data_source large_stringclasses 3
values | content large_stringlengths 131 1.55k | ground_truth large_stringlengths 2 46 |
|---|---|---|
aime2025 | Find the sum of all integer bases $b>9$ for which $17_{b}$ is a divisor of $97_{b}$.
Let's think step by step and output the final answer within \boxed{}. | 70 |
aime2025 | On $\triangle ABC$ points $A,D,E$, and $B$ lie that order on side $\overline{AB}$ with $AD=4, DE=16$, and $EB=8$. Points $A,F,G$, and $C$ lie in that order on side $\overline{AC}$ with $AF=13, FG=52$, and $GC=26$. Let $M$ be the reflection of $D$ through $F$, and let $N$ be the reflection of $G$ through $E$. Quadrilate... | 588 |
aime2025 | The 9 members of a baseball team went to an ice cream parlor after their game. Each player had a singlescoop cone of chocolate, vanilla, or strawberry ice cream. At least one player chose each flavor, and the number of players who chose chocolate was greater than the number of players who chose vanilla, which was great... | 16 |
aime2025 | Find the number of ordered pairs $(x,y)$, where both $x$ and $y$ are integers between $-100$ and $100$, inclusive, such that $12x^{2}-xy-6y^{2}=0$.
Let's think step by step and output the final answer within \boxed{}. | 117 |
aime2025 | There are $8!=40320$ eight-digit positive integers that use each of the digits $1,2,3,4,5,6,7,8$ exactly once. Let $N$ be the number of these integers that are divisible by 22. Find the difference between $N$ and 2025.
Let's think step by step and output the final answer within \boxed{}. | 279 |
aime2025 | An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is 3, and the area of the trapezoid is 72. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^{2}+s^{2}$.
Let's think step by step and output the final answer within \boxed... | 504 |
aime2025 | The twelve letters $A,B,C,D,E,F,G,H,I,J,K$, and $L$ are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and those six words are listed alphabetically. For example, a possible result is $AB,CJ,DG,EK,FL,HI$. The pro... | 821 |
aime2025 | Let $k$ be real numbers such that the system $|25+20i-z|=5$ and $|z-4-k|=|z-3i-k|$ has exactly one complex solution $z$. The sum of all possible values of $k$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. Here $i=\sqrt{-1}$.
Let's think step by step and output t... | 77 |
aime2025 | The parabola with equation $y=x^{2}-4$ is rotated $60^{\circ}$ counterclockwise around the origin. The unique point in the fourth quadrant where the original parabola and its image intersect has $y$-coordinate $\frac{a-\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, and $a$ and $c$ are relatively prime. F... | 62 |
aime2025 | The 27 cells of a $3\times9$ grid are filled in using the numbers 1 through 9 so that each row contains 9 different numbers, and each of the three $3\times3$ blocks heavily outlined in the example below contains 9 different numbers, as in the first three rows of a Sudoku puzzle.
| 4 | 2 | 8 | 9 | 6 | 3 | 1 | 7 | 5 | ... | 81 |
aime2025 | A piecewise linear periodic function is defined by $f(x)=\begin{cases}x&\text{if }x\in[-1,1)\\2-x&\text{if }x\in[1,3)\end{cases}$ and $f(x+4)=f(x)$ for all real numbers $x$. The graph of $f(x)$ has the sawtooth pattern. The parabola $x=34y^2$ intersects the graph of $f(x)$ at finitely many points. The sum of the $y$-co... | 259 |
aime2025 | The set of points in 3-dimensional coordinate space that lie in the plane $x+y+z=75$ whose coordinates satisfy the inequalities $x-yz<y-zx<z-xy$ forms three disjoint convex regions. Exactly one of those regions has finite area. The area of this finite region can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ a... | 510 |
aime2025 | Alex divides a disk into four quadrants with two perpendicular diameters intersecting at the center of the disk. He draws 25 more line segments through the disk, drawing each segment by selecting two points at random on the perimeter of the disk in different quadrants and connecting those two points. Find the expected ... | 204 |
aime2025 | Let $ABCDE$ be a convex pentagon with $AB=14, BC=7, CD=24, DE=13, EA=26,$ and $\angle B=\angle E=60^\circ$. For each point $X$ in the plane, define $f(X)=AX+BX+CX+DX+EX$. The least possible value of $f(X)$ can be expressed as $m+n\sqrt{p}$, where $m$ and $n$ are positive integers and $p$ is not divisible by the square ... | 60 |
aime2025 | Let $N$ denote the number of ordered triples of positive integers $(a,b,c)$ such that $a,b,c\leq3^6$ and $a^3+b^3+c^3$ is a multiple of $3^7$. Find the remainder when $N$ is divided by $1000$.
Let's think step by step and output the final answer within \boxed{}. | 735 |
aime2025 | Six points $ A, B, C, D, E, $ and $ F $ lie in a straight line in that order. Suppose that $ G $ is a point not on the line and that $ AC = 26 $, $ BD = 22 $, $ CE = 31 $, $ DF = 33 $, $ AF = 73 $, $ CG = 40 $, and $ DG = 30 $. Find the area of $ \triangle BGE $.
Let's think step by step and output the final answer wi... | 468 |
aime2025 | Find the sum of all positive integers $ n $ such that $ n + 2 $ divides the product $ 3(n + 3)(n^2 + 9) $.
Let's think step by step and output the final answer within \boxed{}. | 49 |
aime2025 | Four unit squares form a $2 \times 2$ grid. Each of the 12 unit line segments forming the sides of the squares is colored either red or blue in such a way that each unit square has 2 red sides and 2 blue sides. Find the number of such colorings.
Let's think step by step and output the final answer within \boxed{}. | 82 |
aime2025 | The product $ \prod_{k=4}^{63} \frac{\log_k(5^{k^2-1})}{\log_{k+1}(5^{k^2-4})} = \frac{\log_4(5^{15})}{\log_5(5^{12})} \cdot \frac{\log_5(5^{24})}{\log_6(5^{21})} \cdot \frac{\log_6(5^{35})}{\log_7(5^{32})} \cdots \frac{\log_{63}(5^{3968})}{\log_{64}(5^{3965})} $ is equal to $ \frac{m}{n} $, where $ m $ and $ n $ are r... | 106 |
aime2025 | Suppose $ \triangle ABC $ has angles $ \angle BAC = 84^\circ $, $ \angle ABC = 60^\circ $, and $ \angle ACB = 36^\circ $. Let $ D, E, $ and $ F $ be the midpoints of sides $ \overline{BC} $, $ \overline{AC} $, and $ \overline{AB} $, respectively. The circumcircle of $ \triangle DEF $ intersects $ \overline{BD} $, $ \ov... | 336^\circ |
aime2025 | Circle $\omega_1$ with radius 6 centered at point $A$ is internally tangent at point $B$ to circle $\omega_2$ with radius 15. Points $C$ and $D$ lie on $\omega_2$ such that $\overline{BC}$ is a diameter of $\omega_2$ and $\overline{BC} \perp \overline{AD}$. The rectangle $EFGH$ is inscribed in $\omega_1$ such that $\ov... | 293 |
aime2025 | Let $ A $ be the set of positive integer divisors of 2025. Let $ B $ be a randomly selected subset of $ A $. The probability that $ B $ is a nonempty set with the property that the least common multiple of its elements is 2025 is $ \frac{m}{n} $, where $ m $ and $ n $ are relatively prime positive integers. Find $ m + ... | 237 |
aime2025 | From an unlimited supply of 1-cent coins, 10-cent coins, and 25-cent coins, Silas wants to find a collection of coins that has a total value of $ N $ cents, where $ N $ is a positive integer. He uses the so-called **greedy algorithm**, successively choosing the coin of greatest value that does not cause the value of hi... | 610 |
aime2025 | There are $ n $ values of $ x $ in the interval $ 0 < x < 2\pi $ where $ f(x) = \sin(7\pi \cdot \sin(5x)) = 0 $. For $ t $ of these $ n $ values of $ x $, the graph of $ y = f(x) $ is tangent to the $ x $-axis. Find $ n + t $.
Let's think step by step and output the final answer within \boxed{}. | 149 |
aime2025 | Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $ N $ be the number of subsets of 16 chairs that could be selected. Find the remainder when $ N $ is divided by 1000.
Let's think step by step and output the final answer within \... | 907 |
aime2025 | Let $ S $ be the set of vertices of a regular 24-gon. Find the number of ways to draw 12 segments of equal lengths so that each vertex in $ S $ is an endpoint of exactly one of the 12 segments.
Let's think step by step and output the final answer within \boxed{}. | 113 |
aime2025 | Let $ A_1A_2 \ldots A_{11} $ be an 11-sided non-convex simple polygon with the following properties:
* The area of $ A_iA_1A_{i+1} $ is 1 for each $ 2 \leq i \leq 10 $,
* $ \cos(\angle A_iA_1A_{i+1}) = \frac{12}{13} $ for each $ 2 \leq i \leq 10 $,
* The perimeter of $ A_1A_2 \ldots A_{11} $ is 20.
If $ A_1A_2 + A_1A_{... | 19 |
aime2025 | Let the sequence of rationals $ x_1, x_2, \ldots $ be defined such that $ x_1 = \frac{25}{11} $ and
$ x_{k+1} = \frac{1}{3} \left( x_k + \frac{1}{x_k} - 1 \right). $
$ x_{2025} $ can be expressed as $ \frac{m}{n} $ for relatively prime positive integers $ m $ and $ n $. Find the remainder when $ m + n $ is divided by 1... | 248 |
aime2025 | Let $ \triangle ABC $ be a right triangle with $ \angle A = 90^\circ $ and $ BC = 38 $. There exist points $ K $ and $ L $ inside the triangle such that $ AK = AL = BK = CL = KL = 14. $ The area of the quadrilateral $ BKLC $ can be expressed as $ n \sqrt{3} $ for some positive integer $ n $. Find $ n $.
Let's think st... | 104 |
aime2025 | There are exactly three positive real numbers $ k $ such that the function
$ f(x) = \frac{(x - 18)(x - 72)(x - 98)(x - k)}{x} $
defined over the positive real numbers achieves its minimum value at exactly two positive real numbers $ x $. Find the sum of these three values of $ k $.
Let's think step by step and output ... | 240 |
aime2026 | Patrick started walking at a constant rate along a straight road from school to the park. One hour after Patrick left, Tanya started running along the same road from school to the park. One hour after Tanya left, Jose started bicycling along the same road from school to the park. Tanya ran at a constant rate of $2$ mil... | 277 |
aime2026 | Find the number of positive integer palindromes written in base $10$ with no zero digits, and whose digits add up to $13$. For example, $42124$ has these properties. Recall that a palindrome is a number whose representation reads the same from left to right as from right to left.
Let's think step by step and output th... | 62 |
aime2026 | A hemisphere with radius $200$ sits on top of a horizontal circular disk with radius $200,$ and the hemisphere and disk have the same center. Let $\mathcal T$ be the region of points P in the disk such that a sphere of radius $42$ can be placed on top of the disk at $P$ and lie completely inside the hemisphere. The are... | 79 |
aime2026 | Find the number of integers less than or equal to 100 that are equal to $a+b+ab$ for some choice of distinct positive integers a and b.
Let's think step by step and output the final answer within \boxed{}. | 70 |
aime2026 | A plane contains points $A$ and $B$ with $AB = 1$. Point $A$ is rotated in the plane counterclockwise through an acute angle $\theta$ around point $B$ to point $A^\prime$. Then $B$ is rotated in the plane clockwise through angle $\theta$ around point $A^\prime$ to point $B^\prime$. Suppose that $AB^\prime = \frac{4}{3}... | 65 |
aime2026 | A real number $x$ satisfies $\sqrt[20]{x^{\log_{2026}x}}=26x$. What is the number of positive divisors of the product of all possible positive values of $x$?
Let's think step by step and output the final answer within \boxed{}. | 441 |
aime2026 |
Find the number of functions $\pi$ mapping the set $A =\{1,2,3,4,5,6\}$ onto $A$ such that for every $a \in A,$
\[
\pi(\pi(\pi(\pi(\pi(\pi(a)))))) = a.
\]
Let's think step by step and output the final answer within \boxed{}. | 396 |
aime2026 | Let $N$ be the number of positive integer divisors of $17017^{17}$ that leave a remainder of $5$ when divided by $12$. Find the remainder when $N$ is divided by $1000$.
Let's think step by step and output the final answer within \boxed{}. | 244 |
aime2026 | Joanne has a blank fair six-sided die and six stickers each displaying a different integer from 1 to 6. Joanne rolls the die and then places the sticker labeled 1 on the top face of the die. She then rolls the die again, places the sticker labeled 2 on the top face, and continues this process to place the rest of the s... | 29 |
aime2026 | Let $\triangle ABC$ have side lengths $AB = 13, BC = 14,$ and $CA = 15.$ Triangle $\triangle A'B'C'$ is obtained by rotating $\triangle ABC$ about its circumcenter so that ${}\overline{AC}$ is perpendicular $\overline{BC},$ with $A'$ and $B$ not on the same side of line $B'C'.$ Find the integer closest to the area of h... | 156 |
aime2026 | The integers from $1$ to $64$ are placed in some order into an $8 \times 8$ grid of cells with one number in each cell. Let $a_{i,j}$ be the number placed in the cell in row $i$ and column $j,$ and let $M$ be the sum of the absolute differences between adjacent cells. That is,
\[
M = \sum^8_{i=1} \sum^7_{j=1} (|a_{i,j+... | 896 |
aime2026 | Triangle $\triangle ABC$ lies in plane $\mathcal P$ with $AB = 6, AC = 4,$ and $\angle BAC = 90^\circ.$ Let $D$ be the reflection across $\overline{BC}$ of the centroid of $\triangle ABC. {}$ Four spheres, all on the same side of $\mathcal P,$ have radii $1, 2, 3,$ and $r$ and are tangent to $\mathcal P$ at points $A, ... | 161 |
aime2026 | For each positive integer $r$ less than $502,$ define
\[
S_r=\sum_{m\ge 0}\dbinom{10000}{502n+r},
\]
where $\binom{10000}{n}$ is defined to be $0$ when $n>10000.$ That is, $S_r$ is the sum of all binomial coefficients of the form $\binom{10000}{k}$ for which $0\le k\le 10000$ and $k-r$ is a multiple of $502.$ Find the ... | 39 |
aime2026 | In an equiangular pentagon, the sum of the squares of the side lengths equals $308,$ and the sum of the squares of the diagonal lengths equals $800.$ The square of the perimeter of the pentagon can be expressed as $m \sqrt n,$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. ... | 681 |
aime2026 | Let $a, b,$ and $n$ be positive integers with both $a$ and $b$ greater than or equal to $2$ and less than or equal to $2n$. Define an $a \times b$ cell loop in a $2n \times 2n$ grid of cells to be the $2a + 2b - 4$ cells that surround an $(a - 2) \times (b - 2)$ (possibly empty) rectangle of cells in the grid. For exam... | 83 |
aime2026 | Find the sum of the $10$th terms of all arithmetic sequences of integers that have first term equal to $4$ and include both $24$ and $34$ as terms.
Let's think step by step and output the final answer within \boxed{}. | 178 |
aime2026 | The figure below shows a grid of $10$ squares in a row. Each square has a diagonal connecting its lower left vertex to its upper right vertex. A bug moves along the line segments from vertex to vertex, never traversing the same segment twice and never moving from right to left along a horizontal or diagonal segment. Le... | 243 |
aime2026 | Let $ABCDE$ be a nonconvex pentagon with internal angles $\angle A = \angle E = 90^\circ$ and $\angle B = \angle D = 45^\circ.$ Suppose that $DE < AB, AE = 20, BC = 14\sqrt2,$ and points $B,C,$ and $D$ lie on the same side of line $AE.$ Suppose further that $AB$ is an integer with $AB < 2026$ and the area of pentagon $... | 503 |
aime2026 | For each positive integer $n$ let $f(n)$ be the value of the base-ten numeral $n$ viewed in base $b$, where $b$ is the least integer greater than the greatest digit in $n$. For example, if $n=72$, then $b=8$, and $72$ as a numeral in base $8$ equals $7\cdot 8+2=58$; therefore $f(72)=58$. Find the number of positive int... | 279 |
aime2026 | An urn contains $n$ marbles. Each marble is either red or blue, and there are at least $7$ marbles of each color. When $7$ marbles are drawn randomly from the urn without replacement, the probability that exactly $4$ of them are red equals the probability that exactly $5$ of them are red. Find the sum of the five least... | 190 |
aime2026 | Find the sum of all real numbers $r$ such that there is at least one point where the circle with radius $r$ centered at $(4, 39)$ is tangent to the parabola with equation $2y = x^2 - 8x + 12.$
Let's think step by step and output the final answer within \boxed{}. | 50 |
aime2026 | A standard fair six-sided die is rolled repeatedly. Each time the die reads 1 or 2, Alice gets a coin; each time it reads 3 or 4, Bob gets a coin; and each time it reads 5 or 6, Carol gets a coin. The probability that Alice and Bob each receive at least two coins before Carol receives any coins can be written as $\tfra... | 754 |
aime2026 | Isosceles triangle $\triangle ABC$ has $AB = BC.$ Let $I$ be the incenter of $\triangle ABC.$ The perimeters of $\triangle ABC$ and $\triangle AIC$ are in the ratio $125:6,$ and all the sides of both triangles have integer lengths. Find the minimum possible value of $AB.$
Let's think step by step and output the final ... | 245 |
aime2026 | Let $S$ denote the value of the infinite sum
\[
\frac{1}{9} + \frac{1}{99} + \frac{1}{999} + \frac{1}{9999} + \cdots
\]
Find the remainder when the greatest integer less than or equal to $10^{100} S$ is divided by $1000.$
Let's think step by step and output the final answer within \boxed{}. | 669 |
aime2026 | Let $\triangle ABC$ be a triangle with $D$ on $\overline{BC}$ such that $\overline{AD}$ bisects $\angle BAC.$ Let $\omega$ be the circle that passes through $A$ and is tangent to segment $\overline{BC}$ at $D.$ Let $E \neq A$ and $F \neq A$ be the intersections of $\omega$ with segments $\overline{AB}$ and $\overline{A... | 850 |
aime2026 | Find the greatest integer $n$ such that the cubic polynomial
\[
x^{3} - \displaystyle\frac{n}{6}x^{2} + (n - 11)x - 400
\]
has roots $\alpha^{2}$, $\beta^{2}$, and $\gamma^{2}$, where $\alpha$, $\beta$, and $\gamma$ are complex numbers, and there are exactly seven different possible values for $\alpha + \beta + \gamma$... | 132 |
aime2026 | Consider a tetrahedron with two isosceles triangle faces with side lengths $5\sqrt{10}, 5\sqrt{10},$ and $10$ and two isosceles triangle faces with side lengths $5\sqrt{10}, 5\sqrt{10},$ and $18.$ The four vertices of the tetrahedron lie on a sphere with center $S,$ and the four faces of the tetrahedron are tangent to ... | 223 |
aime2026 | Call finite sets of integers $S$ and $T$ cousins if
- $S$ and $T$ have the same number of elements,
- $S$ and $T$ are disjoint, and
- the elements of $S$ can be paired with the elements of $T$ so that the elements in each pair differ by exactly $1$.
For example, $\{1,2,5\}$ and $\{0,3,4\}$ are cousins. Suppose that the... | 107 |
aime2026 | For integers $a$ and $b,$ let $a \circ b = a - b$ if $a$ is odd and $b$ is even, and $a+b$ otherwise. Find the number of sequences $a_1, a_2, a_3, \ldots, a_n$ of positive integers such that
\[
a_1 + a_2 + a_3 + \cdots + a_n = 12 \quad \text{and} \quad a_1 \circ a_2 \circ a_3 \circ \cdots \circ a_n = 0
\]
where the ope... | 157 |
aime2026 | Find the number of ordered 7-tuples $(a_1, a_2, a_3, \ldots, a_7)$ having the following properties:
- $a_k \in \{1,2,3\}$ for all $k.$
- $a_1+a_2+a_3+a_4+a_5+a_6+a_7$ is a multiple of $3.$
- $a_1a_2 a_4 + a_2a_3a_5 + a_3a_4 a_6 + a_4 a_5 a_7 + a_5 a_6 a_1 + a_6 a_7 a_2 + a_7 a_1 a_3$ is a multiple of $3.$
Let's think ... | 393 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Compute the sum of the positive divisors (including $1$) of $9!$ that have units digit $1$. | 103 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Mark writes the expression $\sqrt{\underline{a b c d}}$ on the board, where $\underline{a b c d}$ is a four-digit number and $a \neq 0$. Derek, a toddler, decides to move the $a$, changing Mark's expression to $a \sqrt{\underline{b c d}}$. Surprisingly, these two expressions are ... | 3375 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Given that $x, y$, and $z$ are positive real numbers such that
$$
x^{\log _{2}(y z)}=2^{8} \cdot 3^{4}, \quad y^{\log _{2}(z x)}=2^{9} \cdot 3^{6}, \quad \text { and } \quad z^{\log _{2}(x y)}=2^{5} \cdot 3^{10}
$$
compute the smallest possible value of $x y z$. | \frac{1}{576} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $\lfloor z\rfloor$ denote the greatest integer less than or equal to $z$. Compute
$$
\sum_{j=-1000}^{1000}\left\lfloor\frac{2025}{j+0.5}\right\rfloor
$$ | -984 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $\mathcal{S}$ be the set of all nonconstant monic polynomials $P$ with integer coefficients satisfying $P(\sqrt{3}+\sqrt{2})=$ $P(\sqrt{3}-\sqrt{2})$. If $Q$ is an element of $\mathcal{S}$ with minimal degree, compute the only possible value of $Q(10)-Q(0)$. | 890 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $r$ be the remainder when $2017^{2025!}-1$ is divided by 2025!. Compute $\frac{r}{2025!}$. (Note that $2017$ is prime.) | \frac{1311}{2017} |
hmmt_feb2025 | Put your final answer within \boxed{}.
There exists a unique triple $(a, b, c)$ of positive real numbers that satisfies the equations
$$
2\left(a^{2}+1\right)=3\left(b^{2}+1\right)=4\left(c^{2}+1\right) \quad \text { and } \quad a b+b c+c a=1
$$
Compute $a+b+c$. | \frac{9 \sqrt{23}}{23} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Define $\operatorname{sgn}(x)$ to be $1$ when $x$ is positive, $-1$ when $x$ is negative, and $0$ when $x$ is $0$. Compute
$$
\sum_{n=1}^{\infty} \frac{\operatorname{sgn}\left(\sin \left(2^{n}\right)\right)}{2^{n}}
$$
(The arguments to sin are in radians.) | 1-\frac{2}{\pi} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $f$ be the unique polynomial of degree at most $2026$ such that for all $n \in\{1,2,3, \ldots, 2027\}$,
$$
f(n)= \begin{cases}1 & \text { if } n \text { is a perfect square } \\ 0 & \text { otherwise }\end{cases}
$$
Suppose that $\frac{a}{b}$ is the coefficient of $x^{2025}$... | 1037 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $a, b$, and $c$ be pairwise distinct complex numbers such that
$$
a^{2}=b+6, \quad b^{2}=c+6, \quad \text { and } \quad c^{2}=a+6
$$
Compute the two possible values of $a+b+c$. In your answer, list the two values in a comma-separated list of two valid \LaTeX expressions. | \frac{-1+\sqrt{17}}{2}, \frac{-1-\sqrt{17}}{2} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Compute the number of ways to arrange the numbers $1,2,3,4,5,6$, and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.) | 56 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Kevin the frog in on the bottom-left lily pad of a $3 \times 3$ grid of lily pads, and his home is at the topright lily pad. He can only jump between two lily pads which are horizontally or vertically adjacent. Compute the number of ways to remove $4$ of the lily pads so that the... | 29 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Ben has $16$ balls labeled $1,2,3, \ldots, 16$, as well as $4$ indistinguishable boxes. Two balls are \emph{neighbors} if their labels differ by $1$. Compute the number of ways for him to put $4$ balls in each box such that each ball is in the same box as at least one of its neig... | 105 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Sophie is at $(0,0)$ on a coordinate grid and would like to get to $(3,3)$. If Sophie is at $(x, y)$, in a single step she can move to one of $(x+1, y),(x, y+1),(x-1, y+1)$, or $(x+1, y-1)$. She cannot revisit any points along her path, and neither her $x$-coordinate nor her $y$-... | 2304 |
hmmt_feb2025 | Put your final answer within \boxed{}.
In an $11 \times 11$ grid of cells, each pair of edge-adjacent cells is connected by a door. Karthik wants to walk a path in this grid. He can start in any cell, but he must end in the same cell he started in, and he cannot go through any door more than once (not even in opposite... | 200 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Compute the number of ways to pick two rectangles in a $5 \times 5$ grid of squares such that the edges of the rectangles lie on the lines of the grid and the rectangles do not overlap at their interiors, edges, or vertices. The order in which the rectangles are chosen does not m... | 6300 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Compute the number of ways to arrange $3$ copies of each of the $26$ lowercase letters of the English alphabet such that for any two distinct letters $x_{1}$ and $x_{2}$, the number of $x_{2}$ 's between the first and second occurrences of $x_{1}$ equals the number of $x_{2}$ 's ... | 2^{25} \cdot 26! |
hmmt_feb2025 | Put your final answer within \boxed{}.
Albert writes $2025$ numbers $a_{1}, \ldots, a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1]$. Then, each second, he \emph{simultaneously} replaces $a_{i}$ with $\max \left(a_{i-1}, a_... | \frac{2025}{101} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Two points are selected independently and uniformly at random inside a regular hexagon. Compute the probability that a line passing through both of the points intersects a pair of opposite edges of the hexagon. | \frac{4}{9} |
hmmt_feb2025 | Put your final answer within \boxed{}.
The circumference of a circle is divided into $45$ arcs, each of length $1$. Initially, there are $15$ snakes, each of length $1$, occupying every third arc. Every second, each snake independently moves either one arc left or one arc right, each with probability $\frac{1}{2}$. If... | \frac{448}{3} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Equilateral triangles $\triangle A B C$ and $\triangle D E F$ are drawn such that points $B, E, F$, and $C$ lie on a line in this order, and point $D$ lies inside triangle $\triangle A B C$. If $B E=14, E F=15$, and $F C=16$, compute $A D$. | 26 |
hmmt_feb2025 | Put your final answer within \boxed{}.
In a two-dimensional cave with a parallel floor and ceiling, two stalactites of lengths $16$ and $36$ hang perpendicularly from the ceiling, while two stalagmites of heights $25$ and $49$ grow perpendicularly from the ground. If the tips of these four structures form the vertices... | 63 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Point $P$ lies inside square $A B C D$ such that the areas of $\triangle P A B, \triangle P B C, \triangle P C D$, and $\triangle P D A$ are 1, $2,3$, and $4$, in some order. Compute $P A \cdot P B \cdot P C \cdot P D$. | 8\sqrt{10} |
hmmt_feb2025 | Put your final answer within \boxed{}.
A semicircle is inscribed in another semicircle if the smaller semicircle's diameter is a chord of the larger semicircle, and the smaller semicircle's arc is tangent to the diameter of the larger semicircle.
Semicircle $S_{1}$ is inscribed in a semicircle $S_{2}$, which is inscri... | 20 |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $\triangle A B C$ be an equilateral triangle with side length $6$. Let $P$ be a point inside triangle $\triangle A B C$ such that $\angle B P C=120^{\circ}$. The circle with diameter $\overline{A P}$ meets the circumcircle of $\triangle A B C$ again at $X \neq A$. Given that ... | \sqrt{23}-2 \sqrt{3} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Trapezoid $A B C D$, with $A B \| C D$, has side lengths $A B=11, B C=8, C D=19$, and $D A=4$. Compute the area of the convex quadrilateral whose vertices are the circumcenters of $\triangle A B C, \triangle B C D$, $\triangle C D A$, and $\triangle D A B$. | 9\sqrt{15} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Point $P$ is inside triangle $\triangle A B C$ such that $\angle A B P=\angle A C P$. Given that $A B=6, A C=8, B C=7$, and $\frac{B P}{P C}=\frac{1}{2}$, compute $\frac{[B P C]}{[A B C]}$.
(Here, $[X Y Z]$ denotes the area of $\triangle X Y Z$ ). | \frac{7}{18} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $A B C D$ be an isosceles trapezoid such that $C D>A B=4$. Let $E$ be a point on line $C D$ such that $D E=2$ and $D$ lies between $E$ and $C$. Let $M$ be the midpoint of $\overline{A E}$. Given that points $A, B, C, D$, and $M$ lie on a circle with radius $5$, compute $M D$. | \sqrt{6} |
hmmt_feb2025 | Put your final answer within \boxed{}.
Let $A B C D$ be a rectangle with $B C=24$. Point $X$ lies inside the rectangle such that $\angle A X B=90^{\circ}$. Given that triangles $\triangle A X D$ and $\triangle B X C$ are both acute and have circumradii $13$ and $15$, respectively, compute $A B$. | 14+4\sqrt{37} |
hmmt_feb2025 | Put your final answer within \boxed{}.
A plane $\mathcal{P}$ intersects a rectangular prism at a hexagon which has side lengths $45,66,63,55,54$, and 77, in that order. Compute the distance from the center of the rectangular prism to $\mathcal{P}$. | \sqrt{\frac{95}{24}} |
Math90 entropy prompts
The 90 competition-math problems we use to measure the entropy of a model's own rollouts at each training checkpoint (SFT / on-policy distillation runs): AIME 2025 (30), AIME 2026 (30) and HMMT February 2025 (30), with the prompt text exactly as the model sees it in our math eval (the answer-format instruction included) and the reference answer.
| column | |
|---|---|
data_source |
aime2025, aime2026 or hmmt_feb2025 |
content |
user-turn text (problem + answer-format instruction) |
ground_truth |
reference answer (string) |
Measurement protocol
Code: examples/llm_fusion/analysis/rollout_entropy.py (vLLM).
- chat template of the model, thinking on (template default), one user turn =
content - 4 samples per problem, temperature 1.0, top_p 1.0 (the unmodified model distribution), up to 8,192 new tokens, seed 0
- per generated token, entropy from the top-20 logprobs:
H20 = -sum_top20 p log p(lower bound) andHub = H20 - r log(r / (V-20)),r = 1 - sum_top20 p(upper bound, tail spread uniformly); at T=1 the two bracket the full-vocabulary entropy tightly - reported: token-mean entropy over all 360 rollouts (plus per set, first 1k vs later tokens, p50/p90, mean length, truncation rate)
The 8k budget truncates most thinking-mode rollouts (73% for Klear-Reasoner-8B-SFT), so this measures the entropy of the first 8k tokens of reasoning, not of whole solutions. Example (Klear-Reasoner-8B-SFT): token entropy 0.355 (upper bound 0.358), 0.290 on the first 1k tokens, 0.366 after.
Sources and license
Problems and answers are public competition problems: AIME 2025/2026 (MAA) and HMMT February 2025 (as curated by MathArena, CC BY-NC-SA 4.0). This file is a re-formatting of those problems for evaluation and is released under CC BY-NC-SA 4.0; the problems remain the property of their original authors. These are evaluation problems: do not train on them.
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