problem_statement stringlengths 20 1.41k | source stringclasses 1
value | expected_answer stringlengths 0 480 | original_expected_answer stringlengths 0 3.25k | metadata dict | solutions listlengths 1 89 |
|---|---|---|---|---|---|
$1-\frac{1}{4}\sin ^2 2\alpha -\sin ^2 \beta -\cos ^4 \alpha $ | aops | $\sin(\alpha +\beta )\sin(\alpha -\beta )$ | $\sin(\alpha +\beta )\sin(\alpha -\beta )$ | {
"reason_high_no_tool": {
"accuracy": 0.75,
"count": 8,
"pass": 6
},
"reason_high_with_tool": {
"accuracy": 0.25,
"count": 8,
"pass": 2
},
"reason_low_no_tool": {
"accuracy": 0,
"count": 8,
"pass": 0
},
"reason_low_with_tool": {
"accuracy": 0,
"count": 8,
"... | [
{
"split_origin": "high_part00",
"changed_answer_to_majority": false,
"messages": [
{
"content": "Solve the following math problem. Make sure to put the answer (and only answer) inside \\boxed{}.\n\n$1-\\frac{1}{4}\\sin ^2 2\\alpha -\\sin ^2 \\beta -\\cos ^4 \\alpha $",
"name": "",... |
"$100$ pictures of BdMO math campers were painted using exactly $k$ colors in each picture. There is(...TRUNCATED) | aops | 20 | 20 | {"reason_high_no_tool":{"accuracy":0.0,"count":0,"pass":0},"reason_high_with_tool":{"accuracy":0.0,"(...TRUNCATED) | [{"split_origin":"low","changed_answer_to_majority":false,"messages":[{"content":"Solve the followin(...TRUNCATED) |
"$2023$ wise men are located in a circle. Each wise man either believes the earth is the center of t(...TRUNCATED) | aops | \text{No finite time – unanimity is not guaranteed.} | {"reason_high_no_tool":{"accuracy":0.875,"count":8,"pass":7},"reason_high_with_tool":{"accuracy":1.0(...TRUNCATED) | [{"split_origin":"high_part01","changed_answer_to_majority":true,"messages":[{"content":"Solve the f(...TRUNCATED) | |
$2\sin 17x+\sqrt 3\cos 5x+\sin 5x=0$. | aops | $-\frac{\pi }{66}+\frac{\pi }{11}k;\ \frac{\pi }{9}+\frac{\pi }{6}n$ | $-\frac{\pi }{66}+\frac{\pi }{11}k;\ \frac{\pi }{9}+\frac{\pi }{6}n$ | {"reason_high_no_tool":{"accuracy":0.875,"count":8,"pass":7},"reason_high_with_tool":{"accuracy":1.0(...TRUNCATED) | [{"split_origin":"high_part00","changed_answer_to_majority":false,"messages":[{"content":"Solve the (...TRUNCATED) |
$2\sin x-3\cos x=3$. | aops | $\pi +2\pi k;\ 2\arctan \frac{3}{2}+2\pi n$ | $\pi +2\pi k;\ 2\arctan \frac{3}{2}+2\pi n$ | {"reason_high_no_tool":{"accuracy":0.75,"count":8,"pass":6},"reason_high_with_tool":{"accuracy":0.75(...TRUNCATED) | [{"split_origin":"high_part00","changed_answer_to_majority":false,"messages":[{"content":"Solve the (...TRUNCATED) |
$32\cos ^6 x-\cos 6x=1$. | aops | $\frac{\pi }{2}+\pi k;\ \pm \frac{1}{2}\arccos \left(-\frac{1}{4}\right)+\pi n$ | $\frac{\pi }{2}+\pi k;\ \pm \frac{1}{2}\arccos \left(-\frac{1}{4}\right)+\pi n$ | {"reason_high_no_tool":{"accuracy":0.0,"count":8,"pass":0},"reason_high_with_tool":{"accuracy":0.25,(...TRUNCATED) | [{"split_origin":"high_part00","changed_answer_to_majority":false,"messages":[{"content":"Solve the (...TRUNCATED) |
$4^{\sin x}+m2^{\sin x}+m^2-1=0$, where $-1<m<1$. | aops | "$x=(-1)^k \\arcsin \\log _2\\left[\\left(-m+\\sqrt {4-3m^3}\\right)/2\\right]+k\\pi $. Find the roo(...TRUNCATED) | "$x=(-1)^k \\arcsin \\log _2\\left[\\left(-m+\\sqrt {4-3m^3}\\right)/2\\right]+k\\pi $. Find the roo(...TRUNCATED) | {"reason_high_no_tool":{"accuracy":0.5,"count":8,"pass":4},"reason_high_with_tool":{"accuracy":0.375(...TRUNCATED) | [{"split_origin":"high_part00","changed_answer_to_majority":false,"messages":[{"content":"Solve the (...TRUNCATED) |
"$6$ straight lines are given in the plane. Let's look at all regular triangles whose sides lie on s(...TRUNCATED) | aops | 8 | 8 | {"reason_high_no_tool":{"accuracy":1.0,"count":8,"pass":8},"reason_high_with_tool":{"accuracy":0.875(...TRUNCATED) | [{"split_origin":"high_part00","changed_answer_to_majority":false,"messages":[{"content":"Solve the (...TRUNCATED) |
"$A$ writes, at his choice, $8$ ones and $8$ twos on a $4 \\times 4$ board. Then $B$ covers the boar(...TRUNCATED) | aops | 11 | 8 | {"reason_high_no_tool":{"accuracy":0.5,"count":8,"pass":4},"reason_high_with_tool":{"accuracy":1.0,"(...TRUNCATED) | [{"split_origin":"high_part00","changed_answer_to_majority":true,"messages":[{"content":"Solve the f(...TRUNCATED) |
"$ABC$ and $PQR$ are both equilateral triangles with an area of 1. The center $M$ of $PQR$ lies on t(...TRUNCATED) | aops | \dfrac19 | {"reason_high_no_tool":{"accuracy":0.875,"count":8,"pass":7},"reason_high_with_tool":{"accuracy":1.0(...TRUNCATED) | [{"split_origin":"high_part01","changed_answer_to_majority":true,"messages":[{"content":"Solve the f(...TRUNCATED) |
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