problem stringlengths 40 1.4k | answer stringlengths 1 18 | responses sequencelengths 20 20 | rewards sequencelengths 20 20 | mean_reward float64 0 1 |
|---|---|---|---|---|
"In triangle $ABC$, $\\sin \\angle A = \\frac{4}{5}$ and $\\angle A < 90^\\circ$. Let $D$ be a point(...TRUNCATED) | 34 | ["<think>\nOkay, so I need to solve this geometry problem. Let me try to understand it step by step.(...TRUNCATED) | [
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"Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly(...TRUNCATED) | 113 | ["<think>\nOkay, so I need to find the expected value of the area of triangle AXY, where X and Y are(...TRUNCATED) | [
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"Let $a, b, c$ be distinct numbers such that the equations $x^2 + ax + 1 = 0$ and $x^2 + bx + c = 0$(...TRUNCATED) | -3 | ["<think>\nOkay, so I need to solve this problem where we have three distinct numbers a, b, c. The p(...TRUNCATED) | [
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"There are $7$ boxes arranged in a row and numbered $1$ through $7$. You have a stack of $2015$ card(...TRUNCATED) | 3 | ["<think>\nOkay, so there's this problem about 7 boxes arranged in a row, numbered 1 through 7. Ther(...TRUNCATED) | [
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"Let $S$ be the set of triples $(a,b,c)$ of non-negative integers such that $a+b+c$ is even. The val(...TRUNCATED) | 37 | ["<think>\nOkay, so I need to find the sum of 1/(2^a 3^b 5^c) over all triples (a, b, c) of non-nega(...TRUNCATED) | [
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For which $n$ is $n^4 + 6n^3 + 11n^2 + 3n + 31$ a perfect square? | 10 | ["<think>\nOkay, so I need to figure out for which values of n the expression $ n^4 + 6n^3 + 11n^2 +(...TRUNCATED) | [
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"Four positive integers $a, b, c, d$ satisfy the condition $a < b < c < d$. Determine the smallest p(...TRUNCATED) | 10 | ["<think>\nOkay, so I need to find the smallest possible value of d, where a, b, c, d are four posit(...TRUNCATED) | [
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"$n$ coins are arranged in a circle. If two neighboring coins both show heads or both show tails, yo(...TRUNCATED) | 2 | ["<think>\nOkay, so I need to figure out how many distinct arrangements of n coins arranged in a cir(...TRUNCATED) | [
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"Determine the smallest prime $p$ such that $2018!$ is divisible by $p^3$, but not divisible by $p^4(...TRUNCATED) | 509 | ["<think>\nOkay, so I need to find the smallest prime number p such that 2018! is divisible by p鲁 b(...TRUNCATED) | [
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"Suppose you have a sphere tangent to the $xy$-plane with its center having a positive $z$-coordinat(...TRUNCATED) | 3 | ["<think>\nOkay, so I need to figure out the equation of a sphere that's tangent to the xy-plane wit(...TRUNCATED) | [
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