question string | answer string | difficulty int64 | problem_id string |
|---|---|---|---|
What is the value of $9901\cdot101-99\cdot10101?$
\textbf{(A)}~2\qquad\textbf{(B)}~20\qquad\textbf{(C)}~200\qquad\textbf{(D)}~202\qquad\textbf{(E)}~2020 | A | 2 | 2024A-P1 |
A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T=aL+bG,$ where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to... | B | 2 | 2024A-P2 |
The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$ | B | 2 | 2024A-P3 |
What is the least value of $n$ such that $n!$ is a multiple of $2024$?
$\textbf{(A) }11 \qquad
\textbf{(B) }21 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }23 \qquad
\textbf{(E) }253 \qquad$ | D | 2 | 2024A-P4 |
A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$ | D | 2 | 2024A-P5 |
The product of three integers is $60$. What is the least possible positive sum of the three integers?
$\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$ | B | 2 | 2024A-P6 |
In $\Delta ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \dots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1= P_1P_2 = P_2P_3 = \dots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum
$$
\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + ... | D | 2 | 2024A-P7 |
How many angles $\theta$ with $0\le\theta\le2\pi$ satisfy $\log(\sin(3\theta))+\log(\cos(2\theta))=0$?
$\textbf{(A) }0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4 \qquad$ | A | 2 | 2024A-P8 |
Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$?
$\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }8 \qquad$ | E | 2 | 2024A-P9 |
Let $\alpha$ be the radian measure of the smallest angle in a $3{-}4{-}5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7{-}24{-}25$ right triangle. In terms of $\alpha$, what is $\beta$?
$\textbf{(A) }\frac{\alpha}{3}\qquad
\textbf{(B) }\alpha - \frac{\pi}{8}\qquad
\textbf{(C) }\frac{\... | C | 2 | 2024A-P10 |
There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$?
$\textbf{(A) }16\qquad\textbf{(B) }17\qquad\textbf{(C) }18\qquad\textbf{(D) }20\qquad\textbf{(E) }21$ | D | 3 | 2024A-P11 |
The first three terms of a geometric sequence are the integers $a,\,720,$ and $b,$ where $a<720<b.$ What is the sum of the digits of the least possible value of $b?$
$\textbf{(A) } 9 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 16 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 21$ | E | 3 | 2024A-P12 |
The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?
$\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\f... | D | 3 | 2024A-P13 |
The numbers, in order, of each row and the numbers, in order, of each column of a $5 \times 5$ array of integers form an arithmetic progression of length $5{.}$ The numbers in positions $(5, 5), \,(2,4),\,(4,3),$ and $(3, 1)$ are $0, 48, 16,$ and $12{,}$ respectively. What number is in position $(1, 2)?$
$$
\begin{bmat... | C | 3 | 2024A-P14 |
The roots of $x^3 + 2x^2 - x + 3$ are $p, q,$ and $r.$ What is the value of $$
(p^2 + 4)(q^2 + 4)(r^2 + 4)?
$$
$\textbf{(A) } 64 \qquad \textbf{(B) } 75 \qquad \textbf{(C) } 100 \qquad \textbf{(D) } 125 \qquad \textbf{(E) } 144$ | D | 3 | 2024A-P15 |
A set of $12$ tokens ---- $3$ red, $2$ white, $1$ blue, and $6$ black ---- is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$,... | C | 3 | 2024A-P16 |
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad$ | D | 3 | 2024A-P17 |
On top of a rectangular card with sides of length $1$ and $2+\sqrt{3}$, an identical card is placed so that two of their diagonals line up, as shown ($\overline{AC}$, in this case).
```asy
defaultpen(fontsize(12)+0.85); size(150);
real h=2.25;
pair C=origin,B=(0,h),A=(1,h),D=(1,0),Dp=reflect(A,C)*D,Bp=reflect(A,C)*B;
... | A | 3 | 2024A-P18 |
Cyclic quadrilateral $ABCD$ has lengths $BC=CD=3$ and $DA=5$ with $\angle CDA=120^\circ$. What is the length of the shorter diagonal of $ABCD$?
$\textbf{(A) }\frac{31}7 \qquad
\textbf{(B) }\frac{33}7 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }\frac{39}7 \qquad
\textbf{(E) }\frac{41}7 \qquad$ | D | 3 | 2024A-P19 |
Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline {AB}$ and $\overline{AC},$ respectively, of equilateral triangle $\Delta ABC.$ Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC?$
$\textbf{(A) }... | D | 3 | 2024A-P20 |
Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation $$
\frac{a_n -1}{n-1}=\frac{a_{n-1}+1}{(n-1)+1}
$$for all $n \ge 2.$ What is the greatest integer less than or equal to $$
\sum^{100}_{n=1} a_n^2?
$$
$\textbf{(A) } 338{,}550 \qquad \textbf{(B) } 338{,}551 \qquad \textbf{(C) } 338{,}552 \... | B | 4 | 2024A-P21 |
The figure below shows a dotted grid $8$ cells wide and $3$ cells tall consisting of $1''\times1''$ squares. Carl places $1$-inch toothpicks along some of the sides of the squares to create a closed loop that does not intersect itself. The numbers in the cells indicate the number of sides of that square that are to be ... | C | 4 | 2024A-P22 |
What is the value of
$$
\tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {3\pi}{16}~ + ~ \tan^2 \frac {\pi}{16} \cdot \tan^2 \frac {5\pi}{16} ~+~\tan^2 \frac {3\pi}{16} \cdot \tan^2 \frac {7\pi}{16} ~+~ \tan^2 \frac {5\pi}{16} \cdot \tan^2 \frac {7\pi}{16}?
$$
$\textbf{(A) } 28 \qquad \textbf{(B) } 68 \qquad \textbf{(C) } 7... | B | 4 | 2024A-P23 |
A $\textit{disphenoid}$ is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?
$\textbf{(A) }\sqrt{3}\qquad\textbf{(B) }3\sqrt{15}\qquad\textbf{(C) }15\qquad\textbf{(D) }15\sqrt{7}\qquad\tex... | D | 4 | 2024A-P24 |
A graph is $\textit{symmetric}$ about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a,b,c,d)$, where $|a|,|b|,|c|,|d|\le5$ and $c$ and $d$ are not both $0$, is the graph of $$
y=\frac{ax+b}{cx+d}
$$symmetric about the line $y=x$?
$\textbf{(A) }1282\qquad\tex... | B | 4 | 2024A-P25 |
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
$\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025$ | B | 2 | 2024B-P1 |
What is $10! - 7! \cdot 6!$?
$\textbf{(A) }-120 \qquad\textbf{(B) }0 \qquad\textbf{(C) }120 \qquad\textbf{(D) }600 \qquad\textbf{(E) }720 \qquad$ | B | 2 | 2024B-P2 |
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$ | E | 2 | 2024B-P3 |
Balls numbered $1,2,3,\ldots$ are deposited in $5$ bins, labeled $A,B,C,D,$ and $E$, using the following procedure. Ball $1$ is deposited in bin $A$, and balls $2$ and $3$ are deposited in $B$. The next three balls are deposited in bin $C$, the next $4$ in bin $D$, and so on, cycling back to bin $A$ after balls are dep... | D | 2 | 2024B-P4 |
In the following expression, Melanie changed some of the plus signs to minus signs:$$
1 + 3+5+7+\cdots+97+99
$$When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
$\textbf{(A) }14 \qquad
\textbf{(B) }15 \qquad
\textbf{(C) }16 \q... | B | 2 | 2024B-P5 |
The national debt of the United States is on track to reach $5 \cdot 10^{13}$ dollars by $2033$. How many digits does this number of dollars have when written as a numeral in base $5$? (The approximation of $\log_{10} 5$ as $0.7$ is sufficient for this problem.)
$\textbf{(A) }18 \qquad
\textbf{(B) }20 \qquad
\textbf{(... | B | 2 | 2024B-P6 |
In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$?
```asy
pair X = (0, 0);
pair W = (0, 4);
pair Y = (8, ... | C | 2 | 2024B-P7 |
What value of $x$ satisfies$$
\frac{\log_2x\cdot\log_3x}{\log_2x+\log_3x}=2?
$$
$\textbf{(A) }25\qquad
\textbf{(B) }32\qquad
\textbf{(C) }36\qquad
\textbf{(D) }42\qquad
\textbf{(E) }48\qquad$ | C | 2 | 2024B-P8 |
A dartboard is the region $B$ in the coordinate plane consisting of points $(x,y)$ such that $|x| + |y| \le 8$ . A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49.$ A dart is thrown and lands at a random point in $B$. The probability that the dart lands in $T$ can be expressed as $\frac{m}{n} \cdot \pi,$ wher... | B | 2 | 2024B-P9 |
A list of 9 real numbers consists of $1$, $2.2$, $3.2$, $5.2$, $6.2$, and $7$, as well as $x, y,z$ with $x\leq y\leq z$. The range of the list is $7$, and the mean and median are both positive integers. How many ordered triples $(x,y,z)$ are possible?
$\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }3 \qquad\te... | C | 2 | 2024B-P10 |
Let $x_{n} = \sin^2(n^\circ)$. What is the mean of $x_{1}, x_{2}, x_{3}, \cdots, x_{90}$?
$\textbf{(A) }\frac{11}{45} \qquad
\textbf{(B) }\frac{22}{45} \qquad
\textbf{(C) }\frac{89}{180} \qquad
\textbf{(D) }\frac{1}{2} \qquad
\textbf{(E) }\frac{91}{180} \qquad$ | E | 3 | 2024B-P11 |
Suppose $z$ is a complex number with positive imaginary part, with real part greater than $1$, and with $|z| = 2$. In the complex plane, the four points $0$, $z$, $z^{2}$, and $z^{3}$ are the vertices of a quadrilateral with area $15$. What is the imaginary part of $z$?
\textbf{(A)}~\frac{3}{4}\qquad\textbf{(B)}~1\qqu... | D | 3 | 2024B-P12 |
There are real numbers $x,y,h$ and $k$ that satisfy the system of equations$$
x^2 + y^2 - 6x - 8y = h
$$$$
x^2 + y^2 - 10x + 4y = k
$$What is the minimum possible value of $h+k$?
$\textbf{(A) }-54 \qquad
\textbf{(B) }-46 \qquad
\textbf{(C) }-34 \qquad
\textbf{(D) }-16 \qquad
\textbf{(E) }16 \qquad$ | C | 3 | 2024B-P13 |
How many different remainders can result when the $100$th power of an integer is divided by $125$?
$\textbf{(A) }1 \qquad\textbf{(B) }2 \qquad\textbf{(C) }5 \qquad\textbf{(D) }25 \qquad\textbf{(E) }125 \qquad$ | B | 3 | 2024B-P14 |
A triangle in the coordinate plane has vertices $A(\log_21,\log_22)$, $B(\log_23,\log_24)$, and $C(\log_27,\log_28)$. What is the area of $\triangle ABC$?
$\textbf{(A) }\log_2\frac{\sqrt3}7\qquad
\textbf{(B) }\log_2\frac3{\sqrt7}\qquad
\textbf{(C) }\log_2\frac7{\sqrt3}\qquad
\textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad
... | B | 3 | 2024B-P15 |
A group of $16$ people will be partitioned into $4$ indistinguishable $4$-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as $3^{r}M$, where $r$ and $M$ are positive integers and $M$ is not divisible by $3$. What is $r$... | A | 3 | 2024B-P16 |
Integers $a$ and $b$ are randomly chosen without replacement from the set of integers with absolute value not exceeding $10$. What is the probability that the polynomial $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots?
$\textbf{(A) }\frac{1}{240} \qquad \textbf{(B) }\frac{1}{221} \qquad \textbf{(C) }\frac{1}{105}... | C | 3 | 2024B-P17 |
The Fibonacci numbers are defined by $F_1=1,$ $F_2=1,$ and $F_n=F_{n-1}+F_{n-2}$ for $n\geq 3.$ What is$$
\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?
$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$ | B | 3 | 2024B-P18 |
Equilateral $\triangle ABC$ with side length $14$ is rotated about its center by angle $\theta$, where $0 < \theta < 60^{\circ}$, to form $\triangle DEF$. See the figure. The area of hexagon $ADBECF$ is $91\sqrt{3}$. What is $\tan\theta$?
```asy
// Credit to shihan for this diagram.
defaultpen(fontsize(13)); size(200... | B | 3 | 2024B-P19 |
Suppose $A$, $B$, and $C$ are points in the plane with $AB=40$ and $AC=42$, and let $x$ be the length of the line segment from $A$ to the midpoint of $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p,q)$, and the maximum value $r$ of ... | C | 3 | 2024B-P20 |
The measures of the smallest angles of three different right triangles sum to $90^\circ$. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are $3-4-5$ and $5-12-13$. What is the perimeter of the third triangle?
$\textbf{(A) }40 \qquad
\textbf{(B) }126 \qquad
\textbf{(C) }154 \q... | C | 4 | 2024B-P21 |
Let $\triangle ABC$ be a triangle with integer side lengths and the property that $\angle{B} = 2\angle{A}$. What is the least possible perimeter of such a triangle?
$\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad$ | C | 4 | 2024B-P22 |
A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V.$ Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid?
$\textbf{(A) }1 \qquad
\textbf{(B) }\frac{1+\sqrt2}{2} \qquad
\textbf{(C) }\sqrt2 \qquad
\textbf{(D) }\frac32 \q... | B | 4 | 2024B-P23 |
What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, such that there exists a (non-degenerate) triangle $\triangle ABC$ with an integer inradius for which $a$, $b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overli... | B | 4 | 2024B-P24 |
Pablo will decorate each of $6$ identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the $12$ decisions he must make. After the paint dries, he will place the $6$ balls in an urn. Frida ... | A | 4 | 2024B-P25 |
Andy and Betsy both live in Mathville. Andy leaves Mathville on his bicycle at $1{:}30$, traveling due north at a steady $8$ miles per hour. Betsy leaves on her bicycle from the same point at $2{:}30$, traveling due east at a steady $12$ miles per hour. At what time will they be exactly the same distance from their com... | E | 2 | 2025A-P1 |
A box contains $10$ pounds of a nut mix that is $50$ percent peanuts, $20$ percent cashews, and $30$ percent almonds. A second nut mix containing $20$ percent peanuts, $40$ percent cashews, and $40$ percent almonds is added to the box resulting in a new nut mix that is $40$ percent peanuts. How many pounds of cashews a... | B | 2 | 2025A-P2 |
A team of students is going to compete against a team of teachers in a trivia contest. The total number of students and teachers is $15$. Ash, a cousin of one of the students, wants to join the contest. If Ash plays with the students, the average age on that team will increase from $12$ to $14$. If Ash plays with the t... | A | 2 | 2025A-P3 |
Agnes writes the following four statements on a blank piece of paper.
*At least one of these statements is true.
*At least two of these statements are true.
*At least two of these statements are false.
*At least one of these statements is false.
Each statement is either true or false. How many false statements did Agne... | B | 2 | 2025A-P4 |
In the figure below, the outside square contains infinitely many squares, each of them with the same center and sides parallel to the outside square. The ratio of the side length of a square to the side length of the next inner square is $k$, where $0 < k < 1.$ The spaces between squares are alternately shaded as shown... | D | 2 | 2025A-P5 |
Six chairs are arranged around a round table. Two students and two teachers randomly select four of the chairs to sit in. What is the probability that the two students will sit in two adjacent chairs and the two teachers will also sit in two adjacent chairs?
$\textbf{(A) } \frac 16 \qquad \textbf{(B) } \frac 15 \qquad... | B | 2 | 2025A-P6 |
In a certain alien world, the maximum running speed $v$ of an organism is dependent on its number of toes $n$ and number of eyes $m$. The relationship can be expressed as $v=kn^am^b$ centimeters per hour, where $k$, $a$, and $b$ are integer constants. In a population where all organisms have $5$ toes, $\log v=4+2\log m... | C | 2 | 2025A-P7 |
Pentagon $ABCDE$ is inscribed in a circle, and $\angle BEC = \angle CED = 30^\circ$. Let line $AC$ and line $BD$ intersect at point $F$, and suppose that $AB = 9$ and $AD = 24$. What is $BF$?
$\textbf{(A) } \frac{57}{11} \qquad\textbf{(B) } \frac{59}{11} \qquad\textbf{(C) } \frac{60}{11} \qquad\textbf{(D) } \frac{61}{... | E | 2 | 2025A-P8 |
Let $w$ be the complex number $2+i$, where $i=\sqrt{-1}$. What real number $r$ has the property that $r$, $w$, and $w^2$ are three collinear points in the complex plane?
\textbf{(A)}~\frac34\qquad\textbf{(B)}~1\qquad\textbf{(C)}~\frac75\qquad\textbf{(D)}~\frac32\qquad\textbf{(E)}~\frac53 | E | 2 | 2025A-P9 |
In the figure shown below, major arc $\widehat{AD}$ and minor arc $\widehat{BC}$ have the same center, $O$. Also, $A$ lies between $O$ and $B$, and $D$ lies between $O$ and $C$. Major arc $\widehat{AD}$, minor arc $\widehat{BC}$, and each of the two segments $\overline{AB}$ and $\overline{CD}$ have length $2\pi$. What ... | B | 2 | 2025A-P10 |
The orthocenter of a triangle is the concurrent intersection of the three (possibly extended) altitudes. What is the sum of the coordinates of the orthocenter of the triangle whose vertices are $A(2,31), B(8,27),$ and $C(18,27)$?
\textbf{(A)}~5\qquad\textbf{(B)}~17\qquad\textbf{(C)}~10+4\sqrt{17} +2\sqrt{13}\qquad\tex... | A | 3 | 2025A-P11 |
The ''harmonic mean'' of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of $4,4,$ and $5$ is
$$
\frac{1}{\frac{1}{3}(\frac{1}{4}+\frac{1}{4}+\frac{1}{5})}=\frac{30}{7}.
$$
What is the harmonic mean of all the real ro... | B | 3 | 2025A-P12 |
Let $C = \{1, 2, 3, \dots, 13\}$. Let $N$ be the greatest integer such that there exists a subset of $C$ with $N$ elements that does not contain five consecutive integers. Suppose $N$ integers are chosen at random from $C$ without replacement. What is the probability that the chosen elements do not include five consecu... | D | 3 | 2025A-P13 |
Points $F$, $G$, and $H$ are collinear with $G$ between $F$ and $H$. The ellipse with foci at $G$ and $H$ is internally tangent to the ellipse with foci at $F$ and $G$, as shown below.
```asy
/* Made by Mathemagician108 */
import graph;
unitsize(0.15 inch);
draw(ellipse((0,0), 12, 4));
draw(ellipse((8,0), 4, 1.45));
p... | D | 3 | 2025A-P14 |
A set of numbers is called sum-free if whenever $x$ and $y$ are (not necessarily distinct) elements of the set, $x+y$ is not an element of the set. For example, $\{1,4,6\}$ and the empty set are sum-free, but $\{2,4,5\}$ is not. What is the greatest possible number of elements in a sum-free subset of $\{1,2,3,...,20\}$... | C | 3 | 2025A-P15 |
Triangle $\triangle ABC$ has side lengths $AB = 80$, $BC = 45$, and $AC = 75$. The bisector $\angle B$ and the altitude to side $\overline{AB}$ intersect at point $P$. What is $BP$?
\textbf{(A)}~18\qquad\textbf{(B)}~19\qquad\textbf{(C)}~20\qquad\textbf{(D)}~21\qquad\textbf{(E)}~22 | D | 3 | 2025A-P16 |
The polynomial $(z + i)(z + 2i)(z + 3i) + 10$ has three roots in the complex plane, where $i = \sqrt{-1}$. What is the area of the triangle formed by these three roots?
\textbf{(A)}~6 \qquad \textbf{(B)}~8 \qquad \textbf{(C)}~10 \qquad \textbf{(D)}~12 \qquad \textbf{(E)}~14 | A | 3 | 2025A-P17 |
How many ordered triples $(x, y, z)$ of different positive integers less than or equal to $8$ satisfy $xy > z$, $xz > y$, and $yz > x$?
\textbf{(A)}~36 \qquad \textbf{(B)}~84 \qquad \textbf{(C)}~186 \qquad \textbf{(D)}~336 \qquad \textbf{(E)}~486 | C | 3 | 2025A-P18 |
Let $a$, $b$, and $c$ be the roots of the polynomial $x^3 + kx + 1$. What is the sum $$
a^3b^2 + a^2b^3 + b^3c^2 + b^2c^3 + c^3a^2 + c^2a^3?
$$
\textbf{(A)}~-k\qquad\textbf{(B)}~-k+1\qquad\textbf{(C)}~1\qquad\textbf{(D)}~k-1\qquad\textbf{(E)}~k | E | 3 | 2025A-P19 |
The base of the pentahedron shown below is a $13 \times 8$ rectangle, and its lateral faces are two isosceles triangles with base of length $8$ and congruent sides of length $13$, and two isosceles trapezoids with bases of length $7$ and $13$ and nonparallel sides of length $13$.
```asy
/* Refined by Mathemagician108 ... | C | 3 | 2025A-P20 |
There is a unique ordered triple $(a,k,m)$ of nonnegative integers such that $$
\frac{4^a + 4^{a+k}+4^{a+2k}+\cdots + 4^{a+mk}}{2^a + 2^{a+k} + 2^{a+2k}+ \cdots + 2^{a+mk}} = 964.
$$ What is $a+k+m$?
$\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C)} 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } 12$ | A | 4 | 2025A-P21 |
Three real numbers are chosen independently and uniformly at random between $0$ and $1$. What is the probability that the greatest of these three numbers is greater than $2$ times each of the other two numbers? (In other words, if the chosen numbers are $a \geq b \geq c$, then $a > 2b$.)
\textbf{(A)}~\frac{1}{12}\qqua... | E | 4 | 2025A-P22 |
Call a positive integer fair if no digit is used more than once, it has no $0$s, and no digit is adjacent to two greater digits. For example, $196, 23,$ and $12463$ are fair, but $1546, 320$ and $34321$ are not. How many fair positive integers are there?
$\textbf{(A) } 511 \qquad \textbf{(B) } 2584 \qquad \textbf{(C) ... | C | 4 | 2025A-P23 |
A circle of radius $r$ is surrounded by $12$ circles of radius $1,$ externally tangent to the central circle and sequentially tangent to each other, as shown. Then $r$ can be written as $\sqrt a + \sqrt b + c,$ where $a, b, c$ are integers. What is $a+b+c?$
```asy
defaultpen(fontsize(12)+linewidth(1)); size(200);
real... | C | 4 | 2025A-P24 |
The instructions on a $350$-gram bag of coffee beans say that proper brewing of a large mug of pour-over coffee requires $20$ grams of coffee beans. What is the greatest number of properly brewed large mugs of coffee that can be made from the coffee beans in that bag?
$\textbf{(A) } 16 \qquad\textbf{(B) } 17 \qquad\te... | B | 2 | 2025B-P1 |
Jerry wrote down the ones digit of each of the first 2025 positive squares: $1,4, 9,6,5,6,\dots$. What is the sum of all the numbers Jerry wrote down?
$\textbf{(A) } 9025 \qquad\textbf{(B) } 9070 \qquad\textbf{(C) } 9090 \qquad\textbf{(D) } 9115 \qquad\textbf{(E) } 9160$ | D | 2 | 2025B-P2 |
What is the value of $i(i-1)(i-2)(i-3)$, where $i=\sqrt{-1}$?
$\textbf{(A) } 6-5i \qquad\textbf{(B) } -10i \qquad\textbf{(C) } 10i \qquad\textbf{(D) } -10 \qquad\textbf{(E) } 10$ | D | 2 | 2025B-P3 |
The value of the two-digit number $\underline{a}\;\underline{b}$ in base seven equals the value of the two-digit number $\underline{b}\;\underline{a}$ in base nine. What is $a+b$?
$\textbf{(A) } 7 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 14$ | A | 2 | 2025B-P4 |
Positive integers $x$ and $y$ satisfy the equation $57x+22y = 400$. What is the least possible value of $x+y$?
$\textbf{(A) } 10 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 13 \qquad\textbf{(D) } 14 \qquad\textbf{(E) } 15$ | E | 2 | 2025B-P5 |
Emmy says to Max, "I ordered 36 math club sweatshirts today." Max asks, "How much did each shirt cost?" Emmy responds, "I'll give you a hint. The total cost was $\$\underline{A}\;\underline{B}\;\underline{B}\;.\underline{B}\;\underline{A}$, where $A$ and $B$ are digits and $A \neq 0$." After a pause, Max says, "That wa... | C | 2 | 2025B-P6 |
What is the value of
$$
\sum_{n=2}^{255} \frac{\log_2(1+\frac{1}{n})}{(\log_2 n)(\log_2(n+1))}
$$
$\textbf{(A) } \dfrac{3}{4} \qquad\textbf{(B) } 1-\dfrac{1}{\log_2 255} \qquad\textbf{(C) } \dfrac{7}{8} \qquad\textbf{(D) } \dfrac{15}{16} \qquad\textbf{(E) } 1$ | C | 2 | 2025B-P7 |
There are integers $a$ and $b$ such that the polynomial $x^3 - 5x^2 + ax + b$ has $4+\sqrt{5}$ as a root. What is $a+b$?
$\textbf{(A) } 13 \qquad \textbf{(B) } 17 \qquad \textbf{(C) } 20 \qquad \textbf{(D) } 30 \qquad \textbf{(E) } 68$ | C | 2 | 2025B-P8 |
What is the tens digit of $6^{6^6}$?
\textbf{(A)}~1 \qquad \textbf{(B)}~3 \qquad \textbf{(C)}~5 \qquad \textbf{(D)}~7 \qquad \textbf{(E)}~9 | C | 2 | 2025B-P9 |
The altitude to the hypotenuse of a $30{-}60{-}90^\circ$ is divided into two segments of lengths $x<y$ by the median to the shortest side of the triangle. What is the ratio $\tfrac{x}{x+y}$?
$\textbf{(A) } \dfrac{3}{7} \qquad \textbf{(B) } \dfrac{\sqrt3}{4} \qquad \textbf{(C) } \dfrac{4}{9} \qquad \textbf{(D) } \dfrac... | A | 2 | 2025B-P10 |
Nine athletes, no two of whom are the same height, try out for the basketball team. One at a time, they draw a wristband at random, without replacement, from a bag containing 3 blue bands, 3 red bands, and 3 green bands. They are divided into a blue group, a red group, and a green group. The tallest member of each grou... | C | 3 | 2025B-P11 |
The windshield wiper on the driver's side of a large bus is depicted below.
```asy
/* Made by MRENTHUSIASM */
size(200);
pair A, B, C, D;
A = origin;
B = (3/2,3*sqrt(3)/2);
C = B+(0,1.75);
D = B-(0,1.75);
draw(A--B,linewidth(1.1));
draw(C--D,red+linewidth(2.1));
dot(A^^B,linewidth(5));
label("$A$",A,1.5*W);
label("$B$... | C | 3 | 2025B-P12 |
A circle has been divided into 6 sectors of different sizes. Then 2 of the sectors are painted red, 2 painted green, and 2 painted blue so that no two neighboring sectors are painted the same color. One such coloring is shown below.
```asy
/* Made by Mathemagician108 */
unitsize(1.7cm);
pair O = (0, 0);
real R = 2;
p... | D | 3 | 2025B-P13 |
Consider a decreasing sequence of $n$ positive integers $x_1>x_2>\cdots>x_n$ that satisfies the following conditions:
* The average of the first $3$ terms in the sequence is $2025.$
* For all $4 \leq k \leq n,$ the average of the first $k$ terms is $1$ less than the average of the first $k - 1$ terms.
What is the gre... | B | 3 | 2025B-P14 |
A container has a $1\times 1$ square bottom, a $3 \times {}3$ open square top, and four congruent trapezoidal sides, as shown. Starting when the container is empty, a hose that runs water at a constant rate takes $35$ minutes to fill the container up to the midline of the trapezoids.
```asy
/* Made by Mathemagician108... | D | 3 | 2025B-P15 |
An analog clock starts at midnight and runs for $2025$ minutes before stopping. What is the tangent of the acute angle between the hour hand and the minute hand when the clock stops?
$\textbf{(A) } 0 \qquad \textbf{(B) } \sqrt{2}-1 \qquad \textbf{(C) } 2-\sqrt{2} \qquad \textbf{(D) } \frac{\sqrt{2}}{2} \qquad \textbf{... | B | 3 | 2025B-P16 |
Each of the $9$ squares in a ${3 \times 3}$ grid is to be colored red, blue, or yellow in such a way that each red square shares an edge with at least one blue square, each blue square shares an edge with at least one yellow square, and each yellow square shares an edge with at least one red square. Colorings that can ... | C | 3 | 2025B-P17 |
Awnik repeatedly plays a game that has a probability of winning of $\frac{1}{3}$. The outcomes of the games are independent. What is the expected value of the number of games he will play until he has both won and lost at least once?
$\textbf{(A) } \frac{5}{2} \qquad \textbf{(B) } 3 \qquad \textbf{(C) } \frac{16}{5} \... | D | 3 | 2025B-P18 |
A rectangular grid of squares has $141$ rows and $91$ columns. Each square has room for two numbers. Horace and Vera each fill in the grid by putting the numbers from $1$ through $141 \times 91 = 12{,}831$ into the squares. Horace fills the grid horizontally: he puts $1$ through $91$ in order from left to right into ro... | C | 3 | 2025B-P19 |
A frog hops along the number line according to the following rules:
* It starts at $0$.
* If it is at $0$, then it moves to $1$ with probability $\frac 12$ and disappears with probability $\frac 12$.
* For $n=1,2,3$, if it is at $n$, then it moves to $n+1$ with probability $\frac 14$, to $n-1$ with probability $\frac 1... | E | 3 | 2025B-P20 |
Two non-congruent triangles have the same area. Each triangle has sides of length $8$ and $9$, and the third side of each triangle has integer length. What is the sum of the lengths of the third sides?
$\textbf{(A) } 20 \qquad \textbf{(B) } 22 \qquad \textbf{(C) } 24 \qquad \textbf{(D) } 26 \qquad \textbf{(E) } 28$ | C | 4 | 2025B-P21 |
What is the greatest possible area of the triangle in the complex plane with vertices $2z$, $(1+i)z$, and $(1-i)z$, where $z$ is a complex number satisfying $|4z - 2| = 1$?
\textbf{(A)}~\frac 14 \qquad \textbf{(B)}~\frac 12 \qquad \textbf{(C)}~\frac{9}{16} \qquad \textbf{(D)}~\frac 34 \qquad \textbf{(E)}~1 | C | 4 | 2025B-P22 |
Let $S$ be the set of all integers $z > 1$ such that for all pairs of nonnegative integers $(x, y)$ with $x < y < z$, the remainder when $2025x$ is divided by $z$ is less than the remainder when $2025y$ is divided by $z$. What is the sum of the elements of $S$?
\textbf{(A)}~3041 \qquad \textbf{(B)}~3542 \qquad \textbf... | E | 4 | 2025B-P23 |
How many real numbers satisfy the equation $\sin(20\pi x) = \log_{20}(x)$?
$\textbf{(A) } 199 \qquad \textbf{(B) } 200 \qquad \textbf{(C) } 398 \qquad \textbf{(D) } 399 \qquad \textbf{(E) } 400$ | D | 4 | 2025B-P24 |
Three concentric circles have radii $1,2,3$. An equilateral triangle with side length $s$ has one vertex on each circle. What is $s^2?$
$\textbf{(A) } 6 \qquad\textbf{(B) } \frac{25}{4} \qquad\textbf{(C) } \frac{13}{2} \qquad\textbf{(D) } \frac{27}{4} \qquad\textbf{(E) } 7$ | E | 4 | 2025B-P25 |
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