conversations listlengths 2 2 |
|---|
[
{
"from": "human",
"value": "Let $N$ be the largest integer whose square has exactly $3$ digits when written in base 9.\n\nWhat is $N$, expressed in base 9?"
},
{
"from": "gpt",
"value": "28"
}
] |
[
{
"from": "human",
"value": "A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?\n$\\textbf{(A)}\\ 8 \\qquad \\textbf{(B)}\\ 12 \\qquad \\textbf{(C)}\\ ... |
[
{
"from": "human",
"value": "From the diagram below, find the positive difference in the $x$-coordinates when lines $l$ and $m$ reach $y = 15.$ [asy]\nimport cse5; import olympiad;\nsize(120);\nadd(grid(8,8));\ndraw((0,0)--(8,0),linewidth(1.2));\ndraw((0,0)--(0,8),linewidth(1.2));\nlabel(\"$x$\",(8,0),E);\n... |
[
{
"from": "human",
"value": "Solve the inequality\n\\[\\dfrac{x+1}{x+2}>\\dfrac{3x+4}{2x+9}.\\]"
},
{
"from": "gpt",
"value": "Union(Interval.open(-9/2, -2), Interval.open((1 - sqrt(5))/2, (1 + sqrt(5))/2))"
}
] |
[
{
"from": "human",
"value": "Compute the sum of the geometric series $-1 + 2 - 4 + 8 - \\cdots + 512$."
},
{
"from": "gpt",
"value": "341"
}
] |
[
{
"from": "human",
"value": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$."
},
{
"from": "gpt",
"value": "\\left(\\frac{1}{2},\\frac{2}{3}\\right)"
}
] |
[
{
"from": "human",
"value": "How many natural numbers greater than 6 but less than 60 are relatively prime to 15?"
},
{
"from": "gpt",
"value": "29"
}
] |
[
{
"from": "human",
"value": "Simplify $\\displaystyle\\frac{1-i}{2+3i}$, where $i^2 = -1.$"
},
{
"from": "gpt",
"value": "-1/13 - 5*I/13"
}
] |
[
{
"from": "human",
"value": "One line is described by\n\\[\\begin{pmatrix} 2 \\\\ 3 \\\\ 4 \\end{pmatrix} + t \\begin{pmatrix} 1 \\\\ 1 \\\\ -k \\end{pmatrix}.\\]Another line is described by\n\\[\\begin{pmatrix} 1 \\\\ 4 \\\\ 5 \\end{pmatrix} + u \\begin{pmatrix} k \\\\ 2 \\\\ 1 \\end{pmatrix}.\\]If the lin... |
[
{
"from": "human",
"value": "Solve for $x$: $\\left(\\frac{1}{4}\\right)^{2x+8} = (16)^{2x+5}$."
},
{
"from": "gpt",
"value": "-3"
}
] |
[
{
"from": "human",
"value": "Find the solutions to $z^4 = -4.$ Enter the solutions, separated by commas."
},
{
"from": "gpt",
"value": "[-1 - I, -1 + I, 1 - I, 1 + I]"
}
] |
[
{
"from": "human",
"value": "Find the largest prime divisor of $25^2+72^2$.\n"
},
{
"from": "gpt",
"value": "157"
}
] |
[
{
"from": "human",
"value": "What is the sum of the squares of the coefficients of $4(x^4 + 3x^2 + 1)$?"
},
{
"from": "gpt",
"value": "176"
}
] |
[
{
"from": "human",
"value": "Find all values of $k$ so that\n\\[x^2 - (k - 3) x - k + 6 > 0\\]for all $x.$"
},
{
"from": "gpt",
"value": "Interval.open(-3, 5)"
}
] |
[
{
"from": "human",
"value": "The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0,$ and none of $m,$ $n,$ and $p$ is zero. What is the value of $n/p?$"
},
{
"from": "gpt",
"value": "8"
}
] |
[
{
"from": "human",
"value": "Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$.\n"
},
{
"from": "gpt",
"value": "757"
}
] |
[
{
"from": "human",
"value": "A penny, nickel, dime, and quarter are simultaneously flipped. What is the expected value of the amount of the coins which come up heads? Express your answer in cents, rounded to the nearest tenth of a cent."
},
{
"from": "gpt",
"value": "20.5"
}
] |
[
{
"from": "human",
"value": "Let $P(x)$ be a polynomial such that when $P(x)$ is divided by $x-17$, the remainder is $14$, and when $P(x)$ is divided by $x-13$, the remainder is $6$. What is the remainder when $P(x)$ is divided by $(x-13)(x-17)$?"
},
{
"from": "gpt",
"value": "2x-20"
}
] |
[
{
"from": "human",
"value": "In triangle $ABC$, $AB = 5$, $BC = 4$, and $CA = 3$.\n\n[asy]\ndefaultpen(1);\n\npair C=(0,0), A = (0,3), B = (4,0);\ndraw(A--B--C--cycle);\n\nlabel(\"\\(A\\)\",A,N);\nlabel(\"\\(B\\)\",B,E);\nlabel(\"\\(C\\)\",C,SW);\n[/asy]\n\nPoint $P$ is randomly selected inside triangle $AB... |
[
{
"from": "human",
"value": "In spherical coordinates, the point $\\left( 3, \\frac{2 \\pi}{7}, \\frac{8 \\pi}{5} \\right)$ is equivalent to what other point, in the standard spherical coordinate representation? Enter your answer in the form $(\\rho,\\theta,\\phi),$ where $\\rho > 0,$ $0 \\le \\theta < 2 \... |
[
{
"from": "human",
"value": "A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of the shadow, which does not include the... |
[
{
"from": "human",
"value": "Solve for $z$ in the following equation: $1-iz = -1 + iz$ (where $i^2 = -1$). Simplify your answer as much as possible."
},
{
"from": "gpt",
"value": "-I"
}
] |
[
{
"from": "human",
"value": "Let $\\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\\mathcal{S}_{11}$.\n"
},
{
"from": "gpt",
"v... |
[
{
"from": "human",
"value": "Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x,y)$ such that $41x + y = 2009$. Find the number of such distinct triangles whose area is a positive integer.\n"
},
{
... |
[
{
"from": "human",
"value": "The function $f(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $f(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $f(x) \\le f(y).$\n(iii) $f(1 - x) = 1 - f(x)$ for all $0 \\le x \\le 1.$\n(iv) $f \\left( \\frac{x}{3} \\right) = \\frac{f(x)}{2}$ for $0 \\... |
[
{
"from": "human",
"value": "The parabola with equation $y=ax^2+bx+c$ is graphed below:\n\n[asy]\nxaxis(-3,7);\n\nyaxis(-5,32);\n\nreal g(real x)\n\n{\n\nreturn 4(x-2)^2-4;\n}\n\ndraw(graph(g,-1,5));\ndot((2,-4));\nlabel(\"Vertex: $(2,-4)$\", (2,-4), SE);\ndot((4,12));\nlabel(\"$(4,12)$\", (4,12), E);\n[/as... |
[
{
"from": "human",
"value": "How many integers $n$ satisfy the condition $100 < n < 200$ and the condition $n$ has the same remainder whether it is divided by $6$ or by $8$?"
},
{
"from": "gpt",
"value": "25"
}
] |
[
{
"from": "human",
"value": "If\n\\[\\frac{x}{a} + \\frac{y}{b} + \\frac{z}{c} = 3 \\quad \\text{and} \\quad \\frac{a}{x} + \\frac{b}{y} + \\frac{c}{z} = 0,\\]find $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} + \\frac{z^2}{c^2}.$"
},
{
"from": "gpt",
"value": "9"
}
] |
[
{
"from": "human",
"value": "A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on M... |
[
{
"from": "human",
"value": "Each face of a cube is painted either red or blue, each with probability 1/2. The color of each face is determined independently. What is the probability that the painted cube can be placed on a horizontal surface so that the four vertical faces are all the same color?"
},
... |
[
{
"from": "human",
"value": "Let \\[ f(x) =\n\\begin{cases}\n-x^2 & \\text{if } x \\geq 0,\\\\\nx+8& \\text{if } x <0.\n\\end{cases}\n\\]Compute $f(f(f(f(f(1))))).$"
},
{
"from": "gpt",
"value": "-33"
}
] |
[
{
"from": "human",
"value": "Find the sum of the coefficients in the polynomial $3(3x^{7} + 8x^4 - 7) + 7(x^5 - 7x^2 + 5)$ when it is fully simplified."
},
{
"from": "gpt",
"value": "5"
}
] |
[
{
"from": "human",
"value": "There exists a complex number of the form $z = x + yi,$ where $x$ and $y$ are positive integers, such that\n\\[z^3 = -74 + ci,\\]for some integer $c.$ Find $z.$"
},
{
"from": "gpt",
"value": "1 + 5*I"
}
] |
[
{
"from": "human",
"value": "In rectangle $ABCD$, $P$ is a point on $BC$ so that $\\angle APD=90^{\\circ}$. $TS$ is perpendicular to $BC$ with $BP=PT$, as shown. $PD$ intersects $TS$ at $Q$. Point $R$ is on $CD$ such that $RA$ passes through $Q$. In $\\triangle PQA$, $PA=20$, $AQ=25$ and $QP=15$. Find $... |
[
{
"from": "human",
"value": "Let $x$ and $y$ be real numbers such that $x + y = 3.$ Find the maximum value of\n\\[x^4 y + x^3 y + x^2 y + xy + xy^2 + xy^3 + xy^4.\\]"
},
{
"from": "gpt",
"value": "\\frac{400}{11}"
}
] |
[
{
"from": "human",
"value": "Let $X,$ $Y,$ and $Z$ be points such that $\\frac{XZ}{XY} = \\frac{ZY}{XY} = \\frac{1}{2}.$ If $Y = (1, 7)$, $Z = (-1, -7)$, then what is the sum of the coordinates of $X$?"
},
{
"from": "gpt",
"value": "-24"
}
] |
[
{
"from": "human",
"value": "A parabola with equation $y = x^2 + bx + c$ passes through the points $(2,3)$ and $(4,3)$. What is $c$?"
},
{
"from": "gpt",
"value": "11"
}
] |
[
{
"from": "human",
"value": "Below is a portion of the graph of a quadratic function, $y=q(x)=ax^2+bx+c$:\n\n[asy]\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-0.99,xmax=10.5,ymin=-5.5,ymax=5.5;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n/*g... |
[
{
"from": "human",
"value": "Find a nonzero $p$ such that $px^2-12x+4=0$ has only one solution."
},
{
"from": "gpt",
"value": "9"
}
] |
[
{
"from": "human",
"value": "Find the sum: $(-39) + (-37) + \\cdots + (-1)$."
},
{
"from": "gpt",
"value": "-400"
}
] |
[
{
"from": "human",
"value": "Let $a$, $b$, and $c$ be positive real numbers. What is the smallest possible value of $(a+b+c)\\left(\\frac{1}{a+b}+\\frac{1}{a+c}+\\frac{1}{b+c}\\right)$?"
},
{
"from": "gpt",
"value": "\\frac{9}{2}"
}
] |
[
{
"from": "human",
"value": "If $a,b,c$ satisfy the system of equations \\begin{align*}b + c &= 12-3a \\\\\na+c &= -14 - 3b \\\\\na+b &= 7 - 3c,\n\\end{align*} what is $2a + 2b + 2c$?"
},
{
"from": "gpt",
"value": "2"
}
] |
[
{
"from": "human",
"value": "Given that $\\binom{17}{10}=19448$, $\\binom{17}{11}=12376$ and $\\binom{19}{12}=50388$, find $\\binom{18}{12}$."
},
{
"from": "gpt",
"value": "18564"
}
] |
[
{
"from": "human",
"value": "The sum of three numbers $a$, $b$, and $c$ is 99. If we increase $a$ by 6, decrease $b$ by 6 and multiply $c$ by 5, the three resulting numbers are equal. What is the value of $b$?"
},
{
"from": "gpt",
"value": "51"
}
] |
[
{
"from": "human",
"value": "A positive five-digit integer is in the form $AB,CBA$; where $A$, $B$ and $C$ are each distinct digits. What is the greatest possible value of $AB,CBA$ that is divisible by eleven?"
},
{
"from": "gpt",
"value": "96,\\!569"
}
] |
[
{
"from": "human",
"value": "Let $n = 2^4 \\cdot 3^5 \\cdot 4^6\\cdot 6^7$. How many natural-number factors does $n$ have?"
},
{
"from": "gpt",
"value": "312"
}
] |
[
{
"from": "human",
"value": "If $$1+6+11+16+21+26+\\cdots+91+96+101 \\equiv n\\pmod{15},$$where $0\\le n<15$, what is the value of $n$?"
},
{
"from": "gpt",
"value": "6"
}
] |
[
{
"from": "human",
"value": "A circle of radius 1 is internally tangent to two circles of radius 2 at points $A$ and $B$, where $AB$ is a diameter of the smaller circle. What is the area of the region, shaded in the figure, that is outside the smaller circle and inside each of the two larger circles? Exp... |
[
{
"from": "human",
"value": "How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\\frac14$?"
},
{
"from": "gpt",
"value": "5"
}
] |
[
{
"from": "human",
"value": "A point $(3\\sqrt{5},d+3)$ is $3d$ units away from the origin. What is the smallest possible value of $d$?"
},
{
"from": "gpt",
"value": "3"
}
] |
[
{
"from": "human",
"value": "Solve the inequality\n\\[\\frac{x^2 - 25}{x + 5} < 0.\\]"
},
{
"from": "gpt",
"value": "Union(Interval.open(-oo, -5), Interval.open(-5, 5))"
}
] |
[
{
"from": "human",
"value": "What is the sum of all the positive two-digit integers divisible by both the sum and product of their digits?"
},
{
"from": "gpt",
"value": "72"
}
] |
[
{
"from": "human",
"value": "Krzysztof solved the quadratic equation $11x^2-44x-99=0$ by completing the square. In the process, he came up with the equivalent equation $$(x+r)^2 = s,$$where $r$ and $s$ are constants.\n\nWhat is $r+s$?"
},
{
"from": "gpt",
"value": "11"
}
] |
[
{
"from": "human",
"value": "Two standard dice are rolled. What is the expected number of 1's obtained? Express your answer as a common fraction."
},
{
"from": "gpt",
"value": "\\frac{1}{3}"
}
] |
[
{
"from": "human",
"value": "Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{2}{b}.\\]"
},
{
"from": "gpt",
"value": "9"
}
] |
[
{
"from": "human",
"value": "Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\\sqrt{89}$ units and diagonals that differ by 6 units?"
},
{
"from": "gpt",
"value": "80"
}
] |
[
{
"from": "human",
"value": "What is the positive difference of the solutions of $\\dfrac{r^2-3r-17}{r+4}=2r+7$?"
},
{
"from": "gpt",
"value": "12"
}
] |
[
{
"from": "human",
"value": "Alicia has $n$ candies, where $n$ is a positive integer with three digits. If she buys $5$ more, she will have a multiple of $8$. If she loses $8$, she will have a multiple of $5$. What is the smallest possible value of $n$?"
},
{
"from": "gpt",
"value": "123"
}
] |
[
{
"from": "human",
"value": "Let the reciprocals of the roots of $5x^2 + 3x + 4$ be $\\alpha$ and $\\beta$. Evaluate $\\alpha + \\beta$."
},
{
"from": "gpt",
"value": "-\\dfrac{3}{4}"
}
] |
[
{
"from": "human",
"value": "Compute the unique positive integer $n$ such that\n\\[2 \\cdot 2^2 + 3 \\cdot 2^3 + 4 \\cdot 2^4 + \\dots + n \\cdot 2^n = 2^{n + 10}.\\]"
},
{
"from": "gpt",
"value": "513"
}
] |
[
{
"from": "human",
"value": "In the game Deal or No Deal, participants choose a box at random from a set of $26,$ one containing each of the following values: \\begin{tabular}{|c|c|}\\hline\\$.01&\\$1,000\\\\\\hline\\$1&\\$5,000\\\\\\hline\\$5&\\$10,000\\\\\\hline\\$10&\\$25,000\\\\\\hline\\$25&\\$50,000\\\... |
[
{
"from": "human",
"value": "Let $\\omega$ be a nonreal root of $z^3 = 1.$ Find the number of ordered pairs $(a,b)$ of integers such that $|a \\omega + b| = 1.$"
},
{
"from": "gpt",
"value": "6"
}
] |
[
{
"from": "human",
"value": "Four horizontal lines and four vertical lines are drawn in a plane. In how many ways can four lines be chosen such that a rectangular region is enclosed?"
},
{
"from": "gpt",
"value": "36"
}
] |
[
{
"from": "human",
"value": "Solve\n\\[(x^3 + 3x^2 \\sqrt{2} + 6x + 2 \\sqrt{2}) + (x + \\sqrt{2}) = 0.\\]Enter all the solutions, separated by commas."
},
{
"from": "gpt",
"value": "[-sqrt(2), -sqrt(2) - I, -sqrt(2) + I]"
}
] |
[
{
"from": "human",
"value": "Quadrilateral $ABCD$ is a square. A circle with center $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. If $AB = 2$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined? Express your answer as a decimal to the ne... |
[
{
"from": "human",
"value": "A curve is described parametrically by\n\\[(x,y) = (2 \\cos t - \\sin t, 4 \\sin t).\\]The graph of the curve can be expressed in the form\n\\[ax^2 + bxy + cy^2 = 1.\\]Enter the ordered triple $(a,b,c).$"
},
{
"from": "gpt",
"value": "\\left( \\frac{1}{4}, \\frac{1}{... |
[
{
"from": "human",
"value": "The functions $f(x) = x^2-2x + m$ and $g(x) = x^2-2x + 4m$ are evaluated when $x = 4$. What is the value of $m$ if $2f(4) = g(4)$?"
},
{
"from": "gpt",
"value": "4"
}
] |
[
{
"from": "human",
"value": "Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the... |
[
{
"from": "human",
"value": "The letters of the alphabet are each assigned a random integer value, and $H=10$. The value of a word comes from the sum of its letters' values. If $MATH$ is 35 points, $TEAM$ is 42 points and $MEET$ is 38 points, what is the value of $A$?"
},
{
"from": "gpt",
"value... |
[
{
"from": "human",
"value": "Find $\\left(\\frac{1+i}{\\sqrt{2}}\\right)^{46}$."
},
{
"from": "gpt",
"value": "-I"
}
] |
[
{
"from": "human",
"value": "The numbers 2, 3, 5, 7, 11, 13 are arranged in a multiplication table, with three along the top and the other three down the left. The multiplication table is completed and the sum of the nine entries is tabulated. What is the largest possible sum of the nine entries?\n\n\\[\n... |
[
{
"from": "human",
"value": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt[3]{3} - \\sqrt[3]{2}}$. With your answer in the form $\\displaystyle \\frac{\\sqrt[3]{A} + \\sqrt[3]{B} + \\sqrt[3]{C}}{D}$, and the fraction in lowest terms, what is $A + B + C + D$?"
},
{
"from": "gpt",
... |
[
{
"from": "human",
"value": "The value of the expression \\[(2^{1004}+5^{1005})^2-(2^{1004}-5^{1005})^2\\]is $k\\cdot10^{1004}$ for some positive integer $k$. What is $k$?"
},
{
"from": "gpt",
"value": "20"
}
] |
[
{
"from": "human",
"value": "An engineer invested $\\$10,\\!000$ in a six-month savings certificate that paid a simple annual interest rate of $12\\%$. After six months, she invested the total value of her investment in another six-month certificate. After six more months, the investment was worth $\\$11,\\... |
[
{
"from": "human",
"value": "The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)\n\nWhat is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal tic... |
[
{
"from": "human",
"value": "The rules for a race require that all runners start at $A$, touch any part of the 1200-meter wall, and stop at $B$. What is the number of meters in the minimum distance a participant must run? Express your answer to the nearest meter. [asy]\nimport olympiad; import geometry; siz... |
[
{
"from": "human",
"value": "A pyramid with volume 40 cubic inches has a rectangular base. If the length of the base is doubled, the width tripled and the height increased by $50\\%$, what is the volume of the new pyramid, in cubic inches?"
},
{
"from": "gpt",
"value": "360"
}
] |
[
{
"from": "human",
"value": "What is the value of the sum\n\\[\n \\sum_z \\frac{1}{{\\left|1 - z\\right|}^2} \\, ,\n\\]where $z$ ranges over all 7 solutions (real and nonreal) of the equation $z^7 = -1$?"
},
{
"from": "gpt",
"value": "\\frac{49}{4}"
}
] |
[
{
"from": "human",
"value": "Suppose that $f(x)$ is a function such that\n\\[f(xy) + x = xf(y) + f(x)\\]for all real numbers $x$ and $y.$ If $f(-1) = 5$ then compute $f(-1001).$"
},
{
"from": "gpt",
"value": "2005"
}
] |
[
{
"from": "human",
"value": "The graph of the function $y=f(x)$ is shown below. For all $x > 4$, it is true that $f(x) > 0.4$. If $f(x) = \\frac{x^2}{Ax^2 + Bx + C}$, where $A,B,$ and $C$ are integers, then find $A+B+C$. [asy]\nimport graph; size(10.9cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); d... |
[
{
"from": "human",
"value": "Compute the number of ordered pairs of integers $(x,y)$ with $1\\le x<y\\le 100$ such that $i^x+i^y$ is a real number."
},
{
"from": "gpt",
"value": "1850"
}
] |
[
{
"from": "human",
"value": "The function $f$ is graphed below. Each small box has width and height 1.\n\n[asy]\nsize(150);\nreal ticklen=3;\nreal tickspace=2;\n\nreal ticklength=0.1cm;\nreal axisarrowsize=0.14cm;\npen axispen=black+1.3bp;\nreal vectorarrowsize=0.2cm;\nreal tickdown=-0.5;\nreal tickdownlen... |
[
{
"from": "human",
"value": "A circle having center $(0,k)$, with $k>6$, is tangent to the lines $y=x$, $y=-x$ and $y=6$. What is the radius of this circle?"
},
{
"from": "gpt",
"value": "6\\sqrt{2}+6"
}
] |
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