{3}$']		['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+108	"Let $T=5$. The diagram at right consists of $T$ congruent circles, each of radius 1 , whose centers are collinear, and each pair of adjacent circles are externally tangent to each other. Compute the length of the tangent segment $\overline{A B}$.
+
+<image_1>"	['For each point of tangency of consecutive circles, drop a perpendicular from that point to $\\overline{A B}$. For each of the $T-2$ circles between the first and last circles, the distance between consecutive perpendiculars is $2 \\cdot 1=2$. Furthermore, the distance from $A$ to the first perpendicular equals 1 (i.e., the common radius of the circles), which also equals the distance from the last perpendicular to $B$. Thus $A B=1+(T-2) \\cdot 2+1=2(T-1)$. With $T=5$, it follows that $A B=2 \\cdot 4=8$.']	['8']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+109	"Square $A B C D$ has side length 22. Points $G$ and $H$ lie on $\overline{A B}$ so that $A H=B G=5$. Points $E$ and $F$ lie outside square $A B C D$ so that $E F G H$ is a square. Compute the area of hexagon $A E F B C D$.
+
+<image_1>"	['Note that $G H=A B-A H-B G=22-5-5=12$. Thus\n\n$$\n\\begin{aligned}\n{[A E F B C D] } & =[A B C D]+[E F G H]+[A E H]+[B F G] \\\\\n& =22^{2}+12^{2}+\\frac{1}{2} \\cdot 5 \\cdot 12+\\frac{1}{2} \\cdot 5 \\cdot 12 \\\\\n& =484+144+30+30 \\\\\n& =\\mathbf{6 8 8} .\n\\end{aligned}\n$$']	['688']		['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+110	"Suppose that Xena traces a path along the segments in the figure shown, starting and ending at point $A$. The path passes through each of the eleven vertices besides $A$ exactly once, and only visits $A$ at the beginning and end of the path. Compute the number of possible paths Xena could trace.
+
+<image_1>"	['Count the number of complete paths that pass through all vertices exactly once (such a path is called a Hamiltonian path). The set of vertices can be split into two rings:\n\n$$\n\\mathcal{I}=\\left\\{A_{1}, A_{2}, \\ldots, A_{6}\\right\\} \\text { (i.e., the inner ring), } \\quad \\mathcal{O}=\\left\\{B_{1}, B_{2}, \\ldots, B_{6}\\right\\} \\text { (i.e., the outer ring). }\n$$\n\nwhere $A_{1}=A$. The two rings are connected by the edges $E=\\left\\{A_{1} B_{1}, A_{2} B_{2}, \\ldots, A_{6} B_{6}\\right\\}$. Each vertex in the figure has exactly three edges joining it with the neighboring vertices. Also note that any closed loop must use exactly two edges (out of three) for each vertex.\n\nFurther note that a loop must use at least one edge from $E$ to move from one ring to the other. Consider two cases: the loop uses all six edges from $E$, or it uses some but not all of them.\n\nIf all edges from $E$ are used, there are two possible undirected loops. It is not possible to use both edges $A_{1} A_{2}$ and $B_{1} B_{2}$, so either $A_{1} A_{2}$ or $B_{1} B_{2}$ will be used. This choice determines how the entire loop is constructed.\n\nIf not all edges from $E$ are used, then there must be some $i$ for which the loop uses the edge $A_{i} B_{i}$ and does not use $A_{i+1} B_{i+1}$ (where $A_{j} B_{j}$ represents $A_{j-6} B_{j-6}$ if $7 \\leq j \\leq 12$ ). Because $A_{i+1}$ is only connected to three other vertices, the loop must use $A_{i} A_{i+1}$ and $A_{i+1} A_{i+2}$, and similarly must use $B_{i} B_{i+1}$ and $B_{i+1} B_{i+2}$. This also precludes using $A_{i+2} B_{i+2}$, because doing so would close the loop before it visits all 12 vertices. Therefore the loop must also use $A_{i+2} A_{i+3}$ and $B_{i+2} B_{i+3}$, which now precludes using $A_{i+3} B_{i+3}$. This continues to force the structure of the loop until it closes by using $A_{i+5} B_{i+5}=A_{i-1} B_{i-1}$. Hence the loop must use exactly two edges from $E$, and they must be consecutive: $A_{i-1} B_{i-1}$ and $A_{i} B_{i}$. There are 6 ways to choose those two consecutive edges, so there are 6 possible undirected loops in this case.\n\nThe forced path is a loop, and the only way the given conditions are satisfied if $A_{i+k}=A_{i-1}$ and $B_{i+k}=B_{i-1}$. Hence the loop must use (precisely) two consecutive edges from $E: A_{i-1} B_{i-1}$ and $A_{i} B_{i}$. There are 6 ways to choose two consecutive edges, so there are 6 possible undirected loops in this case.\n\nEach undirected loop can be traced in two ways, and thus the number of ways for Xena to trace the path is $(6+2) \\cdot 2=16$.']	['16']		['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+111	"Let $T$ be a rational number. Two coplanar squares $\mathcal{S}_{1}$ and $\mathcal{S}_{2}$ each have area $T$ and are arranged as shown to form a nonconvex octagon. The center of $\mathcal{S}_{1}$ is a vertex of $\mathcal{S}_{2}$, and the center of $\mathcal{S}_{2}$ is a vertex of $\mathcal{S}_{1}$. Compute $\frac{\text { area of the union of } \mathcal{S}_{1} \text { and } \mathcal{S}_{2}}{\text { area of the intersection of } \mathcal{S}_{1} \text { and } \mathcal{S}_{2}}$.
+
+<image_1>"	['Let $2 x$ be the side length of the squares. Then the intersection of $\\mathcal{S}_{1}$ and $\\mathcal{S}_{2}$ is a square of side length $x$, so its area is $x^{2}$. The area of the union of $\\mathcal{S}_{1}$ and $\\mathcal{S}_{2}$ is $(2 x)^{2}+(2 x)^{2}-x^{2}=7 x^{2}$. Thus the desired ratio of areas is $\\frac{7 x^{2}}{x^{2}}=7$ (independent of $T$ ).']	['7']		['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+112	"In acute triangle $I L K$, shown in the figure, point $G$ lies on $\overline{L K}$ so that $\overline{I G} \perp \overline{L K}$. Given that $I L=\sqrt{41}$ and $L G=I K=5$, compute $G K$.
+
+<image_1>"	['Using the Pythagorean Theorem, $I G=\\sqrt{(I L)^{2}-(L G)^{2}}=\\sqrt{41-25}=4$, and $G K=\\sqrt{(I K)^{2}-(I G)^{2}}=$ $\\sqrt{25-16}=3$.']	['3']		['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+113	"This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only if they are connected by an edge in the diagram, also called a hideout map (or map). For the purposes of this Power Question, the map must be connected; that is, given any two hideouts, there must be a path from one to the other. To clarify, the Robber may not stay in the same hideout for two consecutive days, although he may return to a hideout he has previously visited. For example, in the map below, if the Robber holes up in hideout $C$ for day 1 , then he would have to move to $B$ for day 2 , and would then have to move to either $A, C$, or $D$ on day 3.
+
+<image_1>
+
+Every day, each Cop searches one hideout: the Cops know the location of all hideouts and which hideouts are adjacent to which. Cops are thorough searchers, so if the Robber is present in the hideout searched, he is found and arrested. If the Robber is not present in the hideout searched, his location is not revealed. That is, the Cops only know that the Robber was not caught at any of the hideouts searched; they get no specific information (other than what they can derive by logic) about what hideout he was in. Cops are not constrained by edges on the map: a Cop may search any hideout on any day, regardless of whether it is adjacent to the hideout searched the previous day. A Cop may search the same hideout on consecutive days, and multiple Cops may search different hideouts on the same day. In the map above, a Cop could search $A$ on day 1 and day 2, and then search $C$ on day 3 .
+
+The focus of this Power Question is to determine, given a hideout map and a fixed number of Cops, whether the Cops can be sure of catching the Robber within some time limit.
+
+Map Notation: The following notation may be useful when writing your solutions. For a map $M$, let $h(M)$ be the number of hideouts and $e(M)$ be the number of edges in $M$. The safety of a hideout $H$ is the number of hideouts adjacent to $H$, and is denoted by $s(H)$.
+
+The Cop number of a map $M$, denoted $C(M)$, is the minimum number of Cops required to guarantee that the Robber is caught.
+Find $C(M)$ for the map below.
+
+<image_2>"	['First we prove that for all maps $M, C(M)<h(M)$.\n\nProve. Let $n=h(M)$. The following strategy will always catch a Robber within two days using $n-1$ Cops, which proves that $C(M) \\leq n-1$. Choose any subset $\\mathcal{S}$ of $n-1$ hideouts and position $n-1$ Cops at the hideouts of $\\mathcal{S}$ for 2 days. If the Robber is not caught on the first day, he must have been at the hideout not in $\\mathcal{S}$, and therefore must move to a hideout in $\\mathcal{S}$ on the following day.\n\nso $C(M) \\leq 6$. To see that 6 is minimal, note that it is possible for the Robber to go from any hideout to any other hideout in a single night. Suppose that on a given day, the Robber successfully evades capture. Without loss of generality, assume that the Robber was at hideout $A_{1}$. Then the only valid conclusion the Cops can draw is that on the next day, the Robber will not be at $A_{1}$. Thus the only hideout the Cops can afford to leave open is $A_{1}$ itself. If they leave another hideout open, the Robber might have chosen to hide there, and the process will repeat. So we get $C(M)=6$.']	['6']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAVABvwMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKQ0ALketFcB8SfHg8MJYaTp7q2t6nMkcI4PkozbTIR+YHqc9dpFd6p60AOooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKADNZevazDoeltdvG88pYR29vGfnnlY4VF+p79hkngGr9xNFbQPNPKkUUalndyAqgckknoK4e10aPx7NH4hvpr23sUz/ZEdrcvA6xkENOxUg7pAeB2QjuxoA4Txpo0sHjzwDDfSrPqt/qZu9QkXpkPFtRc9EUAqPpnqa93HrXh2q6RBZ/tDeE9Ohur64SO3e4Y3l5JcFW2ytgF2JH3BwPY969xHpQAtFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQBka/4a0zxNaC01WKea2/iiS5liV84+8EYbunfOOam0nRbPRNLj02wWaO0jXaiPcSSlBjAAZ2LADsAcDtWjRQByEnw18Ly6wusPZXjakhyt2dSufMHUcN5mQME8V1cMYhiWNd21VABZix4Hcnk/U81JRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUZxQAUhpayte8Q6V4b0832rX8NpAM4MjcuQM4UDlj7AE+1AGdrnj/wr4dv0stV1u2t7pmCmLl2Qn+8FB2jnqcVv2V5a6haR3Vlcw3NvIMpLC4dGHsRwa+GdbvLjUNbvb+5dnlupmnMhTZu3HcGC9gc8Dng19N/ACC+h+GwN3kQyXcjWgJziPCg49t4c/iT3oA9SooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoozjrSHrjPNAC1FPNFDA00sqRxICzOzbQoHcnoB9a43xZ8S9H8NXK6dbpLqutyHEWm2Q3yFvRsZ2/z9AeawI/BHifx5NHd+Pb37JpobdHoVhIQvsZXycnj178FelAFjUvibd63fSaL8PdN/ti8XCy6jKNtpBnuTxu4Deg443dKl0f4Z20F03iLxtqJ17V1Uu0lyP9GtgOSEQ8YHPJGO4VTXeaXpdjo9hHZadZw2ltGPlihQKo98Dv79TXE/FzU518OWvhzTiP7S8Q3C2MXP3YyR5jH/AGcEA+zUAY/w60a08Z3+seOtY06C5F/d7NNS6hV/JhiO1WUEHaeAD7oT3r1lRgYwAOwFUtF0u20PRbLS7QYgtIViTjkgDGT7nqav0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUVBdXVvZ28lzczxwwRqWeSRwqqB1JJ4FeaXvxH1bxPeyaV8OtN+3Mh2TatdKVtYPp/ePOf5Bh0AO58R+KdF8K2H2zWNQitYyDtUnLyeyqOSenQV5+b/AMb/ABLUDSo5fC/ht/8Al8lH+mXC/wCwB9wHjkH6MeRWz4d+F9nY6ida8R3UniDXWIJubsZji74jQ5AA7enbHSu/Xvnv7UAc14V8CaD4Ot/L0q0/fsMTXc2Glm9dzY/HAAFdIoxTqKAEbpj+VeW+HR/wmfxe1bxGT5mm6Cv9nWBxwZj/AK1x24ywz3DKe1dP8R/Ev/CK+Cb6/ifF46/Z7QDk+c4wuB3IGWx6A1L8PvDI8J+C9P0tlAuQnm3R67pn+Zue+M7QfRRQB0wz3paKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiuf8U+MdD8I2YuNXvo4WIJjhX5pZP91ByfTPT1IoA3zziuE8UfE3S9GvP7J0u3l1zXWO1bCy+faeh3sMhcenJ9h1rB/4rr4m9PN8KeGX/wDA25T9NoI+nX+MV3XhfwZofg+y+zaPYpESB5kzfNJIfVmxnueBwM8AUAcXb/D/AMQeMriPUPiJqG6BW3xaHZOVgjPYuwOWI56HP+1jivS7CxtdNso7Oytora2iXEcMSBVUewHFWBS0AFFFFABSNnHFLWb4g1i28P6BfatdnEFpC0rDOC2Bwo9ycAe5oA8/1lf+Ey+Mem6MmW0zw1GL274+VrlseWh7EgYYf8DFeorXn/wj0e4tvC82u6io/tTX521C4YLjCsfkUe2PmA7bq9BFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFNbkUAOqtf31rp1nJeXtzFbW0Q3SSyuFVR6kmuH8SfFCysdROieHLSXxBrzfKLW0OY4z6yOOAB3x+JXrWdY/DjVfFF5Hq3xE1D7cyt5kOkWrFLWE++D8x7foSwoAbP4/1/xncy6b8PNOBgRvLm1y+UrBGe+xSPmPTqPquOa2fDPww03Rrw6vq00mua85Dvf3nzbT/sKchcdjyR2wOK7S0tYLK3S2tYI4LeJQsccShVUegA4FT0AIOvTtS0UUAFFFFABRRRQAjfzry/4nSSeJfEGgeALZmC30ovNRKcbbaM5xn3IOP9pV9a9OlkSGNpZGCogLMzHAAAySa8x+F8T+JNZ17x9dRsP7SmNrp6sPuW0Zxx6ZIAPuh9aAPTokSKJY40CIoAVQMAD0Ap9IvfrS0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFJketAC0ZqlqmqWOj2Ml7qN3Da20Yy8szhVH59z6V5tJ438TePJXs/AdgbPTt22TXr5CEx38pCDk+/PuF60Adj4t8c6D4Ntll1W8xMwzFaRfPNKewC5/U4Hv0rizp3jf4ljOpvL4W8NSdbOI/6ZcL33kj5AeeCPqG610fhX4a6R4cuDqVyZdW1yQ7pdRvjvk3eq5J2/UZPOMmu0UdTzz60AY3hzwto/hSwFno9hFbRnl2Ay8h9Wbqe/wBO2BW0KWigAooooAKKKKACiiigAoopD/k0Aee/F7V7iDwzB4e05lOp+IJxYQqT0RseYx9sEKT2312eh6Tb6FolnpVopEFpCsSZ6nA6n3PU+5rzzw//AMVp8YNV185fTPDqnT7Lj5WnOfMYfT5h9GU16kowKAFooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAozjrSEgdSOK5bxd4+0LwhGsd7cNNfycQ2FsN88hPT5ewPqce3NAHU5B54rzrXvilD/aP9ieDrJvEOtHIxAf8AR4P9p3zjGe2QODkjjOYdA8Z/EnD+I5X8OeHX5XS7V/8ASJl9JWPQe2PT5cjNeh6F4d0nw3p62OkafFaW46hByx9Wbqx9yaAOH034ZXmuX0esfELUP7XvFOYdOiJW0ts9gvG48DPrjndwa9KgijgiWKKNY4kAVEVQFUegA6U8ZpaACiiigAooooAKKKKACiiigAooooAK5T4i+Jx4T8E6hqUbf6UV8i1UHlpn4XH0+9j0U11Rry7Xj/wmXxh0rQUJbTfDqDULwfwtOQPLQ9iQMEexb0oA6j4deGf+ET8E6fpsi4uinnXTdzM/Lc98cLn0UV1VNXvTqACiiig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+114	"This Question involves one Robber and one or more Cops. After robbing a bank, the Robber retreats to a network of hideouts, represented by dots in the diagram below. Every day, the Robber stays holed up in a single hideout, and every night, the Robber moves to an adjacent hideout. Two hideouts are adjacent if and only if they are connected by an edge in the diagram, also called a hideout map (or map). For the purposes of this Power Question, the map must be connected; that is, given any two hideouts, there must be a path from one to the other. To clarify, the Robber may not stay in the same hideout for two consecutive days, although he may return to a hideout he has previously visited. For example, in the map below, if the Robber holes up in hideout $C$ for day 1 , then he would have to move to $B$ for day 2 , and would then have to move to either $A, C$, or $D$ on day 3.
+
+<image_1>
+
+Every day, each Cop searches one hideout: the Cops know the location of all hideouts and which hideouts are adjacent to which. Cops are thorough searchers, so if the Robber is present in the hideout searched, he is found and arrested. If the Robber is not present in the hideout searched, his location is not revealed. That is, the Cops only know that the Robber was not caught at any of the hideouts searched; they get no specific information (other than what they can derive by logic) about what hideout he was in. Cops are not constrained by edges on the map: a Cop may search any hideout on any day, regardless of whether it is adjacent to the hideout searched the previous day. A Cop may search the same hideout on consecutive days, and multiple Cops may search different hideouts on the same day. In the map above, a Cop could search $A$ on day 1 and day 2, and then search $C$ on day 3 .
+
+The focus of this Power Question is to determine, given a hideout map and a fixed number of Cops, whether the Cops can be sure of catching the Robber within some time limit.
+
+Map Notation: The following notation may be useful when writing your solutions. For a map $M$, let $h(M)$ be the number of hideouts and $e(M)$ be the number of edges in $M$. The safety of a hideout $H$ is the number of hideouts adjacent to $H$, and is denoted by $s(H)$.
+
+The Cop number of a map $M$, denoted $C(M)$, is the minimum number of Cops required to guarantee that the Robber is caught.
+
+
+The police want to catch the Robber with a minimum number of Cops, but time is of the essence. For a map $M$ and a fixed number of Cops $c \geq C(M)$, define the capture time, denoted $D(M, c)$, to be the minimum number of days required to guarantee a capture using $c$ Cops. For example, in the graph below, if three Cops are deployed, they might catch the Robber in the first day, but if they don't, there is a strategy that will guarantee they will capture the Robber within two days. Therefore the capture time is $D\left(\mathcal{C}_{6}, 3\right)=2$.
+
+<image_2>
+Definition: The workday number of $M$, denoted $W(M)$, is the minimum number of Cop workdays needed to guarantee the Robber's capture. For example, a strategy that guarantees capture within three days using 17 Cops on the first day, 11 Cops on the second day, and only 6 Cops on the third day would require a total of $17+11+6=34$ Cop workdays.
+
+Determine $W(M)$ for each of the maps in figure a and figure b.
+<image_3>
+figure a
+
+<image_4>
+figure b"	"['or the map $M$ from figure a, $W(M)=7$. The most efficient strategy is to use 7 Cops to blanket all the hideouts on the first day. Any strategy using fewer than 7 Cops would require 6 Cops on each of two consecutive days: given that any hideout can be reached from any other hideout, leaving more than one hideout unsearched on one day makes it is impossible to eliminate any hideouts the following day. So any other strategy would require a minimum of 12 Cop workdays.\n\nFor the map $M$ from figure b, $W(M)=8$. The strategy outlined in $2 \\mathrm{~b}$ used three Cops for a maximum of four workdays, yielding 12 Cop workdays. The most efficient strategy is to use 4 Cops, positioned at $\\{B, E, H, K\\}$ for 2 days each. The following argument demonstrates that 8 Cop workdays is in fact minimal. First, notice that there is no advantage to searching one of the hideouts between vertices of the square (for example, $C$ ) without searching the other hideout between the same vertices (for example, $D$ ). The Robber can reach $C$ on day $n$ if and only if he is at either $B$ or $E$ on day $n-1$, and in either case he could just as well go to $D$ instead of $C$. So there is no situation in which the Robber is certain to be caught at $C$ rather than at $D$. Additionally, the Robber\'s possible locations on day $n+1$ are the same whether he is at $C$ or $D$ on day $n$, so searching one rather than the other fails to rule out any locations for future days. So any successful strategy that involves searching $C$ should also involve searching $D$ on the same day, and similarly for $F$ and $G, I$ and $J$, and $L$ and $A$. On the other hand, if the Robber must be at one of $C$ and $D$ on day $n$, then he must be at either $B$ or $E$ on day $n+1$, because those are the only adjacent hideouts. So any strategy that involves searching both hideouts of one of the off-the-square pairs on day $n$ is equivalent to a strategy that searches the adjacent on-the-square hideouts on day $n+1$; the two strategies use the same number of Cops for the same number of workdays. Thus the optimal number of Cop workdays can be achieved using strategies that only search the ""corner"" hideouts $B, E, H, K$.\n\nRestricting the search to only those strategies searching corner hideouts $B, E, H, K$, a total of 8 workdays can be achieved by searching all four hideouts on two consecutive days: if the Robber is at one of the other eight hideouts the first day, he must move to one of the two adjacent corner hideouts the second day. But each of these corner hideouts is adjacent to two other corner hideouts. So if only one hideout is searched, for no matter how many consecutive days, the following day, the Robber could either be back at the previously-searched hideout or be at any other hideout: no possibilities are ruled out. If two adjacent corners are searched, the Cops do no better, as the following argument shows. Suppose that $B$ and $E$ are both searched for two consecutive days. Then the Cops can rule out $B, E, C$, and $D$ as possible locations, but if the Cops then switch to searching either $H$ or $K$ instead of $B$ or $E$, the Robber can go back to $C$ or $D$ within two days. So searching two adjacent corner hideouts for two days is fruitless and costs four Cop workdays. Searching diagonally opposite corner hideouts is even less fruitful, because doing so rules out none of the other hideouts as possible Robber locations. Using three Cops each day, it is easy to imagine scenarios in which the Robber evades capture for three days before being caught: for example, if $B, E, H$ are searched for two consecutive days, the Robber goes from $I$ or $J$ to $K$ to $L$. Therefore if three Cops are used, four days are required for a total of 12 Cop workdays.\n\nIf there are more than four Cops, the preceding arguments show that the number of Cops must be even to produce optimal results (because there is no advantage to searching one hideout between vertices of the square without searching the other). Using six Cops with four at corner hideouts yields no improvement, because the following day, the Robber could get to any of the four corner hideouts, requiring at least four Cops the second day, for ten Cop workdays. If two or fewer Cops are at corner hideouts, the situation is even worse, because if the Robber is not caught that day, he has at least nine possible hideouts the following day (depending on whether the unsearched corners are adjacent or diagonally opposite to each other). Using eight Cops (with four at corner vertices) could eliminate one corner vertex as a possible location for the second day (if the non-corner hideouts searched are on adjacent sides of the square), but eight Cop workdays have already been used on the first day. So 8 Cop workdays is minimal.']"	['12, 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']	Multimodal	Competition	True		Numerical		Open-ended	Combinatorics	Math	English
+115	"In $\triangle A B C, \mathrm{~m} \angle A=\mathrm{m} \angle B=45^{\circ}$ and $A B=16$. Mutually tangent circular arcs are drawn centered at all three vertices; the arcs centered at $A$ and $B$ intersect at the midpoint of $\overline{A B}$. Compute the area of the region inside the triangle and outside of the three arcs.
+
+<image_1>"	['Because $A B=16, A C=B C=\\frac{16}{\\sqrt{2}}=8 \\sqrt{2}$. Then each of the large arcs has radius 8 , and the small arc has radius $8 \\sqrt{2}-8$. Each large arc has measure $45^{\\circ}$ and the small arc has measure $90^{\\circ}$. Therefore the area enclosed by each large arc is $\\frac{45}{360} \\cdot \\pi \\cdot 8^{2}=8 \\pi$, and the area enclosed by the small arc is $\\frac{90}{360} \\cdot \\pi \\cdot(8 \\sqrt{2}-8)^{2}=48 \\pi-32 \\pi \\sqrt{2}$. Thus the sum of the areas enclosed by the three arcs is $64 \\pi-32 \\pi \\sqrt{2}$. On the other hand, the area of the triangle is $\\frac{1}{2}(8 \\sqrt{2})^{2}=64$. So the area of the desired region is $64-64 \\pi+32 \\pi \\sqrt{2}$.']	['$\\quad 64-64 \\pi+32 \\pi \\sqrt{2}$']		['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+116	"Given noncollinear points $A, B, C$, segment $\overline{A B}$ is trisected by points $D$ and $E$, and $F$ is the midpoint of segment $\overline{A C} . \overline{D F}$ and $\overline{B F}$ intersect $\overline{C E}$ at $G$ and $H$, respectively. If $[D E G]=18$, compute $[F G H]$.
+
+<image_1>"	"[""Compute the desired area as $[E G F B]-[E H B]$. To compute the area of concave quadrilateral $E G F B$, draw segment $\\overline{B G}$, which divides the quadrilateral into three triangles, $\\triangle D E G, \\triangle B D G$, and $\\triangle B G F$. Then $[B D G]=[D E G]=18$ because the triangles have equal bases and heights. Because $D, G$, and $F$ are collinear, to compute $[B G F]$ it suffices to find the ratio $D G / G F$. Use Menelaus's Theorem on $\\triangle A D F$ with Menelaus Line $\\overline{E C}$ to obtain\n\n$$\n\\frac{A E}{E D} \\cdot \\frac{D G}{G F} \\cdot \\frac{F C}{C A}=1\n$$\n\n\n\nBecause $E$ and $F$ are the midpoints of $\\overline{A D}$ and $\\overline{C A}$ respectively, $A E / E D=1$ and $F C / C A=$ $1 / 2$. Therefore $D G / G F=2 / 1$, and $[B G F]=\\frac{1}{2}[B D G]=9$. Thus $[E G F B]=18+18+9=45$.\n\nTo compute $[E H B]$, consider that its base $\\overline{E B}$ is twice the base of $\\triangle D E G$. The ratio of their heights equals the ratio $E H / E G$ because the altitudes from $H$ and $G$ to $\\overleftrightarrow{B E}$ are parallel to each other. Use Menelaus's Theorem twice more on $\\triangle A E C$ to find these values:\n\n$$\n\\begin{gathered}\n\\frac{A D}{D E} \\cdot \\frac{E G}{G C} \\cdot \\frac{C F}{F A}=1 \\Rightarrow \\frac{E G}{G C}=\\frac{1}{2} \\Rightarrow E G=\\frac{1}{3} E C, \\text { and } \\\\\n\\frac{A B}{B E} \\cdot \\frac{E H}{H C} \\cdot \\frac{C F}{F A}=1 \\Rightarrow \\frac{E H}{H C}=\\frac{2}{3} \\Rightarrow E H=\\frac{2}{5} E C .\n\\end{gathered}\n$$\n\nTherefore $\\frac{E H}{E G}=\\frac{2 / 5}{1 / 3}=\\frac{6}{5}$. Thus $[E H B]=\\frac{6}{5} \\cdot 2 \\cdot[D E G]=\\frac{216}{5}$. Thus $[F G H]=45-\\frac{216}{5}=\\frac{9}{5}$.""
+ ""The method of mass points leads to the same results as Menelaus's Theorem, but corresponds to the physical intuition that masses on opposite sides of a fulcrum balance if and only if the products of the masses and their distances from the fulcrum are equal (in physics-speak, the net torque is zero). If a mass of weight 1 is placed at vertex $B$ and masses of weight 2 are placed at vertices $A$ and $C$, then $\\triangle A B C$ balances on the line $\\overleftrightarrow{B F}$ and also on the line $\\overleftrightarrow{C E}$. Thus it balances on the point $H$ where these two lines intersect. Replacing the masses at $A$ and $C$ with a single mass of weight 4 at their center of mass $F$, the triangle still balances at $H$. Thus $B H / H F=4$.\n\nNext, consider $\\triangle B E F$. Placing masses of weight 1 at the vertices $B$ and $E$ and a mass of weight 4 at $F$, the triangle balances at $G$. A similar argument shows that $D G / G F=2$ and that $E G / G H=5$. Because $\\triangle D E G$ and $\\triangle F H G$ have congruent (vertical) angles at $G$, it follows that $[D E G] /[F H G]=(D G / F G) \\cdot(E G / H G)=2 \\cdot 5=10$. Thus $[F G H]=[D E G] / 10=$ $\\frac{18}{10}=\\frac{9}{5}$.""]"	['$\\frac{9}{5}$']		['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+117	"Let $T=6$. In the square $D E F G$ diagrammed at right, points $M$ and $N$ trisect $\overline{F G}$, points $A$ and $B$ are the midpoints of $\overline{E F}$ and $\overline{D G}$, respectively, and $\overline{E M} \cap \overline{A B}=S$ and $\overline{D N} \cap \overline{A B}=H$. If the side length of square $D E F G$ is $T$, compute $[D E S H]$.
+
+<image_1>"	['Note that $D E S H$ is a trapezoid with height $\\frac{T}{2}$. Because $\\overline{A S}$ and $\\overline{B H}$ are midlines of triangles $E F M$ and $D G N$ respectively, it follows that $A S=B H=\\frac{T}{6}$. Thus $S H=T-2 \\cdot \\frac{T}{6}=\\frac{2 T}{3}$. Thus $[D E S H]=\\frac{1}{2}\\left(T+\\frac{2 T}{3}\\right) \\cdot \\frac{T}{2}=\\frac{5 T^{2}}{12}$. With $T=6$, the desired area is 15 .']	['15']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Geometry	Math	English
+118	"Let $R$ be the larger number you will receive, and let $r$ be the smaller number you will receive. In the diagram at right (not drawn to scale), circle $D$ has radius $R$, circle $K$ has radius $r$, and circles $D$ and $K$ are tangent at $C$. Line $\overleftrightarrow{Y P}$ is tangent to circles $D$ and $K$. Compute $Y P$.
+
+<image_1>"	['Note that $\\overline{D Y}$ and $\\overline{K P}$ are both perpendicular to line $\\overleftrightarrow{Y P}$. Let $J$ be the foot of the perpendicular from $K$ to $\\overline{D Y}$. Then $P K J Y$ is a rectangle and $Y P=J K=\\sqrt{D K^{2}-D J^{2}}=$ $\\sqrt{(R+r)^{2}-(R-r)^{2}}=2 \\sqrt{R r}$. With $R=450$ and $r=\\frac{1}{3}$, the answer is $2 \\sqrt{150}=\\mathbf{1 0} \\sqrt{\\mathbf{6}}$.']	['$10 \\sqrt{6}$']		['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+119	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+Consider the shape below:
+<image_3>
+
+Find the 2-signature that corresponds to this shape."	['The first pair indicates an increase; the next three are decreases, and the last pair is an increase. So the 2-signature is $(12,21,21,21,12)$.']	['$(12,21,21,21,12)$']		['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+120	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+List all 5-labels with 2-signature $(12,12,21,21)$."	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+121	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+Find a formula for the number of $(2 n+1)$-labels with the 2 -signature
+
+$$
+(\underbrace{12,12, \ldots, 12}_{n}, \underbrace{21,21, \ldots, 21}_{n})
+$$"	['The answer is $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$. The shape of this signature is a wedge: $n$ up steps followed by $n$ down steps. The wedge for $n=3$ is illustrated below:\n\n<img_3277>\n\nThe largest number in the label, $2 n+1$, must be placed at the peak in the center. If we choose the numbers to put in the first $n$ spaces, then they must be placed in increasing order. Likewise, the remaining $n$ numbers must be placed in decreasing order on the downward sloping piece of the shape. Thus there are exactly $\\left(\\begin{array}{c}2 n \\\\ n\\end{array}\\right)$ such labels.\n\n']	['$\\binom{2n}{n}$']		['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+122	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+Compute the number of 5-labels with 2 -signature $(12,21,12,21)$."	"[""The answer is 16 . We have a shape with two peaks and a valley in the middle. The 5 must go on one of the two peaks, so we place it on the first peak. By the shape's symmetry, we will double our answer at the end to account for the 5 -labels where the 5 is on the other peak.\n\n<img_3879>\n\nThe 4 can go to the left of the 5 or at the other peak. In the first case, shown below left, the 3 must go at the other peak and the 1 and 2 can go in either order. In the latter case, shown below right, the 1,2 , and 3 can go in any of 3 ! arrangements.\n<img_3481>\n\n\n\nSo there are $2 !+3 !=8$ possibilities. In all, there are 165 -labels (including the ones where the 5 is at the other peak).""]"	['16']		['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+123	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+Determine the number of 9-labels with 2-signature
+
+$$
+(12,21,12,21,12,21,12,21) \text {. }
+$$
+
+Justify your answer."	['The answer is 7936. The shape of this 2-signature has four peaks and three intermediate valleys:\n\n<img_3473>\n\nWe will solve this problem by building up from smaller examples. Let $f_{n}$ equal the number of $(2 n+1)$-labels whose 2 -signature consists of $n$ peaks and $n-1$ intermediate valleys. In part (b) we showed that $f_{2}=16$. In the case where we have one peak, $f_{1}=2$. For the trivial case (no peaks), we get $f_{0}=1$. These cases are shown below.\n\n1\n\n<img_3650>\n\n<img_3227>\n\nSuppose we know the peak on which the largest number, $2 n+1$, is placed. Then that splits our picture into two shapes with fewer peaks. Once we choose which numbers from $1,2, \\ldots, 2 n$ to place each shape, we can compute the number of arrangements of the numbers on each shape, and then take the product. For example, if we place the 9 at the second peak, as shown below, we get a 1-peak shape on the left and a 2-peak shape on the right.\n\n<img_4030>\n\nFor the above shape, there are $\\left(\\begin{array}{l}8 \\\\ 3\\end{array}\\right)$ ways to pick the three numbers to place on the left-hand side, $f_{1}=2$ ways to place them, and $f_{2}=16$ ways to place the remaining five numbers on the right.\n\nThis argument works for any $n>1$, so we have shown the following:\n\n$$\nf_{n}=\\sum_{k=1}^{n}\\left(\\begin{array}{c}\n2 n \\\\\n2 k-1\n\\end{array}\\right) f_{k-1} f_{n-k}\n$$\n\n\n\nSo we have:\n\n$$\n\\begin{aligned}\n& f_{1}=\\left(\\begin{array}{l}\n2 \\\\\n1\n\\end{array}\\right) f_{0}^{2}=2 \\\\\n& f_{2}=\\left(\\begin{array}{l}\n4 \\\\\n1\n\\end{array}\\right) f_{0} f_{1}+\\left(\\begin{array}{l}\n4 \\\\\n3\n\\end{array}\\right) f_{1} f_{0}=16 \\\\\n& f_{3}=\\left(\\begin{array}{l}\n6 \\\\\n1\n\\end{array}\\right) f_{0} f_{2}+\\left(\\begin{array}{l}\n6 \\\\\n3\n\\end{array}\\right) f_{1}^{2}+\\left(\\begin{array}{l}\n6 \\\\\n5\n\\end{array}\\right) f_{2} f_{0}=272 \\\\\n& f_{4}=\\left(\\begin{array}{l}\n8 \\\\\n1\n\\end{array}\\right) f_{0} f_{3}+\\left(\\begin{array}{l}\n8 \\\\\n3\n\\end{array}\\right) f_{1} f_{2}+\\left(\\begin{array}{l}\n8 \\\\\n5\n\\end{array}\\right) f_{2} f_{1}+\\left(\\begin{array}{l}\n8 \\\\\n7\n\\end{array}\\right) f_{3} f_{0}=7936 .\n\\end{aligned}\n$$']	['7936']		['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+124	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+For a general $n$, determine the number of distinct possible $p$-signatures."	"['The answer is $p ! \\cdot p^{n-p}$.\n\nCall two consecutive windows in a $p$-signature compatible if the last $p-1$ numbers in the first label and the first $p-1$ numbers in the second label (their ""overlap"") describe the same ordering. For example, in the $p$-signature $(. ., 2143,2431, \\ldots), 2143$ and 2431 are compatible. Notice that the last three digits of 2143 and the first three digits of 2431 can be described by the same 3-label, 132 .\n\nTheorem: A signature $\\sigma$ is possible if and only if every pair of consecutive windows is compatible.\n\nProof: $(\\Rightarrow)$ Consider a signature $\\sigma$ describing a $p$-label $L$. If some pair in $\\sigma$ is not compatible, then there is some string of $p-1$ numbers in our label $L$ that has two different $(p-1)$-signatures. This is impossible, since the $p$-signature is well-defined.\n\n$(\\Leftarrow)$ Now suppose $\\sigma$ is a $p$-signature such that that every pair of consecutive windows is compatible. We need to show that there is at least one label $L$ with $S_{p}[L]=\\sigma$. We do so by induction on the number of windows in $\\sigma$, using the results from $5(\\mathrm{~b})$.\n\nLet $\\sigma=\\left\\{\\omega_{1}, \\omega_{2}, \\ldots, \\omega_{k+1}\\right\\}$, and suppose $\\omega_{1}=a_{1}, a_{2}, \\ldots, a_{p}$. Set $L_{1}=\\omega_{1}$.\n\nSuppose that $L_{k}$ is a $(p+k-1)$-label such that $S_{p}\\left[L_{k}\\right]=\\left\\{\\omega_{1}, \\ldots, \\omega_{k}\\right\\}$. We will construct $L_{k+1}$ for which $S_{p}\\left[L_{k+1}\\right]=\\left\\{\\omega_{1}, \\ldots, \\omega_{k+1}\\right\\}$.\n\nAs in $5(\\mathrm{~b})$, denote by $L_{k}^{(j)}$ the label $L_{k}$ with a $j+0.5$ appended; we will eventually renumber the elements in the label to make them all integers. Appending $j+0.5$ does not affect any of the non-terminal windows of $S_{p}\\left[L_{k}\\right]$, and as $j$ varies from 0 to $p-k+1$ the final window of $S_{p}\\left[L_{k}^{(j)}\\right]$ varies over each of the $p$ windows compatible with $\\omega_{k}$. Since $\\omega_{k+1}$ is compatible with $\\omega_{k}$, there exists some $j$ for which $S_{p}\\left[L_{k}^{(j)}\\right]=\\left\\{\\omega_{1}, \\ldots, \\omega_{k+1}\\right\\}$. Now we renumber as follows: set $L_{k+1}=S_{k+p}\\left[L_{k}^{(j)}\\right]$, which replaces $L_{k}^{(j)}$ with the integers 1 through $k+p$ and preserves the relative order of all integers in the label.\n\nBy continuing this process, we conclude that the $n$-label $L_{n-p+1}$ has $p$-signature $\\sigma$, so $\\sigma$ is possible.\n\nTo count the number of possible $p$-signatures, we choose the first window ( $p$ ! choices), then choose each of the remaining $n-p$ compatible windows ( $p$ choices each). In all, there are $p ! \\cdot p^{n-p}$ possible $p$-signatures.']"	['$p ! \\cdot 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+125	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+If a randomly chosen $p$-signature is 575 times more likely of being impossible than possible, determine $p$ and $n$."	['The answer is $n=7, p=5$.\n\nLet $P$ denote the probability that a randomly chosen $p$-signature is possible. We are\n\n\n\ngiven that $1-P=575$, so $P=\\frac{1}{576}$. We want to find $p$ and $n$ for which\n\n$$\n\\begin{aligned}\n\\frac{p ! \\cdot p^{n-p}}{(p !)^{n-p+1}} & =\\frac{1}{576} \\\\\n\\frac{p^{n-p}}{(p !)^{n-p}} & =\\frac{1}{576} \\\\\n((p-1) !)^{n-p} & =576\n\\end{aligned}\n$$\n\nThe only factorial that has 576 as an integer power is $4 !=\\sqrt{576}$. Thus $p=5$ and $n-p=2 \\Rightarrow n=7$.']	['7,5']		['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+126	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+Find two 5-labels with unique 2-signatures."	['12345 and 54321 are the only ones.']	['12345, 54321']		['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+127	"An $\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and $L=263415$, the blocks are 263,634,341, and 415. We can associate to each of these blocks a $p$-label that corresponds to the relative order of the numbers in that block. For $L=263415$, we get the following:
+
+$$
+\underline{263} 415 \rightarrow 132 ; \quad 2 \underline{63415} \rightarrow 312 ; \quad 26 \underline{341} 5 \rightarrow 231 ; \quad 263 \underline{415} \rightarrow 213
+$$
+
+Moving from left to right in the $n$-label, there are $n-p+1$ such blocks, which means we obtain an $(n-p+1)$-tuple of $p$-labels. For $L=263415$, we get the 4 -tuple $(132,312,231,213)$. We will call this $(n-p+1)$-tuple the $\boldsymbol{p}$-signature of $L$ (or signature, if $p$ is clear from the context) and denote it by $S_{p}[L]$; the $p$-labels in the signature are called windows. For $L=263415$, the windows are $132,312,231$, and 213 , and we write
+
+$$
+S_{3}[263415]=(132,312,231,213)
+$$
+
+More generally, we will call any $(n-p+1)$-tuple of $p$-labels a $p$-signature, even if we do not know of an $n$-label to which it corresponds (and even if no such label exists). A signature that occurs for exactly one $n$-label is called unique, and a signature that doesn't occur for any $n$-labels is called impossible. A possible signature is one that occurs for at least one $n$-label.
+
+In this power question, you will be asked to analyze some of the properties of labels and signatures.
+
+
+We can associate a shape to a given 2-signature: a diagram of up and down steps that indicates the relative order of adjacent numbers. For example, the following shape corresponds to the 2-signature $(12,12,12,21,12,21)$ :
+
+<image_1>
+
+
+
+A 7-label with this 2-signature corresponds to placing the numbers 1 through 7 at the nodes above so that numbers increase with each up step and decrease with each down step. The 7-label 2347165 is shown below:
+
+<image_2>
+Determine the smallest $p$ for which the 20-label
+
+
+$$
+L=3,11,8,4,17,7,15,19,6,2,14,1,10,16,5,12,20,9,13,18
+$$
+
+has a unique $p$-signature."	"['The answer is $p=16$. To show this fact we will need to extend the idea from part 8(b) about ""linking"" inequalities forced by the various windows:\n\nTheorem: A $p$-signature for an $n$-label $L$ is unique if and only if for every $k<n, k$ and $k+1$ are in at least one window together. That is, the distance between them in the $n$-label is less than $p$.\n\nProof: Suppose that for some $k$, the distance between $k$ and $k+1$ is $p$ or greater. Then the label $L^{\\prime}$ obtained by swapping $k$ and $k+1$ has the same $p$-signature, because there are no numbers between $k$ and $k+1$ in any window and because the two numbers never appear in the same window.\n\nIf the distance between all such pairs is less than $k$, we need to show that $S_{p}[L]$ is unique. For $i=1,2, \\ldots, n$, let $r_{i}$ denote the position where $i$ appears in $L$. For example, if $L=4123$, then $r_{1}=2, r_{2}=3, r_{3}=4$, and $r_{4}=1$.\n\nLet $L=a_{1}, a_{2}, \\ldots, a_{n}$. Since 1 and 2 are in some window together, $a_{r_{1}}<a_{r_{2}}$. Similarly, for any $k$, since $k$ and $k+1$ are in some window together, $a_{r_{k}}<a_{r_{k+1}}$. We then get a linked inequality $a_{r_{1}}<a_{r_{2}}<\\cdots<a_{r_{n}}$, which can only be satisfied if $a_{r_{1}}=1, a_{r_{2}}=$ $2, \\ldots, a_{r_{n}}=n$. Therefore, $S_{p}[L]$ is unique.\n\nFrom the proof above, we know that the signature is unique if and only if every pair of consecutive integers coexists in at least one window. Therefore, we seek the largest distance between consecutive integers in $L$. That distance is 15 (from 8 to 9 , and from 17 to 18). Thus the smallest $p$ is 16 .']"	['16']		['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+128	"In rectangle $M N P Q$, point $A$ lies on $\overline{Q N}$. Segments parallel to the rectangle's sides are drawn through point $A$, dividing the rectangle into four regions. The areas of regions I, II, and III are integers in geometric progression. If the area of $M N P Q$ is 2009 , compute the maximum possible area of region I.
+
+<image_1>"	['Because $A$ is on diagonal $\\overline{N Q}$, rectangles $N X A B$ and $A C Q Y$ are similar. Thus $\\frac{A B}{A X}=\\frac{Q Y}{Q C}=$ $\\frac{A C}{A Y} \\Rightarrow A B \\cdot A Y=A C \\cdot A X$. Therefore, we have $2009=[\\mathrm{I}]+2[\\mathrm{II}]+[\\mathrm{III}]$.\n\nLet the common ratio of the geometric progression be $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers ( $q$ may equal 1 ). Then [I] must be some integer multiple of $q^{2}$, which we will call $a q^{2}$. This gives $[\\mathrm{II}]=a p q$ and [III $=a p^{2}$. By factoring, we get\n\n$$\n2009=a q^{2}+2 a p q+a p^{2} \\Rightarrow 7^{2} \\cdot 41=a(p+q)^{2}\n$$\n\nThus we must have $p+q=7$ and $a=41$. Since $[\\mathrm{I}]=a q^{2}$ and $p, q>0$, the area is maximized when $\\frac{p}{q}=\\frac{1}{6}$, giving $[\\mathrm{I}]=41 \\cdot 36=\\mathbf{1 4 7 6}$. The areas of the other regions are 246,246, and 41 .']	['1476']		['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+129	"The numbers $1,2, \ldots, 8$ are placed in the $3 \times 3$ grid below, leaving exactly one blank square. Such a placement is called okay if in every pair of adjacent squares, either one square is blank or the difference between the two numbers is at most 2 (two squares are considered adjacent if they share a common side). If reflections, rotations, etc. of placements are considered distinct, compute the number of distinct okay placements.
+
+<image_1>"	"[""We say that two numbers are neighbors if they occupy adjacent squares, and that $a$ is a friend of $b$ if $0<|a-b| \\leq 2$. Using this vocabulary, the problem's condition is that every pair of neighbors must be friends of each other. Each of the numbers 1 and 8 has two friends, and each number has at most four friends.\n\nIf there is no number written in the center square, then we must have one of the cycles in the figures below. For each cycle, there are 8 rotations. Thus there are 16 possible configurations with no number written in the center square.\n\n| 2 | 1 | 3 |\n| :--- | :--- | :--- |\n| 4 | - | 5 |\n| 6 | 8 | 7 |\n\n\n| 3 | 1 | 2 |\n| :--- | :--- | :--- |\n| 5 | - | 4 |\n| 7 | 8 | 6 |\n\nNow assume that the center square contains the number $n$. Because $n$ has at least three neighbors, $n \\neq 1$ and $n \\neq 8$. First we show that 1 must be in a corner. If 1 is a neighbor of $n$, then one of the corners neighboring 1 must be empty, because 1 has only two friends ( 2 and $3)$. If $c$ is in the other corner neighboring 1 , then $\\{n, c\\}=\\{2,3\\}$. But then $n$ must have three\n\n\n\nmore friends $\\left(n_{1}, n_{2}, n_{3}\\right)$ other than 1 and $c$, for a total of five friends, which is impossible, as illustrated below. Therefore 1 must be in a corner.\n\n| - | 1 | $c$ |\n| :--- | :--- | :--- |\n| $n_{1}$ | $n$ | $n_{2}$ |\n|  | $n_{3}$ |  |\n\nNow we show that 1 can only have one neighbor, i.e., one of the squares adjacent to 1 is empty. If 1 has two neighbors, then we have, up to a reflection and a rotation, the configuration shown below. Because 2 has only one more friend, the corner next to 2 is empty and $n=4$. Consequently, $m_{1}=5$ (refer to the figure below). Then 4 has one friend (namely 6) left with two neighbors $m_{2}$ and $m_{3}$, which is impossible. Thus 1 must have exactly one neighbor. An analogous argument shows that 8 must also be at a corner with exactly one neighbor.\n\n| 1 | 2 | - |\n| :--- | :--- | :--- |\n| 3 | $n$ | $m_{3}$ |\n| $m_{1}$ | $m_{2}$ |  |\n\nTherefore, 8 and 1 must be in non-opposite corners, with the blank square between them. Thus, up to reflections and rotations, the only possible configuration is the one shown at left below.\n\n| 1 | - | 8 |\n| :--- | :--- | :--- |\n| $m$ |  |  |\n|  |  |  |\n\n\n| 1 | - | 8 |\n| :---: | :---: | :---: |\n| $2 / 3$ | $4 / 5$ | $6 / 7$ |\n| $3 / 2$ | $5 / 4$ | $7 / 6$ |\n\nThere are two possible values for $m$, namely 2 and 3 . For each of the cases $m=2$ and $m=3$, the rest of the configuration is uniquely determined, as illustrated in the figure above right. We summarize our process: there are four corner positions for 1; two (non-opposite) corner positions for 8 (after 1 is placed); and two choices for the number in the square neighboring 1 but not neighboring 8 . This leads to $4 \\cdot 2 \\cdot 2=16$ distinct configurations with a number written in the center square.\n\nTherefore, there are 16 configurations in which the center square is blank and 16 configurations with a number in the center square, for a total of $\\mathbf{3 2}$ distinct configurations.""]"	['32']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAPwBBwMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiue8Wa9qOh21iNJ0f8Ata+vbv7NHbfalt/+WckhbewI4EZ4P/6+fHiz4hgY/wCFY/8Alet/8KAPQaK8/wD+Et+If/RMf/K/b/4Uf8Jb8Q/+iY/+V+3/AMKAPQKK8/8A+Et+If8A0TH/AMr9v/hR/wAJb8Q/+iY/+V+3/wAKAPQKK8//AOEt+If/AETH/wAr9v8A4Uf8Jb8Q/wDomP8A5X7f/CgD0CivP/8AhLfiH/0TH/yv2/8AhR/wlvxD/wCiY/8Alft/8KAPQKK8vPxH8Y/8JD/YP/Cusan9k+2eR/bcP+p37N27bj73GM59qvjxb8Qsf8kx/wDK9b/4UAeg0V5//wAJb8Q/+iY/+V+3/wAKP+Et+If/AETH/wAr9v8A4UAegUV5/wD8Jb8Q/wDomP8A5X7f/Cj/AIS34h/9Ex/8r9v/AIUAegUV5/8A8Jb8Q/8AomP/AJX7f/Cj/hLfiH/0TH/yv2/+FAHoFFef/wDCW/EP/omP/lft/wDCj/hLfiH/ANEx/wDK/b/4UAegUV5tb+OvHV1cXUMHw23SWkohnX+3YBscorgcrz8roePWrA8W/ELA/wCLY59/7et/8KAPQaK8/wD+Et+If/RMf/K/b/4Uf8Jb8Q/+iY/+V+3/AMKAPQKK8/8A+Et+If8A0TH/AMr9v/hR/wAJb8Q/+iY/+V+3/wAKAPQKK8//AOEt+If/AETH/wAr9v8A4Uf8Jb8Q/wDomP8A5X7f/CgD0CivP/8AhLfiH/0TH/yv2/8AhSHxX8QmPPwzx/3Hrf8A+JoA9BorL8O6s2t6JDfSWb2U5eSKe2d1cxSxu0ci7l4YBkbBHUYOB0rUoAKKKKAOf8Q/8hzwn/2FZP8A0iua6Cuf8Q/8hzwn/wBhWT/0iua6CgAooooAKKKKACiiigAooooA8/P/ACcL/wByp/7d16BXn5/5OF/7lT/27r0CgAooooAKKKKACiiigAooooA5/wAPf8h3xZ/2FY//AEitq6Cuf8Pf8h3xZ/2FY/8A0itq6CgAooooAKKKKACiiigAooooA5/wb/yA7j/sK6l/6WzV0Fc/4N/5Adx/2FdS/wDS2augoAKKKKAOf8Q/8hzwn/2FZP8A0iua6Cuf8Q/8hzwn/wBhWT/0iua6CgAooooAKKKKACiiigAooooA8/P/ACcL/wByp/7d16BXn5/5OF/7lT/27r0CgAooooAKKKKACiiigAooooA5/wAPf8h3xZ/2FY//AEitq6Cuf8Pf8h3xZ/2FY/8A0itq6CgAooooAKKKKACiiigAooooA5/wb/yA7j/sK6l/6WzV0Fc/4N/5Adx/2FdS/wDS2augoAKKKKAOf8Q/8hzwn/2FZP8A0iua6Cuf8Q/8hzwn/wBhWT/0iua6CgAooooAKKKKACiiigAooooA8/P/ACcL/wByp/7d16BXn5/5OF/7lT/27r0CgAooooAKKKKACiiigAooooA5/wAPf8h3xZ/2FY//AEitq6Cuf8Pf8h3xZ/2FY/8A0itq6CgAooooAKKKKACiiigAooooA5/wb/yA7j/sK6l/6WzV0Fc/4N/5Adx/2FdS/wDS2augoAKKKKAOf8Q/8hzwn/2FZP8A0iua6Cuf8Q/8hzwn/wBhWT/0iua6CgAooooAKKKKACiiigAooooA8/P/ACcL/wByp/7d16BXn5/5OF/7lT/27r0CgAooooAKKKKACiiigAooooA5/wAPf8h3xZ/2FY//AEitq6Cuf8Pf8h3xZ/2FY/8A0itq6CgAooooAKKKKACiiigAooooA5/wb/yA7j/sK6l/6WzV0Fc/4N/5Adx/2FdS/wDS2augoAKKKKAOf8Q/8hzwn/2FZP8A0iua6CuP8ef2v5nhv+wfsP8Aaf8AareT9u3+T/x6XG7ds+b7ucY74rP/AOLv/wDUjf8Ak3QB6BRXn/8Axd//AKkb/wAm6P8Ai7//AFI3/k3QB6BRXn//ABd//qRv/Juj/i7/AP1I3/k3QB6BRXn/APxd/wD6kb/ybo/4u/8A9SN/5N0AegUV5/8A8Xf/AOpG/wDJuj/i7/8A1I3/AJN0AB/5OF/7lT/27r0CvDz/AMLH/wCFxf8AMqf23/YH/Tx9m+z/AGj/AL68zd+GPeuwH/C3scf8IPj/ALe6APQKK8//AOLv/wDUjf8Ak3R/xd//AKkb/wAm6APQKK8//wCLv/8AUjf+TdH/ABd//qRv/JugD0CivP8A/i7/AP1I3/k3R/xd/wD6kb/yboA9Aorz/wD4u/8A9SN/5N0f8Xf/AOpG/wDJugDoPD3/ACHfFn/YVj/9IraugrybR/8AhaX9q+IPs3/CHed/aCfavM+1bfM+zQY2Y/h2bOvO7d2xWyP+FvY4/wCEG/8AJugD0CivP/8Ai7//AFI3/k3R/wAXf/6kb/yboA9Aorz/AP4u/wD9SN/5N0f8Xf8A+pG/8m6APQKK8/8A+Lv/APUjf+TdH/F3/wDqRv8AyboA9Aorz/8A4u//ANSN/wCTdIf+Fvf9SPn2+10AdD4N/wCQHcf9hXUv/S2augrl/h/9r/4RMf2h5H23+0L/AO0fZ8+X5n2ubds3c7c5xnnFdRQAUUUUAc/4h/5DnhP/ALCsn/pFc10Fc/4h/wCQ54T/AOwrJ/6RXNdBQAUUUUAFFFFABRRRQAUUUUAefn/k4X/uVP8A27r0CvPz/wAnC/8Acqf+3degUAFFFFABRRRQAUUUUAFFFFAHP+Hv+Q74s/7Csf8A6RW1dBXP+Hv+Q74s/wCwrH/6RW1dBQAUUUUAFFFFABRRRQAUUUUAc/4N/wCQHcf9hXUv/S2augrn/Bv/ACA7j/sK6l/6WzV0FABRRRQBz/iH/kOeE/8AsKyf+kVzXQVz/iH/AJDnhP8A7Csn/pFc10FABRRRQAUUUUAFFFFABRRRQB5+f+Thf+5U/wDbuvQK8/P/ACcL/wByp/7d16BQAUUUUAFFFFABRRRQAUUUUAc/4e/5Dviz/sKx/wDpFbV0Fc/4e/5Dviz/ALCsf/pFbV0FABRRRQAUUUUAFFFFABRRRQBz/g3/AJAdx/2FdS/9LZq6Cuf8G/8AIDuP+wrqX/pbNXQUAFFFFAHP+If+Q54T/wCwrJ/6RXNdBXP+If8AkOeE/wDsKyf+kVzXQUAFFFFABRRRQAUUUUAFFFFAHn5/5OF/7lT/ANu69Arz8/8AJwv/AHKn/t3XoFABRRRQAUUUUAFFFFABRRRQBz/h7/kO+LP+wrH/AOkVtXQVz/h7/kO+LP8AsKx/+kVtXQUAFFFFABRRRQAUUUUAFFFFAHP+Df8AkB3H/YV1L/0tmroK5/wb/wAgO4/7Cupf+ls1dBQAUUUUAc/4h/5DnhP/ALCsn/pFc10Fc/4h/wCQ54T/AOwrJ/6RXNdBQAUUUUAFFFFABRRRQAUUUUAefn/k4X/uVP8A27r0CvPz/wAnC/8Acqf+3degUAFFFFABRRRQAUUUUAFFFFAHP+Hv+Q74s/7Csf8A6RW1dBXP+Hv+Q74s/wCwrH/6RW1dBQAUUUUAFFFFABRRRQAUUUUAc/4N/wCQHcf9hXUv/S2augrn/Bv/ACA7j/sK6l/6WzV0FABRRRQBz/iH/kOeE/8AsKyf+kVzXQVy3jC3lutQ8MQwXk1nI2qPtuIAhdMWdyeA6svOMcg9T0OCLA8Papj/AJHLXPxhsv8A5HoA6Giuf/4R7VP+hz1z/vzZf/I9H/CPap/0Oeuf9+bL/wCR6AOgorn/APhHtU/6HPXP+/Nl/wDI9H/CPap/0Oeuf9+bL/5HoA6Ciuf/AOEe1T/oc9c/782X/wAj0f8ACPap/wBDnrn/AH5sv/kegDoKK5//AIR7VP8Aoc9c/wC/Nl/8j0f8I9qn/Q565/35sv8A5HoA58/8nC/9yp/7d16BXk50a+/4Xl9n/wCEl1Xzh4b8w3fl2vmbftONmPJ2bc8/d3Z744rt/wDhHtU/6HLXP+/Nl/8AI9AHQ0Vz/wDwj2qf9Dnrn/fmy/8Akej/AIR7VP8Aoc9c/wC/Nl/8j0AdBRXP/wDCPap/0Oeuf9+bL/5Ho/4R7VP+hz1z/vzZf/I9AHQUVz//AAj2qf8AQ565/wB+bL/5Ho/4R7VP+hz1z/vzZf8AyPQB0FFc/wD8I9qn/Q565/35sv8A5Ho/4R7VP+hz1z/vzZ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+130	"Let $T=80$. In circle $O$, diagrammed at right, minor arc $\widehat{A B}$ measures $\frac{T}{4}$ degrees. If $\mathrm{m} \angle O A C=10^{\circ}$ and $\mathrm{m} \angle O B D=5^{\circ}$, compute the degree measure of $\angle A E B$. Just pass the number without the units.
+
+<image_1>"	['Note that $\\mathrm{m} \\angle A E B=\\frac{1}{2}(\\mathrm{~m} \\widehat{A B}-m \\widehat{C D})=\\frac{1}{2}(\\mathrm{~m} \\widehat{A B}-\\mathrm{m} \\angle C O D)$. Also note that $\\mathrm{m} \\angle C O D=$ $360^{\\circ}-(\\mathrm{m} \\angle A O C+\\mathrm{m} \\angle B O D+\\mathrm{m} \\angle A O B)=360^{\\circ}-\\left(180^{\\circ}-2 \\mathrm{~m} \\angle O A C\\right)-\\left(180^{\\circ}-2 \\mathrm{~m} \\angle O B D\\right)-$ $\\mathrm{m} \\widehat{A B}=2(\\mathrm{~m} \\angle O A C+\\mathrm{m} \\angle O B D)-\\mathrm{m} \\widehat{A B}$. Thus $\\mathrm{m} \\angle A E B=\\mathrm{m} \\widehat{A B}-\\mathrm{m} \\angle O A C-\\mathrm{m} \\angle O B D=$ $\\frac{T}{4}-10^{\\circ}-5^{\\circ}$, and with $T=80$, the answer is 5 .']	['5']		['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+131	"Let $T=24$. A regular $n$-gon is inscribed in a circle; $P$ and $Q$ are consecutive vertices of the polygon, and $A$ is another vertex of the polygon as shown. If $\mathrm{m} \angle A P Q=\mathrm{m} \angle A Q P=T \cdot \mathrm{m} \angle Q A P$, compute the value of $n$.
+
+<image_1>"	['Let $\\mathrm{m} \\angle A=x$. Then $\\mathrm{m} \\angle P=\\mathrm{m} \\angle Q=T x$, and $(2 T+1) x=180^{\\circ}$, so $x=\\frac{180^{\\circ}}{2 T+1}$. Let $O$ be the center of the circle, as shown below.\n\n<img_3423>\n\nThen $\\mathrm{m} \\angle P O Q=2 \\mathrm{~m} \\angle P A Q=2\\left(\\frac{180^{\\circ}}{2 T+1}\\right)=\\frac{360^{\\circ}}{2 T+1}$. Because $\\mathrm{m} \\angle P O Q=\\frac{360^{\\circ}}{n}$, the denominators must be equal: $n=2 T+1$. Substitute $T=24$ to find $n=\\mathbf{4 9}$.']	['49']		['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+132	"A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circles; in what follows, we will call such regions curvilinear triangles, or $c$-triangles ( $\mathrm{c} \triangle$ ) for short.
+
+This sad day marks day 0 of a new fiscal era. Unfortunately, these drastic measures are not enough, and so each day thereafter, court geometers mark off the largest possible circle contained in each c-triangle in the remaining property. This circle is tangent to all three arcs of the c-triangle, and will be referred to as the incircle of the c-triangle. At the end of the day, all incircles demarcated that day are sold off, and the following day, the remaining c-triangles are partitioned in the same manner.
+
+Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle $A$ of radius $r_{a}$ is externally tangent to circle $B$ of radius $r_{b}$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_{a}}$ and $\frac{1}{r_{b}}$. If circle $A$ is internally tangent to circle $B$, however, as in the right diagram below, the curvature of circle $A$ is still $\frac{1}{r_{a}}$, while the curvature of circle $B$ is $-\frac{1}{r_{b}}$, the opposite of the reciprocal of its radius.
+
+<image_1>
+
+Circle $A$ has curvature 2; circle $B$ has curvature 1 .
+
+<image_2>
+
+Circle $A$ has curvature 2; circle $B$ has curvature -1 .
+
+Using these conventions allows us to express a beautiful theorem of Descartes: when four circles $A, B, C, D$ are pairwise tangent, with respective curvatures $a, b, c, d$, then
+
+$$
+(a+b+c+d)^{2}=2\left(a^{2}+b^{2}+c^{2}+d^{2}\right),
+$$
+
+where (as before) $a$ is taken to be negative if $B, C, D$ are internally tangent to $A$, and correspondingly for $b, c$, or $d$.
+Without using Descartes' Circle Formula, Find the combined area of the six remaining curvilinear territories after day 1."	"['The four ""removed"" circles have radii $\\frac{1}{2}, \\frac{1}{2}, \\frac{1}{3}, \\frac{1}{3}$ so the combined area of the six remaining curvilinear territories is:\n\n$$\n\\pi\\left(1^{2}-\\left(\\frac{1}{2}\\right)^{2}-\\left(\\frac{1}{2}\\right)^{2}-\\left(\\frac{1}{3}\\right)^{2}-\\left(\\frac{1}{3}\\right)^{2}\\right)=\\frac{5 \\pi}{18}\n$$']"	['$\\frac{5 \\pi}{18}$']		['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Vz/jTw2vinwne6ZnbcMnmWsgODHMvKMD25Az7ZpngTxC/ibwlZ6hcLsvQDDeRkYKTodrjHbkZA7BhQB0tFFFABRRRQAUUUUAFIeATS0HpQByHibxufDuvaLpX9k3M39qXkdqtyXVY1LFQcckkjd6Ae9daOteb/FL/AJD/AIB/7D8P8xXpVABRRRQAUUUUAFFFFAHM/EDRf+Eg8BazpyKWle2aSEA4PmJ86f8AjyirnhLWR4h8JaTq2QWurVJJMdA+MMPwYEfhWyfunjPtXB/CoCx0fWNAC7E0fV7m1hXP/LIt5iH8d5/KgDvaKKKACkPApaRiFUknAHJNAHh3w7tX8R/GnxTrlxseDS7qZIGA53ufKU59PLiI/wCBV7lXlvwHtHHgi51abBm1S/luGb2Hy4/NW/OvUqAEPSvO4Vlt/j40H2u6lgk8PNP5U0pZEczqp2r0HCjpXoteeyEf8NBxDPP/AAjB4/7eaAOm8U3M0WgyW1s5S7vmWzgYdUeQ7S4/3VJf/gNZuqeBdP12VIdUkuDpltEkVnYwXDxRoFH322kFm7DJIAAx1NXJ/wDiY+NLaAcw6Vb/AGlv+u0u6NPxCLL/AN9iovEniqy0qaPSl1OxtNSuVLK11KqJAnQyMCRn2Xqx9AGZQDE+Fy6jaDxDo91ezXtjpepPa2VxO259oAJQnvtyPzPoAPQq53wtd+H0g/sjQtRgvfsy+ZM8MwmJZySXkZeN7Nub1PPFdFQAUUUUAFFFFABRRRQAUUUUAIelcB4c/wCJD8V/EehgBbXVYU1i2VRgK/8Aq5vxLbWr0A9K4Lxp/wAS7x34I1rfsQ3kunSc43CaM7Af+BJQB3tFNGc4p1ABRRRQAUUUUAFIehpaKAPMfiYbm41/wf8AZdO1K5Fjq8dzctbWM0ixxgrkkqpB+gJr0iCQTRJIofDLkb0KH8VPI+h5qaigAooooAKKKKACiiigBD0rhPDyGw+L3i+1LYS9trO9jTHAwrRsfzAruz0rhJ4/s/x0s7jkC78Pyw+xKTq354NAHeUUUUAB6VzvjPxPpvhTw3c3+oXXkh0aOAAEmSQqSqrjvx16DvgV0J6V5b8bfBWoeLPDtpdaYwa40xpJWgZ9qyRlcnAPBYbRj6tQBY+B+r6XefDqx02yulkurDcLqLBDRl5HYZyOc54Iz068GvS6+dvhD8LrzUdHk1y513VdKgvFUW39kXnkySKGYMZPlPGQMD6+1ek/8Ks/6nzxx/4OP/sKAPQDWW3h3RDqP9pHR9P+3A7vtP2ZPNz67sZ/WuT/AOFWf9T544/8HH/2FH/CrP8AqfPHH/g4/wDsKAO4jtYIZ5po4lWWdg0rAcuQoUZ/AD8qzL3wn4c1K7ku7/QNKu7mQ/PNPZxu7Y4GWKknAAH4VzX/AAqz/qfPHH/g4/8AsKP+FWf9T544/wDBx/8AYUAdfpeh6VowkXS9MsrFZDmRbWBYgxHTO0DPWtA9K4D/AIVZ/wBT544/8HH/ANhR/wAKtx/zPnjj/wAHH/2FAG14u8caH4KtYp9Zumjafd5EUaF3lx1AHTjI5JA5q34Y8U6R4u0z+0dGuvtEAcxvlSrI2M7SCBg4I9vQmvHfiP8ABjWruKzudD1XVdcljVlli1W+EkijjHlkgDHXIz6Ve+H3wa1PT9HlfV9f1vR7ydwTb6RqAjUKBgbyFIZs7uhwBjBNAHt9Fef/APCrP+p88cf+Dj/7Cj/hVn/U+eOP/Bx/9hQB6BRXn/8Awqz/AKnzxx/4OP8A7Cj/AIVZ/wBT544/8HH/ANhQB6BRXn//AAqz/qfPHH/g4/8AsKP+FWf9T544/wDBx/8AYUAd+3CnnFeWfGPxLo2lQaJb3V0q6hbana6jHAEJby0chm46cbvrg1pn4W4/5nzxwf8AuMf/AGFeUfFf4W6ppl5a6pZ6ne6tbTGK1MmpXYkuBIzEKNxCjbyMehznigD6G0TW9P8AEOlwanpVytzZzg7JACMkEg5B5BBHQ1pV494M+Dl5pnh2GHUfFPiHTryRmklt9J1DyoVJOBxtOWwBk/gOma6H/hVn/U+eOP8Awcf/AGFAHoFFef8A/CrP+p88cf8Ag4/+wo/4VZ/1Pnjj/wAHH/2FAHoFFef/APCrP+p88cf+Dj/7Cj/hVn/U+eOP/Bx/9hQB6BRXn/8Awqz/AKnzxx/4OP8A7Cj/AIVZ/wBT544/8HH/ANhQB6BRXn//AAqz/qfPHH/g4/8AsKP+FWf9T544/wDBx/8AYUAegUV5/wD8Ks/6nzxx/wCDj/7Cj/hVn/U+eOP/AAcf/YUAegUV5/8A8Ks/6nzxx/4OP/sKP+FWf9T544/8HH/2FAHoFFef/wDCrP8AqfPHH/g4/wDsKP8AhVn/AFPnjj/wcf8A2FAHft9015B4p8f+GdO+LmhvNqeBp0N1bX7ojMsbOF2KcDnkHOM4OM4xW+fhbgZ/4TzxwfY6v/8AYV4j4p+EXiG18fR6Tayi8GqPLNa3NxMN7qvLGU/3xkZOPmzkd8AH1TFMssSSRsXjdQysvIIPQ0/efRvyrN8P6Y2ieHNM0rzjL9jtY7fzNv3tqhc/TitHLf3v/HDQBJUVzCtxaywuAVkQowPoRipaQ9KAPOvgbdrdfCrTIw2WtpJon9j5jMB+TCvRq8b+Ct+lj4j8Z+Fiyg2upSzxLnkjeY2/Lan517JQAUUUUAFFFFABRRRQAUUUjfdPOKAKmqalb6Tp0t7clvLjAwqLuZ2JwqqO7EkAD1Irjfhbr2peJrLXdV1IlHbVJIIoA+5YY0RAFH4k5Pc5Pelm1a+1LVxq6aFeX2jWQL2LxzQqsj8hpiHcEgDIXjGCWGcjFL4GRMPhpa3D8tdXM8xOep3lf/ZTQB6VRRRQAUUUUAB6VwXxIVru98HaZG3zz69BM68fNHEGdv6V3p6V5/dFdc+NlhAvzQ+H9NkndgeFmnwgU++wE0Ad8cYrz34v3t1ZeE7caZdXdvq13ew2tm1vcPGS7NkghSN2QCOQeteh9q8u8VW//CWfGDQ9BE8sdvo9pJqU7wttYSMQqc9mB2n6NQBF8QHvvh9oGnatpGtajNfC6jga2u7p7hLzIO4bXJIPHVcEV6lCxaNCy7GKjK+ntXkF/bL4Z+M+g217LPrkOpxnyH1FzLLYuM/NH/CBwM8Z6817Ep56/hQA6iiigAooooAKKKKACiiigAooooAKKKKAA9K4bU8XHxn8Pxct9l0q6nx027mRM+9dwehrhLJFvPjlqdypyLDQobVh6NJK0n8hQB3lFFFABQaKQnAoA8E1GVvB3xp1TXApSyiuoDdnput7mMKzn2SVB+de9j2rzPxnpVtN8SNJS9jLWPiDTbjR7hs8KR+9jP8AvZ3AfStP4bavdNYXPhnV3H9taEwtpuf9dFj91KPUFcDPXjnrQB3VFFFABRRRQAUUUUAB6Vz3ivRdV1/SnsLDWI9NhlG2dvshld17qDvXbkcH2PauhooAx9T06+vPD0mnWd7b2c0sBhM/2YsqgqQSibxjHbJOPeqHgXwvdeD/AA5b6JJqMN7b227ymW1MLjcxY7vnYHlu2K6eigAooooAKKKD0oAhvLmGysp7q5lWKCGNpJJGOAiqMkn6AZriPhdbTXenal4rvIil34gujdKGHzJbr8sKn2C8j/eqH4g3EviDUbDwHp7sr6hifUpUPMFmp5+hcjaPXkd67y2ghtLeK3t41ihiQIkaDCqoGAB6AAUASTBzC4iZVkI+VmXIB7ZGRn8xXB6d4J8Q6Z4l1jXYvEmnSXWqlPN83SXIjVBhVTE4OAMdc9BXf0UAcfongSOz8St4m1fU59X1ryzFFNJGI44EPGI0HTgkZyTz7knsKKKACiiigAooooAKKKKACiiigAooooAKKKKAEPSuC+H4W/8AFPjjWwD+/wBVFkCe4t0CZHtkmuw1nUY9H0S/1OYZjtLd52HqFUnH6Vzfws0+Ww+HWkm5Yvc3aNezOerNKxkyffDD8qAOyooooAKDRRQBw3xVs55PBp1W0TdeaJcxanCM4/1bZb/xwtVXxZY3Nwum+PvCqm41C2twzQIP+P8As2AYx8fxAHcvXnPXiu/uIIrq2lt50WSGVCjo3RlIwQfwrhfhdcS2elaj4UvHZrzw/dtagv8AeeBiXhc+xXp7LQB1Ph/XrDxLo9tq2mTebbXC5GeCh7qR2YHOR+vStWvNtZ0rU/A2t3Pijw3bvdaVdMZNX0pG6t3nhHZxzuHGf1XttE1vTvEWmwanpV0lxaSjKsv6gjqCOhBoA06KKKACiiigAooooAKKKKACiig9KAEPSsDxd4ptPCehPf3MbTzORFa2sfL3MzfdjUe/48Z61L4m8T6Z4U0p7/VJiqk7IokGZJ3PREXqT/Lqa5nwx4d1PW9bTxl4uiEd4EK6bph5WwiPOW9ZD3Pbp6BQDR8CeGrvSLa61bW3WbxDqzie+kHSMY+WJf8AZQcfXPtXYU0dRTqACiiigAooooAKKKKACiiigAooooAKKKKACiiigAoPSig9KAOB+K00t14fsfDdszC416+isgUPKRZDSP8AQKuD7NXdQxJDGkUahY0UKqjooHAArgrPPiX4xXV5jdYeGrX7LCexuphmQg+yYUj1Ir0GgAooooAKKKKAEb7przvxQT4T+IWj+KF+TTtSUaVqTDOEYnMEhA44OVJPQcd69FrL8SaFa+JfDl/o14P3F3EYy2MlD1Vh7ggEe4oA0TwCAM8dK4PV/BWoaTq0/iDwPcRWd/Md15p0xP2W+PqR/A/+0MZJ5xkk3Phzr11quhyadqxI1zR5DZXyk5LMvCye4ZRnPcg12ZoA4/w78QNN1i9OlahDLo+ux8Sade/KxPrG3Rx3BHbnGK68VjeIvC2ieKbL7LrOnRXSL9wtwyH/AGWHIrlR4f8AHHhVf+Kf1uPXbBOlhrJxMq/7M68k9AA3AFAHolFefL8UI9MBTxX4d1jQXX787QG4tgfQSxg5/Kt2w8feEtTCfZf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+133	"A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circles; in what follows, we will call such regions curvilinear triangles, or $c$-triangles ( $\mathrm{c} \triangle$ ) for short.
+
+This sad day marks day 0 of a new fiscal era. Unfortunately, these drastic measures are not enough, and so each day thereafter, court geometers mark off the largest possible circle contained in each c-triangle in the remaining property. This circle is tangent to all three arcs of the c-triangle, and will be referred to as the incircle of the c-triangle. At the end of the day, all incircles demarcated that day are sold off, and the following day, the remaining c-triangles are partitioned in the same manner.
+
+Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle $A$ of radius $r_{a}$ is externally tangent to circle $B$ of radius $r_{b}$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_{a}}$ and $\frac{1}{r_{b}}$. If circle $A$ is internally tangent to circle $B$, however, as in the right diagram below, the curvature of circle $A$ is still $\frac{1}{r_{a}}$, while the curvature of circle $B$ is $-\frac{1}{r_{b}}$, the opposite of the reciprocal of its radius.
+
+<image_1>
+
+Circle $A$ has curvature 2; circle $B$ has curvature 1 .
+
+<image_2>
+
+Circle $A$ has curvature 2; circle $B$ has curvature -1 .
+
+Using these conventions allows us to express a beautiful theorem of Descartes: when four circles $A, B, C, D$ are pairwise tangent, with respective curvatures $a, b, c, d$, then
+
+$$
+(a+b+c+d)^{2}=2\left(a^{2}+b^{2}+c^{2}+d^{2}\right),
+$$
+
+where (as before) $a$ is taken to be negative if $B, C, D$ are internally tangent to $A$, and correspondingly for $b, c$, or $d$.
+Determine the number of curvilinear territories remaining at the end of day 3."	"[""At the beginning of day 2, there are six c-triangles, so six incircles are sold, dividing each of the six territories into three smaller curvilinear triangles. So a total of 18 curvilinear triangles exist at the start of day 3, each of which is itself divided into three pieces that day (by the sale of a total of 18 regions bounded by the territories' incircles). Therefore there are 54 regions at the end of day 3.""]"	['54']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIATABxwMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKQ8CgBaKztU13StFhEuqanZ2SHobiZY930yefwrlH+LXh+dnTRbbVtekQ4ZdMsJJNp/3iAP1oA7yiuD/wCEr8b3rgaf4BeGJhkT6jqMcWPrGoZqao+K1wTvfwnZxkcBVnlcdOvIHrQB31FcN/YnxFlO5/GWmQHpti0gMPzZ6aPDvxDDs/8AwntqQQMIdFjwv/j2aAO7orhv7I+I8GfL8V6Rc9x5+llB9PlemLP8U7VyZrXwpfxjosEs8Ln/AL6BAoA7yiuB/wCE38VWAZtZ+HupLGvRtNuYrwsPZQQamtfix4UluEtr27n0m6brDqds9uR9SRtH50AdxRVazv7TUbdbiyuoLmBxlZYZFdT+IOKsA84PWgBaKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooPtTXcKpYnAHc9qAFJwKhuLqG0t3uLmWOGGMbpJJGCqo9STgAVxGofEV7+9fSvBennXr9Tte6DbLO3Pq8vRj0O1evOD2qO0+HEuszx33jnVZNbuVYOlkmYrOE/wCygwWxyMt1BORQBJcfE6G+uHs/COk3niS6Q7Glth5dqjejTN8o45GM5qM+HvHfiMf8T3xNHotq3Wz0OPEhXtmd+Q3rtGK7y2tbezt47e2gjghjXakcShVQegA4FS4HpQBx+l/DDwjp0pnfSY7+7bBe51BjcyO3qS5IB+gFdekaRoqIiqqjCqowB9KdRQAhAPUUYFLRQAYowKKKAEwD2pcUUUAJgVDd2drewNBd20M8LcNHKgZT+BqeigDiLz4VeGnuGu9KjutDvSAPtGk3DQED/dHyY/Cqn2b4jeGsNa3lj4qshn9zcj7JdY7AOMofq3NehYpNoNAHF6X8TdEurwadqy3Og6nwPsuqR+Vu7fK/3WGeBzk+ldmDk1R1fRdM12xaz1WxgvIG/gmQNj3BPQ+45riW8G+IfCDGfwPqhmsxnOiapI0kWPSKQ/Mh9Acgk8mgD0aiuQ8P/ECw1a/Gj6lbTaNroGW0684ZveNuki8HGOeDxXWq2SeelADqKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApCcUprkfFnjVNAlt9L0+1bU/EF3/x7afCfmA/56Of4EGDyf5AkAGl4k8VaV4V043uqXIRSdsUKDdLM/ZUXqxP5euK5BND8R/ET994nMui+HnwYtGgciacf9PD9h/sDHXkArzreGfBD2+pf8JD4mul1bxC4+WUr+5tAf4IVPQDpu6n25z2mBnOOaAKmm6ZY6RYx2WnWkNraxjCxQqFUfh3PvVvFLRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFJtU54HPX3paKAMbxH4W0bxTp/2PV7GO4Qf6tzkSRH1Rxyp+n45FcZ/aPiT4bYGqyXHiHwwM4v1G67sk/6agf6xMfxDkYOccCvTKawBGMDB4oAq6ZqdnrFhDf6fdxXVpMu6OWI5Df8A1+xH8quV5zqfhTVPCOoT694GRSkjeZfaExxDcju0f/POTH4Hj0weo8L+LNO8WaZ9ssHZWjbZcW8o2yW8g6o47Ec+1AG9RSDOaWgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACkPShjhcj+dct408Wnw9ZQ2lhb/bdd1BvJ0+yx99/7zeiLnJPHHcdQAVvGHjC50y6tdC0KCK+8RX5Hk27Z2QJ3llxyEH69ulWvB3g6Hw5FNd3c51DXL0iS+1GUZeVuOF/uoMYCjHQU3wX4PHhyCe7v5/t2u3zebfX7Dl2/ur6IOgHH07DqwMUAJgZzS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFADSOK4bxZ4QvF1UeK/CZjt/EMI/fQk4i1CMY/dyf7RAwG9hz0I7sjPWkI4oA57wj4tsvFmltc2yvBdQN5V3ZyjEltIOqsODjg4Pf8CB0I61wXjHw5f2Wqr4z8LR51i3QC7tBwuoQDqh4++APlPtjngV03hnxDYeKNEg1XTZN0EowVbho2HVGHZgf84IJANiiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopCaAFopOc0ZP+RQAtFIKWgAooooAKKKKACiiigApDxS0jHAzQBl+INdsvDeh3Wr6lJstbZN7cZZj2Ue5JAFcz4G0C9uL2bxl4kixrmoJiC3Y5FhbdVjX/AGjnLH1OMA5znon/AAsTx80p+fwz4cnwg/hvL0dT7qn6npkE49JXr+Hf/P1oAUAA5xz0zS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUANZeOBz6mvNvEEUvw68SyeLLFCdB1GRU1m2UE+S5OFuVA9zhh3z6nI9KIBGCM1Dd21vdWk1tcxJJbzIySxuPldSDkH2IzQA6CZLiJJonWSJ1DI6kEMDyCD3BqWvOfBdxceEvEVx4Cv5JHtwpudEuJTky2+TuiJ7sh/TsBivRQc0ALRRRQAUUUUAFFFFABRRRQAUh4FDZ4x61zHijxxpvhlorMrLfavcD/RtNtF3zSe5H8K8H5j6HGcYoA6YtjJrkNU+JvhzT7s2FpNPrGpD/lz0qI3Mnv8Ad+UY4zkislfCXiPxkfO8aag1jp78roemybVx1xNKDlz6gHHAI9K7bSND0rQbQWmlafb2cA6pDGFz7n1PuaAOQGs/EjWudN8O6XosOeJNWuTLIy+oSL7p9iaePC3jq7lL33xAMKMP9TY6XEgX6OxLV3uBRigDhh8PdQb5pfHvigyHqUuI0X8tlNbwR4igjIsPiDrEbD7puoIbgfkVH867ujFAHBfY/idpYHkavoOtoB832y1e2kb6eWSufrTB8SLzSBjxf4V1PR0HLXkAF5bKPVnj5XPpg16BikIB6igDP0nXdM12z+16VqFvewdC8Dhtp9CByD7HmtAEnqK4vWfhro95eHU9HeXQNYwdt5p3yBj1+dB8rgnBI4JwOaz4fGOt+EJ47Lx3bxm0ZgkOvWaEwtnp5yAZjY+uNuenAJoA9FoqK3niuYUmgkSSKRQyOhDAg8ggjgipaACiiigBDXHfELXrzTtHt9K0dv8AieazMLOyA6x5+/L7BFyc9jiuxbpXnnhHb4u8cat4vkUNZWedM0kkcFVOZZl/3mOAR2yD3oA63w1oFl4Z8P2ejWC/6PaxhNx6u3UsfcnJ/GtfFGMUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABQQD1oooA5H4g+HLjWtCjvNKPl65pUgvNOkAyTIvJQ+oYcY6ZxngVqeE/EVt4r8N2Ws2oCpcxgtHnJjccMh9wQR+RrZPSvPNKx4P+KN7o2CmmeI1a+s/7qXS/65B/vDDfyoA9EopBS0AFFFFABRRRQAUjHC5oPArjfG/ie+042ug6CqzeItUytspGVtk/infrhV7ZByR3wRQBB4o8XX8uqnwr4Thju9edc3E7jMOnxn+OT1bHRe/XHQHS8JeC7Dwwks/myX+r3Q3XmpXBzLOx64P8K+ijsBkkjNT+EPClp4T0n7JDJJcXUredeXkpzJczH7zsTk/QZ49+p6EADtQAmBnNLRRQAUUUUAFFFFABRRRQAYzUN1bw3NtJBcRRzQSKVkjlUMrqRggg8EVNQQD1oA8vn03VPhfO+o6HHcaj4VJLXmlA75bIdTJASeV7lSfx5yPQtJ1ax1zTYNS026jubO4TfHIh6jp+BByCDyCCO1XGAwa8x1CKT4W+IG1izD/8IfqEwGo2wyRYTMcCZB2QkgEe/wDugAHqFIelMilWZFkR1ZHAZWU5BHqDTz0oA5H4k63caL4NuEsedT1CRbCxAOCZpTtBHuBkj3Ara8N6Hb+G/Dtho1qP3VpCI92OXPVmPuSST7muT1If8JF8YdMsfvW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+134	"A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circles; in what follows, we will call such regions curvilinear triangles, or $c$-triangles ( $\mathrm{c} \triangle$ ) for short.
+
+This sad day marks day 0 of a new fiscal era. Unfortunately, these drastic measures are not enough, and so each day thereafter, court geometers mark off the largest possible circle contained in each c-triangle in the remaining property. This circle is tangent to all three arcs of the c-triangle, and will be referred to as the incircle of the c-triangle. At the end of the day, all incircles demarcated that day are sold off, and the following day, the remaining c-triangles are partitioned in the same manner.
+
+Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle $A$ of radius $r_{a}$ is externally tangent to circle $B$ of radius $r_{b}$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_{a}}$ and $\frac{1}{r_{b}}$. If circle $A$ is internally tangent to circle $B$, however, as in the right diagram below, the curvature of circle $A$ is still $\frac{1}{r_{a}}$, while the curvature of circle $B$ is $-\frac{1}{r_{b}}$, the opposite of the reciprocal of its radius.
+
+<image_1>
+
+Circle $A$ has curvature 2; circle $B$ has curvature 1 .
+
+<image_2>
+
+Circle $A$ has curvature 2; circle $B$ has curvature -1 .
+
+Using these conventions allows us to express a beautiful theorem of Descartes: when four circles $A, B, C, D$ are pairwise tangent, with respective curvatures $a, b, c, d$, then
+
+$$
+(a+b+c+d)^{2}=2\left(a^{2}+b^{2}+c^{2}+d^{2}\right),
+$$
+
+where (as before) $a$ is taken to be negative if $B, C, D$ are internally tangent to $A$, and correspondingly for $b, c$, or $d$.
+Determine the total number of plots sold up to and including day $n$."	['The total number of plots sold up to and including day $n$ is\n\n$$\n\\begin{aligned}\n2+\\sum_{k=1}^{n} X_{k} & =2+2 \\sum_{k=1}^{n} 3^{k-1} \\\\\n& =2+2 \\cdot\\left(1+3+3^{2}+\\ldots+3^{n-1}\\right) \\\\\n& =3^{n}+1\n\\end{aligned}\n$$\n\nAlternatively, proceed by induction: on day 0 , there are $2=3^{0}+1$ plots sold, and for $n \\geq 0$,\n\n$$\n\\begin{aligned}\n\\left(3^{n}+1\\right)+X_{n+1} & =\\left(3^{n}+1\\right)+2 \\cdot 3^{n} \\\\\n& =3 \\cdot 3^{n}+1 \\\\\n& =3^{n+1}+1 .\n\\end{aligned}\n$$']	['$3^{n}+1$']		['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+135	"A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circles; in what follows, we will call such regions curvilinear triangles, or $c$-triangles ( $\mathrm{c} \triangle$ ) for short.
+
+This sad day marks day 0 of a new fiscal era. Unfortunately, these drastic measures are not enough, and so each day thereafter, court geometers mark off the largest possible circle contained in each c-triangle in the remaining property. This circle is tangent to all three arcs of the c-triangle, and will be referred to as the incircle of the c-triangle. At the end of the day, all incircles demarcated that day are sold off, and the following day, the remaining c-triangles are partitioned in the same manner.
+
+Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle $A$ of radius $r_{a}$ is externally tangent to circle $B$ of radius $r_{b}$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_{a}}$ and $\frac{1}{r_{b}}$. If circle $A$ is internally tangent to circle $B$, however, as in the right diagram below, the curvature of circle $A$ is still $\frac{1}{r_{a}}$, while the curvature of circle $B$ is $-\frac{1}{r_{b}}$, the opposite of the reciprocal of its radius.
+
+<image_1>
+
+Circle $A$ has curvature 2; circle $B$ has curvature 1 .
+
+<image_2>
+
+Circle $A$ has curvature 2; circle $B$ has curvature -1 .
+
+Using these conventions allows us to express a beautiful theorem of Descartes: when four circles $A, B, C, D$ are pairwise tangent, with respective curvatures $a, b, c, d$, then
+
+$$
+(a+b+c+d)^{2}=2\left(a^{2}+b^{2}+c^{2}+d^{2}\right),
+$$
+
+where (as before) $a$ is taken to be negative if $B, C, D$ are internally tangent to $A$, and correspondingly for $b, c$, or $d$.
+Two unit circles and a circle of radius $\frac{2}{3}$ are mutually externally tangent. Compute all possible values of $r$ such that a circle of radius $r$ is tangent to all three circles."	"[""Use Descartes' Circle Formula with $a=b=1$ and $c=\\frac{3}{2}$ to solve for $d$ :\n\n$$\n\\begin{aligned}\n2 \\cdot\\left(1^{2}+1^{2}+\\left(\\frac{3}{2}\\right)^{2}+d^{2}\\right) & =\\left(1+1+\\frac{3}{2}+d\\right)^{2} \\\\\n\\frac{17}{2}+2 d^{2} & =\\frac{49}{4}+7 d+d^{2} \\\\\nd^{2}-7 d-\\frac{15}{4} & =0\n\\end{aligned}\n$$\n\nfrom which $d=\\frac{15}{2}$ or $d=-\\frac{1}{2}$. These values correspond to radii of $\\frac{2}{15}$, a small circle nestled between the other three, or 2 , a large circle enclosing the other three.\n\nAlternatively, start by scaling the kingdom with the first four circles removed to match the situation given. Thus the three given circles are internally tangent to a circle of radius $r=2$ and curvature $d=-\\frac{1}{2}$. Descartes' Circle Formula gives a quadratic equation for $d$, and the sum of the roots is $2 \\cdot\\left(1+1+\\frac{3}{2}\\right)=7$, so the second root is $7+\\frac{1}{2}=\\frac{15}{2}$, corresponding to a circle of radius $r=\\frac{2}{15}$.""]"	['$2$, 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+136	"A king strapped for cash is forced to sell off his kingdom $U=\left\{(x, y): x^{2}+y^{2} \leq 1\right\}$. He sells the two circular plots $C$ and $C^{\prime}$ centered at $\left( \pm \frac{1}{2}, 0\right)$ with radius $\frac{1}{2}$. The retained parts of the kingdom form two regions, each bordered by three arcs of circles; in what follows, we will call such regions curvilinear triangles, or $c$-triangles ( $\mathrm{c} \triangle$ ) for short.
+
+This sad day marks day 0 of a new fiscal era. Unfortunately, these drastic measures are not enough, and so each day thereafter, court geometers mark off the largest possible circle contained in each c-triangle in the remaining property. This circle is tangent to all three arcs of the c-triangle, and will be referred to as the incircle of the c-triangle. At the end of the day, all incircles demarcated that day are sold off, and the following day, the remaining c-triangles are partitioned in the same manner.
+
+Some notation: when discussing mutually tangent circles (or arcs), it is convenient to refer to the curvature of a circle rather than its radius. We define curvature as follows. Suppose that circle $A$ of radius $r_{a}$ is externally tangent to circle $B$ of radius $r_{b}$. Then the curvatures of the circles are simply the reciprocals of their radii, $\frac{1}{r_{a}}$ and $\frac{1}{r_{b}}$. If circle $A$ is internally tangent to circle $B$, however, as in the right diagram below, the curvature of circle $A$ is still $\frac{1}{r_{a}}$, while the curvature of circle $B$ is $-\frac{1}{r_{b}}$, the opposite of the reciprocal of its radius.
+
+<image_1>
+
+Circle $A$ has curvature 2; circle $B$ has curvature 1 .
+
+<image_2>
+
+Circle $A$ has curvature 2; circle $B$ has curvature -1 .
+
+Using these conventions allows us to express a beautiful theorem of Descartes: when four circles $A, B, C, D$ are pairwise tangent, with respective curvatures $a, b, c, d$, then
+
+$$
+(a+b+c+d)^{2}=2\left(a^{2}+b^{2}+c^{2}+d^{2}\right),
+$$
+
+where (as before) $a$ is taken to be negative if $B, C, D$ are internally tangent to $A$, and correspondingly for $b, c$, or $d$.
+Find the areas of the circles removed on day 3."	['Day 3 begins with two circles of curvature 15 from the configuration $(2,2,3,15)$, and four circles of curvature 6 from the configuration $(-1,2,3,6)$. Consider the following two cases:\n\nCase 1: $(a, b, c, d)=(2,2,3,15), s=22$\n\n- $a=2: a^{\\prime}=2 s-3 a=\\mathbf{3 8}$\n- $b=2: b^{\\prime}=2 s-3 b=\\mathbf{3 8}$\n- $c=3: c^{\\prime}=2 s-3 c=\\mathbf{3 5}$\n- $d=15: d^{\\prime}=2 s-3 d=-1$, which is the configuration from day 1 .\n\nCase 2: $(a, b, c, d)=(-1,2,3,6), s=10$\n\n- $a=-1: a^{\\prime}=2 s-3 a=\\mathbf{2 3}$\n- $b=2: b^{\\prime}=2 s-3 b=\\mathbf{1 4}$\n- $c=3: c^{\\prime}=2 s-3 c=\\mathbf{1 1}$\n- $d=6: d^{\\prime}=2 s-3 d=2$, which is the configuration from day 1 .\n\n\n\nSo the areas of the plots removed on day 3 are:\n\n$$\n\\frac{\\pi}{38^{2}}, \\frac{\\pi}{35^{2}}, \\frac{\\pi}{23^{2}}, \\frac{\\pi}{14^{2}}, \\text { and } \\frac{\\pi}{11^{2}}\n$$\n\nThere are two circles with area $\\frac{\\pi}{35^{2}}$, and four circles with each of the other areas, for a total of 18 plots.']	['$\\frac{\\pi}{38^{2}}, \\frac{\\pi}{35^{2}}, \\frac{\\pi}{23^{2}}, \\frac{\\pi}{14^{2}}, \\frac{\\pi}{11^{2}}$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIATABxwMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKQ8CgBaKztU13StFhEuqanZ2SHobiZY930yefwrlH+LXh+dnTRbbVtekQ4ZdMsJJNp/3iAP1oA7yiuD/wCEr8b3rgaf4BeGJhkT6jqMcWPrGoZqao+K1wTvfwnZxkcBVnlcdOvIHrQB31FcN/YnxFlO5/GWmQHpti0gMPzZ6aPDvxDDs/8AwntqQQMIdFjwv/j2aAO7orhv7I+I8GfL8V6Rc9x5+llB9PlemLP8U7VyZrXwpfxjosEs8Ln/AL6BAoA7yiuB/wCE38VWAZtZ+HupLGvRtNuYrwsPZQQamtfix4UluEtr27n0m6brDqds9uR9SRtH50AdxRVazv7TUbdbiyuoLmBxlZYZFdT+IOKsA84PWgBaKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooPtTXcKpYnAHc9qAFJwKhuLqG0t3uLmWOGGMbpJJGCqo9STgAVxGofEV7+9fSvBennXr9Tte6DbLO3Pq8vRj0O1evOD2qO0+HEuszx33jnVZNbuVYOlkmYrOE/wCygwWxyMt1BORQBJcfE6G+uHs/COk3niS6Q7Glth5dqjejTN8o45GM5qM+HvHfiMf8T3xNHotq3Wz0OPEhXtmd+Q3rtGK7y2tbezt47e2gjghjXakcShVQegA4FS4HpQBx+l/DDwjp0pnfSY7+7bBe51BjcyO3qS5IB+gFdekaRoqIiqqjCqowB9KdRQAhAPUUYFLRQAYowKKKAEwD2pcUUUAJgVDd2drewNBd20M8LcNHKgZT+BqeigDiLz4VeGnuGu9KjutDvSAPtGk3DQED/dHyY/Cqn2b4jeGsNa3lj4qshn9zcj7JdY7AOMofq3NehYpNoNAHF6X8TdEurwadqy3Og6nwPsuqR+Vu7fK/3WGeBzk+ldmDk1R1fRdM12xaz1WxgvIG/gmQNj3BPQ+45riW8G+IfCDGfwPqhmsxnOiapI0kWPSKQ/Mh9Acgk8mgD0aiuQ8P/ECw1a/Gj6lbTaNroGW0684ZveNuki8HGOeDxXWq2SeelADqKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigApCcUprkfFnjVNAlt9L0+1bU/EF3/x7afCfmA/56Of4EGDyf5AkAGl4k8VaV4V043uqXIRSdsUKDdLM/ZUXqxP5euK5BND8R/ET994nMui+HnwYtGgciacf9PD9h/sDHXkArzreGfBD2+pf8JD4mul1bxC4+WUr+5tAf4IVPQDpu6n25z2mBnOOaAKmm6ZY6RYx2WnWkNraxjCxQqFUfh3PvVvFLRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFJtU54HPX3paKAMbxH4W0bxTp/2PV7GO4Qf6tzkSRH1Rxyp+n45FcZ/aPiT4bYGqyXHiHwwM4v1G67sk/6agf6xMfxDkYOccCvTKawBGMDB4oAq6ZqdnrFhDf6fdxXVpMu6OWI5Df8A1+xH8quV5zqfhTVPCOoT694GRSkjeZfaExxDcju0f/POTH4Hj0weo8L+LNO8WaZ9ssHZWjbZcW8o2yW8g6o47Ec+1AG9RSDOaWgAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACkPShjhcj+dct408Wnw9ZQ2lhb/bdd1BvJ0+yx99/7zeiLnJPHHcdQAVvGHjC50y6tdC0KCK+8RX5Hk27Z2QJ3llxyEH69ulWvB3g6Hw5FNd3c51DXL0iS+1GUZeVuOF/uoMYCjHQU3wX4PHhyCe7v5/t2u3zebfX7Dl2/ur6IOgHH07DqwMUAJgZzS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFADSOK4bxZ4QvF1UeK/CZjt/EMI/fQk4i1CMY/dyf7RAwG9hz0I7sjPWkI4oA57wj4tsvFmltc2yvBdQN5V3ZyjEltIOqsODjg4Pf8CB0I61wXjHw5f2Wqr4z8LR51i3QC7tBwuoQDqh4++APlPtjngV03hnxDYeKNEg1XTZN0EowVbho2HVGHZgf84IJANiiiigAooooAKKKKACiiigAooooAKKKKACiiigAoopCaAFopOc0ZP+RQAtFIKWgAooooAKKKKACiiigApDxS0jHAzQBl+INdsvDeh3Wr6lJstbZN7cZZj2Ue5JAFcz4G0C9uL2bxl4kixrmoJiC3Y5FhbdVjX/AGjnLH1OMA5znon/AAsTx80p+fwz4cnwg/hvL0dT7qn6npkE49JXr+Hf/P1oAUAA5xz0zS0UUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUANZeOBz6mvNvEEUvw68SyeLLFCdB1GRU1m2UE+S5OFuVA9zhh3z6nI9KIBGCM1Dd21vdWk1tcxJJbzIySxuPldSDkH2IzQA6CZLiJJonWSJ1DI6kEMDyCD3BqWvOfBdxceEvEVx4Cv5JHtwpudEuJTky2+TuiJ7sh/TsBivRQc0ALRRRQAUUUUAFFFFABRRRQAUh4FDZ4x61zHijxxpvhlorMrLfavcD/RtNtF3zSe5H8K8H5j6HGcYoA6YtjJrkNU+JvhzT7s2FpNPrGpD/lz0qI3Mnv8Ad+UY4zkislfCXiPxkfO8aag1jp78roemybVx1xNKDlz6gHHAI9K7bSND0rQbQWmlafb2cA6pDGFz7n1PuaAOQGs/EjWudN8O6XosOeJNWuTLIy+oSL7p9iaePC3jq7lL33xAMKMP9TY6XEgX6OxLV3uBRigDhh8PdQb5pfHvigyHqUuI0X8tlNbwR4igjIsPiDrEbD7puoIbgfkVH867ujFAHBfY/idpYHkavoOtoB832y1e2kb6eWSufrTB8SLzSBjxf4V1PR0HLXkAF5bKPVnj5XPpg16BikIB6igDP0nXdM12z+16VqFvewdC8Dhtp9CByD7HmtAEnqK4vWfhro95eHU9HeXQNYwdt5p3yBj1+dB8rgnBI4JwOaz4fGOt+EJ47Lx3bxm0ZgkOvWaEwtnp5yAZjY+uNuenAJoA9FoqK3niuYUmgkSSKRQyOhDAg8ggjgipaACiiigBDXHfELXrzTtHt9K0dv8AieazMLOyA6x5+/L7BFyc9jiuxbpXnnhHb4u8cat4vkUNZWedM0kkcFVOZZl/3mOAR2yD3oA63w1oFl4Z8P2ejWC/6PaxhNx6u3UsfcnJ/GtfFGMUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABQQD1oooA5H4g+HLjWtCjvNKPl65pUgvNOkAyTIvJQ+oYcY6ZxngVqeE/EVt4r8N2Ws2oCpcxgtHnJjccMh9wQR+RrZPSvPNKx4P+KN7o2CmmeI1a+s/7qXS/65B/vDDfyoA9EopBS0AFFFFABRRRQAUjHC5oPArjfG/ie+042ug6CqzeItUytspGVtk/infrhV7ZByR3wRQBB4o8XX8uqnwr4Thju9edc3E7jMOnxn+OT1bHRe/XHQHS8JeC7Dwwks/myX+r3Q3XmpXBzLOx64P8K+ijsBkkjNT+EPClp4T0n7JDJJcXUredeXkpzJczH7zsTk/QZ49+p6EADtQAmBnNLRRQAUUUUAFFFFABRRRQAYzUN1bw3NtJBcRRzQSKVkjlUMrqRggg8EVNQQD1oA8vn03VPhfO+o6HHcaj4VJLXmlA75bIdTJASeV7lSfx5yPQtJ1ax1zTYNS026jubO4TfHIh6jp+BByCDyCCO1XGAwa8x1CKT4W+IG1izD/8IfqEwGo2wyRYTMcCZB2QkgEe/wDugAHqFIelMilWZFkR1ZHAZWU5BHqDTz0oA5H4k63caL4NuEsedT1CRbCxAOCZpTtBHuBkj3Ara8N6Hb+G/Dtho1qP3VpCI92OXPVmPuSST7muT1If8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+137	"Points $A$ and $L$ lie outside circle $\omega$, whose center is $O$, and $\overline{A L}$ contains diameter $\overline{R M}$, as shown below. Circle $\omega$ is tangent to $\overline{L K}$ at $K$. Also, $\overline{A K}$ intersects $\omega$ at $Y$, which is between $A$ and $K$. If $K L=3, M L=2$, and $\mathrm{m} \angle A K L-\mathrm{m} \angle Y M K=90^{\circ}$, compute $[A K M]$ (i.e., the area of $\triangle A K M$ ).
+
+<image_1>"	"['Notice that $\\overline{O K} \\perp \\overline{K L}$, and let $r$ be the radius of $\\omega$.\n\n<img_3784>\n\nThen consider right triangle $O K L$. Because $M L=2, O K=r$, and $O L=r+2$, it follows that $r^{2}+3^{2}=(r+2)^{2}$, from which $r=\\frac{5}{4}$.\n\nBecause $\\mathrm{m} \\angle Y K L=\\frac{1}{2} \\mathrm{~m} \\widehat{Y R K}$ and $\\mathrm{m} \\angle Y M K=\\frac{1}{2} \\mathrm{~m} \\widehat{Y K}$, it follows that $\\mathrm{m} \\angle Y K L+\\mathrm{m} \\angle Y M K=$ $180^{\\circ}$. By the given condition, $\\mathrm{m} \\angle Y K L-\\mathrm{m} \\angle Y M K=90^{\\circ}$. It follows that $\\mathrm{m} \\angle Y M K=45^{\\circ}$ and $\\mathrm{m} \\angle Y K L=135^{\\circ}$. hence $\\mathrm{m} \\widehat{Y K}=90^{\\circ}$. Thus,\n\n$$\n\\overline{Y O} \\perp \\overline{O K} \\quad \\text { and } \\quad \\overline{Y O} \\| \\overline{K L}\n\\tag{*}\n$$\nCompute $[A K M]$ as $\\frac{1}{2}$ base $\\cdot$ height, using base $\\overline{A M}$.\n\n<img_3921>\n\nBecause of (*), $\\triangle A Y O \\sim \\triangle A K L$. To compute $A M$, notice that in $\\triangle A Y O, A O=A M-r$, while in $\\triangle A K L$, the corresponding side $A L=A M+M L=A M+2$. Therefore:\n\n$$\n\\begin{aligned}\n\\frac{A O}{A L} & =\\frac{Y O}{K L} \\\\\n\\frac{A M-\\frac{5}{4}}{A M+2} & =\\frac{5 / 4}{3}\n\\end{aligned}\n$$\n\nfrom which $A M=\\frac{25}{7}$. Draw the altitude of $\\triangle A K M$ from vertex $K$, and let $h$ be its length. In right triangle $O K L, h$ is the altitude to the hypotenuse, so $\\frac{h}{3}=\\sin (\\angle K L O)=\\frac{r}{r+2}$. Hence $h=\\frac{15}{13}$. Therefore $[A K M]=\\frac{1}{2} \\cdot \\frac{25}{7} \\cdot \\frac{15}{13}=\\frac{375}{182}$.'
+ 'Notice that $\\overline{O K} \\perp \\overline{K L}$, and let $r$ be the radius of $\\omega$.\n\n<img_3784>\n\nThen consider right triangle $O K L$. Because $M L=2, O K=r$, and $O L=r+2$, it follows that $r^{2}+3^{2}=(r+2)^{2}$, from which $r=\\frac{5}{4}$.\n\nBecause $\\mathrm{m} \\angle Y K L=\\frac{1}{2} \\mathrm{~m} \\widehat{Y R K}$ and $\\mathrm{m} \\angle Y M K=\\frac{1}{2} \\mathrm{~m} \\widehat{Y K}$, it follows that $\\mathrm{m} \\angle Y K L+\\mathrm{m} \\angle Y M K=$ $180^{\\circ}$. By the given condition, $\\mathrm{m} \\angle Y K L-\\mathrm{m} \\angle Y M K=90^{\\circ}$. It follows that $\\mathrm{m} \\angle Y M K=45^{\\circ}$ and $\\mathrm{m} \\angle Y K L=135^{\\circ}$. hence $\\mathrm{m} \\widehat{Y K}=90^{\\circ}$. Thus,\n\n$$\n\\overline{Y O} \\perp \\overline{O K} \\quad \\text { and } \\quad \\overline{Y O} \\| \\overline{K L}\n\\tag{*}\n$$\nBy the Power of the Point Theorem, $L K^{2}=L M \\cdot L R$, so\n\n$$\n\\begin{aligned}\nL R & =\\frac{9}{2} \\\\\nR M & =L R-L M=\\frac{5}{2} \\\\\nO L & =r+M L=\\frac{13}{4}\n\\end{aligned}\n$$\n\nFrom (*), we know that $\\triangle A Y O \\sim \\triangle A K L$. Hence by $(\\dagger)$,\n\n$$\n\\frac{A L}{A O}=\\frac{A L}{A L-O L}=\\frac{K L}{Y O}=\\frac{3}{5 / 4}=\\frac{12}{5}, \\quad \\text { thus } \\quad A L=\\frac{12}{7} \\cdot O L=\\frac{12}{7} \\cdot \\frac{13}{4}=\\frac{39}{7}\n$$\n\nHence $A M=A L-2=\\frac{25}{7}$. The ratio between the areas of triangles $A K M$ and $R K M$ is equal to\n\n$$\n\\frac{[A K M]}{[R K M]}=\\frac{A M}{R M}=\\frac{25 / 7}{5 / 2}=\\frac{10}{7}\n$$\n\nThus $[A K M]=\\frac{10}{7} \\cdot[R K M]$.\n\nBecause $\\angle K R L$ and $\\angle M K L$ both subtend $\\widehat{K M}, \\triangle K R L \\sim \\triangle M K L$. Therefore $\\frac{K R}{M K}=\\frac{L K}{L M}=$ $\\frac{3}{2}$. Thus let $K R=3 x$ and $M K=2 x$ for some positive real number $x$. Because $R M$ is a diameter of $\\omega$ (see left diagram below), $\\mathrm{m} \\angle R K M=90^{\\circ}$. Thus triangle $R K M$ is a right triangle with hypotenuse $\\overline{R M}$. In particular, $13 x^{2}=K R^{2}+M K^{2}=R M^{2}=\\frac{25}{4}$, so $x^{2}=\\frac{25}{52}$ and $[R K M]=\\frac{R K \\cdot K M}{2}=3 x^{2}$. Therefore\n\n$$\n[A K M]=\\frac{10}{7} \\cdot[R K M]=\\frac{10}{7} \\cdot 3 \\cdot \\frac{25}{52}=\\frac{\\mathbf{3 7 5}}{\\mathbf{1 8 2}}\n$$\n\n<img_3387>'
+ 'Notice that $\\overline{O K} \\perp \\overline{K L}$, and let $r$ be the radius of $\\omega$.\n\n<img_3784>\n\nThen consider right triangle $O K L$. Because $M L=2, O K=r$, and $O L=r+2$, it follows that $r^{2}+3^{2}=(r+2)^{2}$, from which $r=\\frac{5}{4}$.\n\nBecause $\\mathrm{m} \\angle Y K L=\\frac{1}{2} \\mathrm{~m} \\widehat{Y R K}$ and $\\mathrm{m} \\angle Y M K=\\frac{1}{2} \\mathrm{~m} \\widehat{Y K}$, it follows that $\\mathrm{m} \\angle Y K L+\\mathrm{m} \\angle Y M K=$ $180^{\\circ}$. By the given condition, $\\mathrm{m} \\angle Y K L-\\mathrm{m} \\angle Y M K=90^{\\circ}$. It follows that $\\mathrm{m} \\angle Y M K=45^{\\circ}$ and $\\mathrm{m} \\angle Y K L=135^{\\circ}$. hence $\\mathrm{m} \\widehat{Y K}=90^{\\circ}$. Thus,\n\n$$\n\\overline{Y O} \\perp \\overline{O K} \\quad \\text { and } \\quad \\overline{Y O} \\| \\overline{K L}\n\\tag{*}\n$$\n\nLet $U$ and $V$ be the respective feet of the perpendiculars dropped from $A$ and $M$ to $\\overleftrightarrow{K L}$. From (*), $\\triangle A K L$ can be dissected into two infinite progressions of triangles: one progression of triangles similar to $\\triangle O K L$ and the other similar to $\\triangle Y O K$, as shown in the right diagram above. In both progressions, the corresponding sides of the triangles have common ratio equal to\n\n$$\n\\frac{Y O}{K L}=\\frac{5 / 4}{3}=\\frac{5}{12}\n$$\n\n\n\nThus\n\n$$\nA U=\\frac{5}{4}\\left(1+\\frac{5}{12}+\\left(\\frac{5}{12}\\right)^{2}+\\cdots\\right)=\\frac{5}{4} \\cdot \\frac{12}{7}=\\frac{15}{7}\n$$\n\nBecause $\\triangle L M V \\sim \\triangle L O K$, and because $L O=\\frac{13}{4}$ by $(\\dagger)$,\n\n$$\n\\frac{M V}{O K}=\\frac{L M}{L O}, \\quad \\text { thus } \\quad M V=\\frac{O K \\cdot L M}{L O}=\\frac{\\frac{5}{4} \\cdot 2}{\\frac{13}{4}}=\\frac{10}{13}\n$$\n\nFinally, note that $[A K M]=[A K L]-[K L M]$. Because $\\triangle A K L$ and $\\triangle K L M$ share base $\\overline{K L}$,\n\n$$\n[A K M]=\\frac{1}{2} \\cdot 3 \\cdot\\left(\\frac{15}{7}-\\frac{10}{13}\\right)=\\frac{\\mathbf{3 7 5}}{\\mathbf{1 8 2}}\n$$']"	['$\\frac{375}{182}$']		['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+138	"Let $T=8 \sqrt{2}$. In the diagram at right, the smaller circle is internally tangent to the larger circle at point $O$, and $\overline{O P}$ is a diameter of the larger circle. Point $Q$ lies on $\overline{O P}$ such that $P Q=T$, and $\overline{P Q}$ does not intersect the smaller circle. If the larger circle's radius is three times the smaller circle's radius, find the least possible integral radius of the larger circle.
+
+<image_1>"	['Let $r$ be the radius of the smaller circle. Then the conditions defining $Q$ imply that $P Q=$ $T<4 r$. With $T=8 \\sqrt{2}$, note that $r>2 \\sqrt{2} \\rightarrow 3 r>6 \\sqrt{2}=\\sqrt{72}$. The least integer greater than $\\sqrt{72}$ is 9 .']	['9']		['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+139	"The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it the Meruprastāra, or ""Mountain of Gems"". In this Power Question, we'll explore some properties of Pingala's/Pascal's Triangle (""PT"") and its variants.
+
+Unless otherwise specified, the only definition, notation, and formulas you may use for PT are the definition, notation, and formulas given below.
+
+PT consists of an infinite number of rows, numbered from 0 onwards. The $n^{\text {th }}$ row contains $n+1$ numbers, identified as $\mathrm{Pa}(n, k)$, where $0 \leq k \leq n$. For all $n$, define $\mathrm{Pa}(n, 0)=\operatorname{Pa}(n, n)=1$. Then for $n>1$ and $1 \leq k \leq n-1$, define $\mathrm{Pa}(n, k)=\mathrm{Pa}(n-1, k-1)+\mathrm{Pa}(n-1, k)$. It is convenient to define $\mathrm{Pa}(n, k)=0$ when $k<0$ or $k>n$. We write the nonzero values of $\mathrm{PT}$ in the familiar pyramid shown below.
+
+<image_1>
+
+As is well known, $\mathrm{Pa}(n, k)$ gives the number of ways of choosing a committee of $k$ people from a set of $n$ people, so a simple formula for $\mathrm{Pa}(n, k)$ is $\mathrm{Pa}(n, k)=\frac{n !}{k !(n-k) !}$. You may use this formula or the recursive definition above throughout this Power Question.
+For $n=1,2,3,4$, and $k=4$, find $\mathrm{Pa}(n, n)+\mathrm{Pa}(n+1, n)+\cdots+\operatorname{Pa}(n+k, n)$."	['$$\n\\begin{aligned}\n& \\mathrm{Pa}(1,1)+\\mathrm{Pa}(2,1)+\\mathrm{Pa}(3,1)+\\mathrm{Pa}(4,1)+\\mathrm{Pa}(5,1)=1+2+3+4+5=\\mathbf{1 5} \\\\\n& \\mathrm{Pa}(2,2)+\\mathrm{Pa}(3,2)+\\mathrm{Pa}(4,2)+\\mathrm{Pa}(5,2)+\\mathrm{Pa}(6,2)=1+3+6+10+15=\\mathbf{3 5} \\\\\n& \\mathrm{Pa}(3,3)+\\mathrm{Pa}(4,3)+\\mathrm{Pa}(5,3)+\\mathrm{Pa}(6,3)+\\mathrm{Pa}(7,3)=1+4+10+20+35=\\mathbf{7 0} \\\\\n& \\mathrm{Pa}(4,4)+\\mathrm{Pa}(5,4)+\\mathrm{Pa}(6,4)+\\mathrm{Pa}(7,4)+\\mathrm{Pa}(8,4)=1+5+15+35+70=\\mathbf{1 2 6}\n\\end{aligned}\n$$']	['15, 35, 70, 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+140	"The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it the Meruprastāra, or ""Mountain of Gems"". In this Power Question, we'll explore some properties of Pingala's/Pascal's Triangle (""PT"") and its variants.
+
+Unless otherwise specified, the only definition, notation, and formulas you may use for PT are the definition, notation, and formulas given below.
+
+PT consists of an infinite number of rows, numbered from 0 onwards. The $n^{\text {th }}$ row contains $n+1$ numbers, identified as $\mathrm{Pa}(n, k)$, where $0 \leq k \leq n$. For all $n$, define $\mathrm{Pa}(n, 0)=\operatorname{Pa}(n, n)=1$. Then for $n>1$ and $1 \leq k \leq n-1$, define $\mathrm{Pa}(n, k)=\mathrm{Pa}(n-1, k-1)+\mathrm{Pa}(n-1, k)$. It is convenient to define $\mathrm{Pa}(n, k)=0$ when $k<0$ or $k>n$. We write the nonzero values of $\mathrm{PT}$ in the familiar pyramid shown below.
+
+<image_1>
+
+As is well known, $\mathrm{Pa}(n, k)$ gives the number of ways of choosing a committee of $k$ people from a set of $n$ people, so a simple formula for $\mathrm{Pa}(n, k)$ is $\mathrm{Pa}(n, k)=\frac{n !}{k !(n-k) !}$. You may use this formula or the recursive definition above throughout this Power Question.
+If $\mathrm{Pa}(n, n)+\mathrm{Pa}(n+1, n)+\cdots+\mathrm{Pa}(n+k, n)=\mathrm{Pa}(m, j)$, find and justify formulas for $m$ and $j$ in terms of $n$ and $k$."	['Notice that $\\mathrm{Pa}(n, n)+\\operatorname{Pa}(n+1, n)+\\cdots+\\operatorname{Pa}(n+k, n)=\\mathrm{Pa}(n+k+1, n+1)$, so $m=n+k+1$ and $j=n+1$. (By symmetry, $j=k$ is also correct.) The equation is true for all $n$ when $k=0$, because the sum is simply $\\mathrm{Pa}(n, n)$ and the right side is $\\mathrm{Pa}(n+1, n+1)$, both of which are 1 . Proceed by induction on $k$. If $\\mathrm{Pa}(n, n)+\\mathrm{Pa}(n+1, n)+\\cdots+\\mathrm{Pa}(n+k, n)=\\mathrm{Pa}(n+k+1, n+1)$, then adding $\\mathrm{Pa}(n+k+1, n)$ to both sides yields $\\mathrm{Pa}(n+k+1, n)+\\mathrm{Pa}(n+k+1, n+1)=$ $\\mathrm{Pa}(n+k+2, n+1)$ by the recursive rule for $\\mathrm{Pa}$.']	['$m=n+k+1$, 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+141	"The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it the Meruprastāra, or ""Mountain of Gems"". In this Power Question, we'll explore some properties of Pingala's/Pascal's Triangle (""PT"") and its variants.
+
+Unless otherwise specified, the only definition, notation, and formulas you may use for PT are the definition, notation, and formulas given below.
+
+PT consists of an infinite number of rows, numbered from 0 onwards. The $n^{\text {th }}$ row contains $n+1$ numbers, identified as $\mathrm{Pa}(n, k)$, where $0 \leq k \leq n$. For all $n$, define $\mathrm{Pa}(n, 0)=\operatorname{Pa}(n, n)=1$. Then for $n>1$ and $1 \leq k \leq n-1$, define $\mathrm{Pa}(n, k)=\mathrm{Pa}(n-1, k-1)+\mathrm{Pa}(n-1, k)$. It is convenient to define $\mathrm{Pa}(n, k)=0$ when $k<0$ or $k>n$. We write the nonzero values of $\mathrm{PT}$ in the familiar pyramid shown below.
+
+<image_1>
+
+As is well known, $\mathrm{Pa}(n, k)$ gives the number of ways of choosing a committee of $k$ people from a set of $n$ people, so a simple formula for $\mathrm{Pa}(n, k)$ is $\mathrm{Pa}(n, k)=\frac{n !}{k !(n-k) !}$. You may use this formula or the recursive definition above throughout this Power Question.
+
+Clark's Triangle: If the left side of PT is replaced with consecutive multiples of 6 , starting with 0 , but the right entries (except the first) and the generating rule are left unchanged, the result is called Clark's Triangle. If the $k^{\text {th }}$ entry of the $n^{\text {th }}$ row is denoted by $\mathrm{Cl}(n, k)$, then the formal rule is:
+
+$$
+\begin{cases}\mathrm{Cl}(n, 0)=6 n & \text { for all } n \\ \mathrm{Cl}(n, n)=1 & \text { for } n \geq 1 \\ \mathrm{Cl}(n, k)=\mathrm{Cl}(n-1, k-1)+\mathrm{Cl}(n-1, k) & \text { for } n \geq 1 \text { and } 1 \leq k \leq n-1\end{cases}
+$$
+
+The first four rows of Clark's Triangle are given below.
+
+<image_2>
+If $\mathrm{Cl}(n, 1)=a n^{2}+b n+c$, determine the values of $a, b$, and $c$."	['Using the given values yields the system of equations below.\n\n$$\n\\left\\{\\begin{array}{l}\n\\mathrm{Cl}(1,1)=1=a(1)^{2}+b(1)+c \\\\\n\\mathrm{Cl}(2,1)=7=a(2)^{2}+b(2)+c \\\\\n\\mathrm{Cl}(3,1)=19=a(3)^{2}+b(3)+c\n\\end{array}\\right.\n$$\n\nSolving this system, $a=3, b=-3, c=1$.']	['$3,-3,1$']		['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']	Multimodal	Competition	True		Numerical		Open-ended	Combinatorics	Math	English
+142	"The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it the Meruprastāra, or ""Mountain of Gems"". In this Power Question, we'll explore some properties of Pingala's/Pascal's Triangle (""PT"") and its variants.
+
+Unless otherwise specified, the only definition, notation, and formulas you may use for PT are the definition, notation, and formulas given below.
+
+PT consists of an infinite number of rows, numbered from 0 onwards. The $n^{\text {th }}$ row contains $n+1$ numbers, identified as $\mathrm{Pa}(n, k)$, where $0 \leq k \leq n$. For all $n$, define $\mathrm{Pa}(n, 0)=\operatorname{Pa}(n, n)=1$. Then for $n>1$ and $1 \leq k \leq n-1$, define $\mathrm{Pa}(n, k)=\mathrm{Pa}(n-1, k-1)+\mathrm{Pa}(n-1, k)$. It is convenient to define $\mathrm{Pa}(n, k)=0$ when $k<0$ or $k>n$. We write the nonzero values of $\mathrm{PT}$ in the familiar pyramid shown below.
+
+<image_1>
+
+As is well known, $\mathrm{Pa}(n, k)$ gives the number of ways of choosing a committee of $k$ people from a set of $n$ people, so a simple formula for $\mathrm{Pa}(n, k)$ is $\mathrm{Pa}(n, k)=\frac{n !}{k !(n-k) !}$. You may use this formula or the recursive definition above throughout this Power Question.
+
+Clark's Triangle: If the left side of PT is replaced with consecutive multiples of 6 , starting with 0 , but the right entries (except the first) and the generating rule are left unchanged, the result is called Clark's Triangle. If the $k^{\text {th }}$ entry of the $n^{\text {th }}$ row is denoted by $\mathrm{Cl}(n, k)$, then the formal rule is:
+
+$$
+\begin{cases}\mathrm{Cl}(n, 0)=6 n & \text { for all } n \\ \mathrm{Cl}(n, n)=1 & \text { for } n \geq 1 \\ \mathrm{Cl}(n, k)=\mathrm{Cl}(n-1, k-1)+\mathrm{Cl}(n-1, k) & \text { for } n \geq 1 \text { and } 1 \leq k \leq n-1\end{cases}
+$$
+
+The first four rows of Clark's Triangle are given below.
+
+<image_2>
+Compute $\mathrm{Cl}(11,2)$."	['$\\mathrm{Cl}(11,2)=1000$.']	['1000']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Combinatorics	Math	English
+143	"The arrangement of numbers known as Pascal's Triangle has fascinated mathematicians for centuries. In fact, about 700 years before Pascal, the Indian mathematician Halayudha wrote about it in his commentaries to a then-1000-year-old treatise on verse structure by the Indian poet and mathematician Pingala, who called it the Meruprastāra, or ""Mountain of Gems"". In this Power Question, we'll explore some properties of Pingala's/Pascal's Triangle (""PT"") and its variants.
+
+Unless otherwise specified, the only definition, notation, and formulas you may use for PT are the definition, notation, and formulas given below.
+
+PT consists of an infinite number of rows, numbered from 0 onwards. The $n^{\text {th }}$ row contains $n+1$ numbers, identified as $\mathrm{Pa}(n, k)$, where $0 \leq k \leq n$. For all $n$, define $\mathrm{Pa}(n, 0)=\operatorname{Pa}(n, n)=1$. Then for $n>1$ and $1 \leq k \leq n-1$, define $\mathrm{Pa}(n, k)=\mathrm{Pa}(n-1, k-1)+\mathrm{Pa}(n-1, k)$. It is convenient to define $\mathrm{Pa}(n, k)=0$ when $k<0$ or $k>n$. We write the nonzero values of $\mathrm{PT}$ in the familiar pyramid shown below.
+
+<image_1>
+
+As is well known, $\mathrm{Pa}(n, k)$ gives the number of ways of choosing a committee of $k$ people from a set of $n$ people, so a simple formula for $\mathrm{Pa}(n, k)$ is $\mathrm{Pa}(n, k)=\frac{n !}{k !(n-k) !}$. You may use this formula or the recursive definition above throughout this Power Question.
+
+Clark's Triangle: If the left side of PT is replaced with consecutive multiples of 6 , starting with 0 , but the right entries (except the first) and the generating rule are left unchanged, the result is called Clark's Triangle. If the $k^{\text {th }}$ entry of the $n^{\text {th }}$ row is denoted by $\mathrm{Cl}(n, k)$, then the formal rule is:
+
+$$
+\begin{cases}\mathrm{Cl}(n, 0)=6 n & \text { for all } n \\ \mathrm{Cl}(n, n)=1 & \text { for } n \geq 1 \\ \mathrm{Cl}(n, k)=\mathrm{Cl}(n-1, k-1)+\mathrm{Cl}(n-1, k) & \text { for } n \geq 1 \text { and } 1 \leq k \leq n-1\end{cases}
+$$
+
+The first four rows of Clark's Triangle are given below.
+
+<image_2>
+Compute $\mathrm{Cl}(11,3)$."	['$\\mathrm{Cl}(11,3)=2025$.']	['2025']		['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']	Multimodal	Competition	False		Numerical		Open-ended	Combinatorics	Math	English
+144	"Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
+
+$$
+\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text { and } 0 \leq k \leq n\end{cases}
+$$
+
+This triangle, discovered first by Leibniz, consists of reciprocals of integers as shown below.
+
+<image_1>
+
+For this contest, you may assume that $\operatorname{Le}(n, k)>0$ whenever $0 \leq k \leq n$, and that $\operatorname{Le}(n, k)$ is undefined if $k<0$ or $k>n$.
+Compute Le(17,1)."	['$\\operatorname{Le}(17,1)=\\operatorname{Le}(16,0)-\\operatorname{Le}(17,0)=\\frac{1}{17}-\\frac{1}{18}=\\frac{1}{306}$.']	['$\\frac{1}{306}$']		['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+145	"Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
+
+$$
+\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text { and } 0 \leq k \leq n\end{cases}
+$$
+
+This triangle, discovered first by Leibniz, consists of reciprocals of integers as shown below.
+
+<image_1>
+
+For this contest, you may assume that $\operatorname{Le}(n, k)>0$ whenever $0 \leq k \leq n$, and that $\operatorname{Le}(n, k)$ is undefined if $k<0$ or $k>n$.
+Compute $\operatorname{Le}(17,2)$."	['$\\operatorname{Le}(17,2)=\\operatorname{Le}(16,1)-\\operatorname{Le}(17,1)=\\operatorname{Le}(15,0)-\\operatorname{Le}(16,0)-\\operatorname{Le}(17,1)=\\frac{1}{2448}$.']	['$\\frac{1}{2448}$']		['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+146	"Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
+
+$$
+\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text { and } 0 \leq k \leq n\end{cases}
+$$
+
+This triangle, discovered first by Leibniz, consists of reciprocals of integers as shown below.
+
+<image_1>
+
+For this contest, you may assume that $\operatorname{Le}(n, k)>0$ whenever $0 \leq k \leq n$, and that $\operatorname{Le}(n, k)$ is undefined if $k<0$ or $k>n$.
+Compute $\sum_{n=1}^{2011} \operatorname{Le}(n, 1)$."	['Because $\\operatorname{Le}(n, 1)=\\frac{1}{n}-\\frac{1}{n+1}$,\n\n$$\n\\begin{aligned}\n\\sum_{i=1}^{2011} \\operatorname{Le}(i, 1) & =\\sum_{i=1}^{2011}\\left(\\frac{1}{n}-\\frac{1}{n+1}\\right) \\\\\n& =\\left(\\frac{1}{1}-\\frac{1}{2}\\right)+\\left(\\frac{1}{2}-\\frac{1}{3}\\right)+\\cdots+\\left(\\frac{1}{2010}-\\frac{1}{2011}\\right)+\\left(\\frac{1}{2011}-\\frac{1}{2012}\\right) \\\\\n& =1-\\frac{1}{2012} \\\\\n& =\\frac{2011}{2012} .\n\\end{aligned}\n$$']	['$\\frac{2011}{2012}$']		['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+147	"Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
+
+$$
+\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text { and } 0 \leq k \leq n\end{cases}
+$$
+
+This triangle, discovered first by Leibniz, consists of reciprocals of integers as shown below.
+
+<image_1>
+
+For this contest, you may assume that $\operatorname{Le}(n, k)>0$ whenever $0 \leq k \leq n$, and that $\operatorname{Le}(n, k)$ is undefined if $k<0$ or $k>n$.
+If $\sum_{i=1}^{\infty} \operatorname{Le}(i, 1)=\operatorname{Le}(n, k)$, determine the values of $n$ and $k$."	['Extending the result of $8 \\mathrm{~b}$ gives\n\n$$\n\\sum_{i=1}^{n} \\operatorname{Le}(i, 1)=\\frac{1}{1}-\\frac{1}{n}\n$$\n\nso as $n \\rightarrow \\infty, \\sum_{i=1}^{n} \\operatorname{Le}(i, 1) \\rightarrow 1$. This value appears as $\\operatorname{Le}(0,0)$, so $n=k=0$.']	['$0,0$']		['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+148	"Leibniz's Harmonic Triangle: Consider the triangle formed by the rule
+
+$$
+\begin{cases}\operatorname{Le}(n, 0)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, n)=\frac{1}{n+1} & \text { for all } n \\ \operatorname{Le}(n, k)=\operatorname{Le}(n+1, k)+\operatorname{Le}(n+1, k+1) & \text { for all } n \text { and } 0 \leq k \leq n\end{cases}
+$$
+
+This triangle, discovered first by Leibniz, consists of reciprocals of integers as shown below.
+
+<image_1>
+
+For this contest, you may assume that $\operatorname{Le}(n, k)>0$ whenever $0 \leq k \leq n$, and that $\operatorname{Le}(n, k)$ is undefined if $k<0$ or $k>n$.
+If $\sum_{i=m}^{\infty} \operatorname{Le}(i, m)=\operatorname{Le}(n, k)$, compute expressions for $n$ and $k$ in terms of $m$."	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+149	"$\quad$ Let $T=12$. As shown, three circles are mutually externally tangent. The large circle has a radius of $T$, and the smaller two circles each have radius $\frac{T}{2}$. Compute the area of the triangle whose vertices are the centers of the three circles.
+
+<image_1>"	['The desired triangle is an isosceles triangle whose base vertices are the centers of the two smaller circles. The congruent sides of the triangle have length $T+\\frac{T}{2}$. Thus the altitude to the base has length $\\sqrt{\\left(\\frac{3 T}{2}\\right)^{2}-\\left(\\frac{T}{2}\\right)^{2}}=T \\sqrt{2}$. Thus the area of the triangle is $\\frac{1}{2} \\cdot\\left(\\frac{T}{2}+\\frac{T}{2}\\right) \\cdot T \\sqrt{2}=\\frac{T^{2} \\sqrt{2}}{2}$. With $T=12$, the area is $\\mathbf{7 2} \\sqrt{\\mathbf{2}}$.']	['$72 \\sqrt{2}$']		['/9j/2wCEAAgGBgcGBQgHBwcJCQgKDBQNDAsLDBkSEw8UHRofHh0aHBwgJC4nICIsIxwcKDcpLDAxNDQ0Hyc5PTgyPC4zNDIBCQkJDAsMGA0NGDIhHCEyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMjIyMv/AABEIAbQCeAMBIgACEQEDEQH/xAGiAAABBQEBAQEBAQAAAAAAAAAAAQIDBAUGBwgJCgsQAAIBAwMCBAMFBQQEAAABfQECAwAEEQUSITFBBhNRYQcicRQygZGhCCNCscEVUtHwJDNicoIJChYXGBkaJSYnKCkqNDU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6g4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2drh4uPk5ebn6Onq8fLz9PX29/j5+gEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoLEQACAQIEBAMEBwUEBAABAncAAQIDEQQFITEGEkFRB2FxEyIygQgUQpGhscEJIzNS8BVictEKFiQ04SXxFxgZGiYnKCkqNTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqCg4SFhoeIiYqSk5SVlpeYmZqio6Slpqeoqaqys7S1tre4ubrCw8TFxsfIycrS09TV1tfY2dri4+Tl5ufo6ery8/T19vf4+fr/2gAMAwEAAhEDEQA/APf6KKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACijNFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUZoAKKMjGc1z3iDxp4c8Lq39saxa2sgGfKLbpT6fIuWP5YoA6GivMB8U9V13jwd4L1TUlYkLd3YFtAfcMeD9CQaeNE+Kevc6l4k0zQYGGDDp1t5z492fkH3DGgD0p3WNC7sFUDJJOAK5vUfiB4Q0veLvxJpqsn3kS4WRh/wFcn9K5qP4L6JdSibxDqut69KO19eNtH0C4I/Ouj0/4deDtLVVtfDWmgryrywLKwP+8+T+tAGBL8b/BQfy7K4vtQk7R2llISf++gtMPxZmmGbDwF4tuM9Gaw2qfxBOBXosUEcCBIYkjX0Rdo/SpMUAean4g+M5cG2+GGouvQma9WI5+hQ8U//hMviKeR8Mjg9M6xFn+VekUUAebN42+IUS5k+GEjA9PL1aIn9FNJ/wALI8UW3N98NNaTH3vssgn/ACwozXpVFAHmh+MVpa4/tLwl4qsV/ieXTvlX6/MD+lWrP40+AryRYzrRt5DxtuLaRMfU7do/Ou/wc1UvNK0/UozHfWFrdRnjZPCrj8iKAKum+KdA1htmm63p94+MlIblHYfUA5Fa9cVqfwm8DaqP33h61ibs1qWgx+CECsf/AIVNd6UAfC/jTXdLKnKQzSi4gX22HaPzzQB6bRXmIm+LXh/mW20fxPbKeTEfs1ww/EBR+ANSWvxi0eC4Sz8UaXqnhy6PGL23Yxsf9lgOR74AoA9KoqnpuradrFqLnTb+2vIDx5lvKrrn0yD19quUAFFGRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRmjvVHVNW07RbJ73U72C0tk6yzOFAPpz1PsKAL1FeXv8AE3V/EsrW3gDw5PqKg7W1O+BhtV5AyMkFu/GQ3saX/hW/iXxGN/jPxleTRMPm0/SwIIR7E4y4+q596AOt1nx14W0Aump69YwSp96HzQ8g/wCALlv0rlz8afD1xL5WjabrmtN62FgWAPvuKn9K3tH+Gvg7QgPsXh6z3qciWdPOfPszkkfhXUrGqIEVQqjgADH8qAPNx8R/FN0u7T/hprLjqDdyi3OO5wVPPtT/APhMviI3K/DFgp7HWIs/yr0gdKKAPNf+FgeNLdibv4Y6iq5xm3vUmJP0C9KQ/GXTrLb/AGz4Z8TaWp6y3NhhB9TnP6V6XSFc5zzQByGk/FDwVrJC2niKyVjwEuGMBJ9AHC5/CuuR1dFZWDKRkEHINYWseCvDWv7zqehWFw7jBlMIWT/vsYb9a5F/hH/Y8jT+DPEuqaFIW3fZy/n2xPujHn6kt9KAPTc4ory0+MPHPg5VXxd4dGqWCcNqej/MwGDy0fHpycKBXaeG/F2g+LLQz6LqMNztGXjziSP/AHkPI+vSgDeooHSjrQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRketISAMkgV57rnxTs49SbRfCljN4i1rn93aH9zF2y8nQDntx6kGgD0GSRIo2kkdURQWZmOAAOpJrzzV/i1pQvG03wxZ3PiXU+nlWAzEvu0mCMe4yPcVST4da/4vdLr4ga08kGQy6NpzGO3Xvh26t9eo7Ma9C0nRdN0KxSy0uxgtLdRjZCgUH3Pcn3OTQB58PDnxG8XfvNf12Lw7YPgmx0jmYjPRpc8HHoxHtW/wCH/hd4S8NlZbXSkuLoYJurz99IT6gtwp/3QK7IdBnrRQA0DHHSnUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFQXVnbX1u1vd28U8DjDRyoGU/UHip6KAPO9T+D+gyXT3+gT3vh3UecT6bKyKeOhTOMey4qidQ+Jfgw/8TGyt/Fmlpkm4sl8q6Ue6Dg49ACf9qvUqTkmgDkfDHxH8M+Kpfs1nem3vwSr2N2vlTK3cYPUjvtJxXXA5A6VzPifwD4c8Xx/8TXTlNwBhLuL5Jkx0ww6gehyPauQNj8Qfh+xNjM3i7QV/wCWFw2LyFRk/K38eOPXpwo60AerZHrRXKeFPH2geLw8dhctHex582xuV8ueMjg5Xvj1GfeuqFAC0UUUAFFFFABRRRQAUUUUAFFFFABRRRkUAFFQXNxDaW81zcSJFBEheSR22qqjkknsAK8puNW1v4t3U2n+H5pdL8Ixv5d1qZXEt4O6RA9B2+nX+4QDW8QfEppNXfw/4KsBrutLxIwP+i2w7mRwQD2yAR3Gc8VFpXwqN9fJrHjvUn8QamACkL5FrbnjIROA34gA91zXZ+HvDOk+F9MWw0iyS2hHLY5aRv7zN1Y+56dsVsjgAUARQwR28KQwRJFEgCoiKAqgdgB0qUdKKKACiiigAooooAKKKKACiiigBCM8HnPWuE8S/C3RtZu/7U01ptD1xCWjv9P+Q7ufvKCAc5OSME9z2rvKKAPKrbx3r/gq8i0z4h2e+2dgkGvWaboZD28xQPlP0A+mPmr1C3uIbu2juLeaOaGRQ6SRsGVwehBHBFR3tjbajay2l5bx3FtMNkkUqhlYe4NeV3mia38KbmTUvDiTan4UYl7vR2ctJbZPLxE5JHqPzB+8oB67kUVkeH/EWmeKNIi1PSrhZ7WTIz0KsOqsOxFa/Qc0AFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUZoAK5rxZ410XwXZfadVuQskmfJto/mlmI7Kv5cnAHrXO+JfiHczaw/hjwTbJquuk4lnJzb2Q6Eu3cj0z19T8pt+E/hxb6PfHXdbun1rxJN80l7cDcsR9IlI+UY49cdNo4oAw10bxl8TB5uvzT+GvDcg+XTbdv9JnX/poxHyj2I/4D3r0HQvDmkeGtPWw0fT4rS3HURj5nPqzdWPuTmtcdBmigAHQUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABTSOc/jTqKAOO8WfDnRPFTLdPG9jq0RDQajZny5kYdMkfexgdeR2IrmI/Fvib4eTpY+OIf7R0cnbBr1rGSVGcATIAcHGORz6bzXrFQz28dzDJDNGkkUgKujjIYHsfWgBmn6hZ6pYw3lhcxXNtKu6OWJgysPwqzmvKtR8D6z4Jvp9c+H8mbVz5l5oMxJilHcxehx0H5HHy11Pg7x3pfjC1dbffa6lb5F1p9x8s0LA4Ix3Ge/54PFAHWUUg7CloAKKKKACiiigAooooAKhnmjt4pJpZFjijUu7ucKoA5JJ6ADmpcj1ryrxXe3PxE8WN4H0idk0ezYPrt5EfQ8QA9Mkjn3B/unIBWY3nxm1cojz2ngWxl2vglH1SRT046Rj/Pzfc9XtLSCxtIrW1gSG3hQJHHGuAoHAAApmn6fa6VYQWFlAsNtAgjjjQYAUDA//XVugAooooAKKKKACiiigAooooAKKKKACiiigAooooAKYwzkYyD2x1p9FAHlHiTw9qHgDWJvGPhGEyWLndq+joSElXvIgHQjOfbnqCRXoOgeINO8S6Lb6tpc4ltpl3DplT3VgOjDuK0mUnNeS6pby/CTxSNcsEk/4RHVZQmo2yDctnMekqjsp7j8P7uAD16io4Zop4I5YZFkidQyOjbgwPQg96koAKKKKACiiigAooooAKKKKACiiigAooooAKMj1oqveXdvY2k11dzJDbxKXklkbaqqOSSaAHzzRW8Mk00ixxRqWd3OFUAZJJ9K8ov/ABDrXxP1CfRPCM0mn6BC5jvtaIwZT3jh/DnIxwecD7zGl1P4yagYrWS4sPA1vLiSTlJNTYdQO4jH/wCvLfd9R07TLPSNPg0/T7dYLSBdsUSDAUf/AK+c9zQBQ8NeFNI8JaSmnaRaCKIcu5wXlb+87dz/AC6DA4rdHSgdPSigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKACiiigAooooAKKKKAExXD+M/h7Dr1wmt6N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