problem_id string | dataset string | statement string | final_answer string | domain string | source string | country string | year int64 | date timestamp[s] | task string | answer_type string | subtopic null | url string | solutions list |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
HMMT2024-FEB-ANT-01 | hmmt_feb_2024 | Suppose $r, s$, and $t$ are nonzero reals such that the polynomial $x^{2}+r x+s$ has $s$ and $t$ as roots, and the polynomial $x^{2}+t x+r$ has $5$ as a root. Compute $s$. | 29 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-01:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-02 | hmmt_feb_2024 | Suppose $a$ and $b$ are positive integers. Isabella and Vidur both fill up an $a \times b$ table. Isabella fills it up with numbers $1,2, \ldots, a b$, putting the numbers $1,2, \ldots, b$ in the first row, $b+1, b+2, \ldots, 2 b$ in the second row, and so on. Vidur fills it up like a multiplication table, putting $i j... | 21 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-02:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-03 | hmmt_feb_2024 | Compute the sum of all two-digit positive integers $x$ such that for all three-digit (base 10) positive integers $\underline{a} \underline{b} \underline{c}$, if $\underline{a} \underline{b} \underline{c}$ is a multiple of $x$, then the three-digit (base 10) number $\underline{b} \underline{c} \underline{a}$ is also a m... | 64 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-03:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-04 | hmmt_feb_2024 | Let $f(x)$ be a quotient of two quadratic polynomials. Given that $f(n)=n^{3}$ for all $n \in\{1,2,3,4,5\}$, compute $f(0)$. | \dfrac{24}{17} | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-04:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-05 | hmmt_feb_2024 | Compute the unique ordered pair $(x, y)$ of real numbers satisfying the system of equations
$$
\frac{x}{\sqrt{x^{2}+y^{2}}}-\frac{1}{x}=7 \quad \text { and } \quad \frac{y}{\sqrt{x^{2}+y^{2}}}+\frac{1}{y}=4
$$ | -\dfrac{13}{96}, \dfrac{13}{40} | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | set | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-05:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-06 | hmmt_feb_2024 | Compute the sum of all positive integers $n$ such that $50 \leq n \leq 100$ and $2 n+3$ does not divide $2^{n!}-1$. | 222 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-06:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-07 | hmmt_feb_2024 | Let $P(n)=\left(n-1^{3}\right)\left(n-2^{3}\right) \ldots\left(n-40^{3}\right)$ for positive integers $n$. Suppose that $d$ is the largest positive integer that divides $P(n)$ for every integer $n>2023$. If $d$ is a product of $m$ (not necessarily distinct) prime numbers, compute $m$. | 48 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-07:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-08 | hmmt_feb_2024 | Let $\zeta=\cos \frac{2 \pi}{13}+i \sin \frac{2 \pi}{13}$. Suppose $a>b>c>d$ are positive integers satisfying
$$
\left|\zeta^{a}+\zeta^{b}+\zeta^{c}+\zeta^{d}\right|=\sqrt{3}
$$
Compute the smallest possible value of $1000 a+100 b+10 c+d$. | 7521 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-08:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-09 | hmmt_feb_2024 | Suppose $a, b$, and $c$ are complex numbers satisfying
$$\begin{aligned}
a^{2} \& =b-c <br>
b^{2} \& =c-a, and <br>
c^{2} \& =a-b
\end{aligned}$$
Compute all possible values of $a+b+c$. | 0,i\sqrt{6},-i\sqrt{6} | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | set | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-09:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-ANT-10 | hmmt_feb_2024 | A polynomial $f \in \mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_{p}, h_{p} \in \mathbb{Z}[x]$ with $\operatorname{deg} g_{p}, \operatorname{deg} h_{p}<\operatorname{deg} f$ and all coefficients of $f-g_{p} h_{p}$ are divisible by $p$. Compute the sum of all positive i... | 693 | algebra | HMMT 2024 ANT | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-ANT-10:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-COMB-01 | hmmt_feb_2024 | Compute the number of ways to divide a $20 \times 24$ rectangle into $4 \times 5$ rectangles. (Rotations and reflections are considered distinct.) | 6 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-01:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-02 | hmmt_feb_2024 | A lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell.
A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twic... | 43 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-02:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-03 | hmmt_feb_2024 | Compute the number of ways there are to assemble $2$ red unit cubes and $25$ white unit cubes into a $3 \times 3 \times 3$ cube such that red is visible on exactly $4$ faces of the larger cube. (Rotations and reflections are considered distinct.) | 114 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-03:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-04 | hmmt_feb_2024 | Sally the snail sits on the $3 \times 24$ lattice of points $(i, j)$ for all $1 \leq i \leq 3$ and $1 \leq j \leq 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2,1)$, compute the number of possible... | 4096 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-04:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-05 | hmmt_feb_2024 | The country of HMMTLand has $8$ cities. Its government decides to construct several two-way roads between pairs of distinct cities. After they finish construction, it turns out that each city can reach exactly $3$ other cities via a single road, and from any pair of distinct cities, either exactly $0$ or $2$ other citi... | 875 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-05:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-06 | hmmt_feb_2024 | In each cell of a $4 \times 4$ grid, one of the two diagonals is drawn uniformly at random. Compute the probability that the resulting $32$ triangular regions can be colored red and blue so that any two regions sharing an edge have different colors. | \dfrac{1}{512} | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-06:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-07 | hmmt_feb_2024 | There is a grid of height $2$ stretching infinitely in one direction. Between any two edge-adjacent cells of the grid, there is a door that is locked with probability $\frac{1}{2}$ independent of all other doors. Philip starts in a corner of the grid (in the starred cell). Compute the expected number of cells that Phil... | \dfrac{32}{7} | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-07:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-08 | hmmt_feb_2024 | Rishabh has $2024$ pairs of socks in a drawer. He draws socks from the drawer uniformly at random, without replacement, until he has drawn a pair of identical socks. Compute the expected number of unpaired socks he has drawn when he stops. | \dfrac{4^{2024}}{\binom{4048}{2024}}-2 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-08:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-09 | hmmt_feb_2024 | Compute the number of triples $(f, g, h)$ of permutations on $\{1,2,3,4,5\}$ such that
$$\begin{aligned}
\& f(g(h(x)))=h(g(f(x)))=g(x), <br>
\& g(h(f(x)))=f(h(g(x)))=h(x), and <br>
\& h(f(g(x)))=g(f(h(x)))=f(x)
\end{aligned}$$
for all $x \in\{1,2,3,4,5\}$. | 146 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-09:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-COMB-10 | hmmt_feb_2024 | A peacock is a ten-digit positive integer that uses each digit exactly once. Compute the number of peacocks that are exactly twice another peacock. | 184320 | combinatorics | HMMT 2024 COMB | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-COMB-10:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step ... |
HMMT2024-FEB-GEO-02 | hmmt_feb_2024 | Let $A B C$ be a triangle with $\angle B A C=90^{\circ}$. Let $D, E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $B C$, respectively. If $D E=3$ and $E F=5$, compute the length of $B C$. | 20 | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-02:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-03 | hmmt_feb_2024 | Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1: 2: 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of ... | 42 | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-03:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-04 | hmmt_feb_2024 | Let $A B C D$ be a square, and let $\ell$ be a line passing through the midpoint of segment $\overline{A B}$ that intersects segment $\overline{B C}$. Given that the distances from $A$ and $C$ to $\ell$ are $4$ and $7$, respectively, compute the area of $A B C D$. | 185 | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-04:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-05 | hmmt_feb_2024 | Let $A B C D$ be a convex trapezoid such that $\angle D A B=\angle A B C=90^{\circ}, D A=2, A B=3$, and $B C=8$. Let $\omega$ be a circle passing through $A$ and tangent to segment $\overline{C D}$ at point $T$. Suppose that the center of $\omega$ lies on line $B C$. Compute $C T$. | 4\sqrt{5}-\sqrt{7} | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-05:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-06 | hmmt_feb_2024 | In triangle $A B C$, a circle $\omega$ with center $O$ passes through $B$ and $C$ and intersects segments $\overline{A B}$ and $\overline{A C}$ again at $B^{\prime}$ and $C^{\prime}$, respectively. Suppose that the circles with diameters $B B^{\prime}$ and $C C^{\prime}$ are externally tangent to each other at $T$. If ... | \dfrac{65}{3} | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-06:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-07 | hmmt_feb_2024 | Let $A B C$ be an acute triangle. Let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $\overline{B C}$, $\overline{C A}$, and $\overline{A B}$, respectively, and let $Q$ be the foot of altitude from $A$ to line $E F$. Given that $A Q=20$, $B C=15$, and $A D=24$, compute the perimeter of triangle ... | 8\sqrt{11} | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-07:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-08 | hmmt_feb_2024 | Let $A B T C D$ be a convex pentagon with area $22$ such that $A B=C D$ and the circumcircles of triangles $T A B$ and $T C D$ are internally tangent. Given that $\angle A T D=90^{\circ}, \angle B T C=120^{\circ}, B T=4$, and $C T=5$, compute the area of triangle $T A D$. | 64(2-\sqrt{3}) | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-08:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-09 | hmmt_feb_2024 | Let $A B C$ be a triangle. Let $X$ be the point on side $\overline{A B}$ such that $\angle B X C=60^{\circ}$. Let $P$ be the point on segment $\overline{C X}$ such that $B P \perp A C$. Given that $A B=6, A C=7$, and $B P=4$, compute $C P$. | \sqrt{38}-3 | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-09:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024-FEB-GEO-10 | hmmt_feb_2024 | Suppose point $P$ is inside quadrilateral $A B C D$ such that
$$\begin{aligned}
\& \angle P A B=\angle P D A <br>
\& \angle P A D=\angle P D C <br>
\& \angle P B A=\angle P C B, and <br>
\& \angle P B C=\angle P C D
\end{aligned}$$
If $P A=4, P B=5$, and $P C=10$, compute the perimeter of $A B C D$. | \dfrac{9\sqrt{410}}{5} | geometry | HMMT 2024 GEO | United States | 2,024 | 2024-02-17T00:00:00 | compute_and_prove | number | null | https://huggingface.co/datasets/MathArena/hmmt_feb_2024/blob/b3cf7fb48814f76142d0847003a5aac0876f3bf4 | [
{
"solution_id": "HMMT2024-FEB-GEO-10:reference",
"role": "reference",
"source_type": "ai_generated",
"source_url": null,
"generated_by": "GPT-6 Astra",
"note": "Generated and model-reviewed by GPT-6 Astra; final answer agrees with the pinned MathArena key. Solution, sketch, and three-step o... |
HMMT2024: solutions and oracle sketches
29 problems, with one GPT-6 Astra reference solution, an 18β24-word strategy sketch, and three 17β20-word outline steps per problem.
Source and selection
Statements and exact final-answer strings are copied from MathArena/hmmt_feb_2024, revision b3cf7fb48814f76142d0847003a5aac0876f3bf4. Original numbering, diagram source, and whitespace are retained.
Round configurations are subsets of the default configuration:
ant: 10 problems.comb: 10 problems.geo: 9 problems.
Excluded problems:
HMMT2024-FEB-GEO-01: User excluded this problem because the MathArena statement omits its diagram.
Fields
Each JSONL record contains problem_id, dataset, statement, final_answer, domain, source, country, year, date, task, answer_type, subtopic, url, and solutions.
final_answer is a string retaining the exact source key, including LaTeX or comma-separated values. Each nested solution contains solution_id, role, source_type, source_url, generated_by, note, solution, sketch, and steps.
Construction and validation
GPT-6 Astra with high reasoning generated solutions from the statements, followed by separate solution reviews, sketch generation/review, and outline generation/review. Final-answer agreement uses the pinned MathArena parser, including its numerical tolerance and unordered-list behavior. Answer agreement alone does not establish proof validity.
Sketch prompts follow the same IMO-aligned examples and protocol used for our AIME datasets. Investigation-guided annotation repairs retain earlier attempts and use unchanged review criteria. Publication was explicitly approved by the curator. No independent expert-panel verification is claimed.
License and attribution
CC BY-NC-SA 4.0. Attribution: MathArena/hmmt_feb_2024; original contest organizers. This release adds generated solutions, sketches, outlines, metadata, and documented exclusions. See provenance.json for source/parser pins and answer-check records.
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