data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $A$ and $B$ be digits from the set $\\{0,1,2, \\ldots, 9\\}$. Let $r$ be the two-digit integer $\\underline{A} \\underline{B}$ and let $s$ be the two-digit integer $\\underline{B} \\underline{A}$, so that $r$ and $s$ are members of the set $\\{00,01, \\ldots, 99\\}$. Comput... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"index": 2851,
"split": "test",
"question": "Let $A$ and $B$ be digits from the set $\\{0,1,2, \\ldots, 9\\}$. Let $r$ be the two-digit integer $\\underline{A} \\underline{B}$ and let $s$ be the two-digit integer $\\underline{B} \\underline{A}$, so that $r$ and $s$ are members of the set $\\{00,01, \\ldots, 99\... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "For $k \\geq 3$, we define an ordered $k$-tuple of real numbers $\\left(x_{1}, x_{2}, \\ldots, x_{k}\\right)$ to be special if, for every $i$ such that $1 \\leq i \\leq k$, the product $x_{1} \\cdot x_{2} \\cdot \\ldots \\cdot x_{k}=x_{i}^{2}$. Compute the smallest value of $k$... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2852,
"split": "test",
"question": "For $k \\geq 3$, we define an ordered $k$-tuple of real numbers $\\left(x_{1}, x_{2}, \\ldots, x_{k}\\right)$ to be special if, for every $i$ such that $1 \\leq i \\leq k$, the product $x_{1} \\cdot x_{2} \\cdot \\ldots \\cdot x_{k}=x_{i}^{2}$. Compute the smallest v... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\\frac{1}{r}+\\frac{1}{h}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2853,
"split": "test",
"question": "A cylinder with radius $r$ and height $h$ has volume 1 and total surface area 12. Compute $\\frac{1}{r}+\\frac{1}{h}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "If $6 \\tan ^{-1} x+4 \\tan ^{-1}(3 x)=\\pi$, compute $x^{2}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{15-8 \\sqrt{3}}{33}$"
} | {
"index": 2854,
"split": "test",
"question": "If $6 \\tan ^{-1} x+4 \\tan ^{-1}(3 x)=\\pi$, compute $x^{2}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "A rectangular box has dimensions $8 \\times 10 \\times 12$. Compute the fraction of the box's volume that is not within 1 unit of any of the box's faces.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{2}$"
} | {
"index": 2855,
"split": "test",
"question": "A rectangular box has dimensions $8 \\times 10 \\times 12$. Compute the fraction of the box's volume that is not within 1 unit of any of the box's faces."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=\\frac{1}{2}$. Compute the largest real solution $x$ to $(\\log x)^{2}-\\log \\sqrt{x}=T$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2856,
"split": "test",
"question": "Let $T=\\frac{1}{2}$. Compute the largest real solution $x$ to $(\\log x)^{2}-\\log \\sqrt{x}=T$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=10$. Kay has $T+1$ different colors of fingernail polish. Compute the number of ways that Kay can paint the five fingernails on her left hand by using at least three colors and such that no two consecutive fingernails have the same color.\n\nLet's think step by step and ... | math | {
"style": "rule",
"ground_truth": "109890"
} | {
"index": 2857,
"split": "test",
"question": "Let $T=10$. Kay has $T+1$ different colors of fingernail polish. Compute the number of ways that Kay can paint the five fingernails on her left hand by using at least three colors and such that no two consecutive fingernails have the same color."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the number of ordered pairs $(x, y)$ of positive integers satisfying $x^{2}-8 x+y^{2}+4 y=5$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 2858,
"split": "test",
"question": "Compute the number of ordered pairs $(x, y)$ of positive integers satisfying $x^{2}-8 x+y^{2}+4 y=5$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=4$ and let $k=21+2 T$. Compute the largest integer $n$ such that $2 n^{2}-k n+77$ is a positive prime number.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2859,
"split": "test",
"question": "Let $T=4$ and let $k=21+2 T$. Compute the largest integer $n$ such that $2 n^{2}-k n+77$ is a positive prime number."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=12$. In triangle $A B C, B C=T$ and $\\mathrm{m} \\angle B=30^{\\circ}$. Compute the number of integer values of $A C$ for which there are two possible values for side length $A B$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2860,
"split": "test",
"question": "Let $T=12$. In triangle $A B C, B C=T$ and $\\mathrm{m} \\angle B=30^{\\circ}$. Compute the number of integer values of $A C$ for which there are two possible values for side length $A B$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "An $\\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \\leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and... | math | {
"style": "rule",
"ground_truth": "$(312,123,231)$"
} | {
"index": 2861,
"split": "test",
"question": "An $\\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \\leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, whe... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "An $\\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \\leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and... | math | {
"style": "rule",
"ground_truth": "$41352,42351,51342$"
} | {
"index": 2862,
"split": "test",
"question": "An $\\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \\leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, whe... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "An $\\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \\leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, when $p=3$ and... | math | {
"style": "rule",
"ground_truth": "$352146,362145,452136,562134$"
} | {
"index": 2863,
"split": "test",
"question": "An $\\boldsymbol{n}$-label is a permutation of the numbers 1 through $n$. For example, $J=35214$ is a 5 -label and $K=132$ is a 3 -label. For a fixed positive integer $p$, where $p \\leq n$, consider consecutive blocks of $p$ numbers in an $n$-label. For example, whe... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In $\\triangle A B C, D$ is on $\\overline{A C}$ so that $\\overline{B D}$ is the angle bisector of $\\angle B$. Point $E$ is on $\\overline{A B}$ and $\\overline{C E}$ intersects $\\overline{B D}$ at $P$. Quadrilateral $B C D E$ is cyclic, $B P=12$ and $P E=4$. Compute the rat... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2884,
"split": "test",
"question": "In $\\triangle A B C, D$ is on $\\overline{A C}$ so that $\\overline{B D}$ is the angle bisector of $\\angle B$. Point $E$ is on $\\overline{A B}$ and $\\overline{C E}$ intersects $\\overline{B D}$ at $P$. Quadrilateral $B C D E$ is cyclic, $B P=12$ and $P E=4$. Comp... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $N$ be a six-digit number formed by an arrangement of the digits $1,2,3,3,4,5$. Compute the smallest value of $N$ that is divisible by 264 .\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "135432"
} | {
"index": 2887,
"split": "test",
"question": "Let $N$ be a six-digit number formed by an arrangement of the digits $1,2,3,3,4,5$. Compute the smallest value of $N$ that is divisible by 264 ."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In triangle $A B C, A B=4, B C=6$, and $A C=8$. Squares $A B Q R$ and $B C S T$ are drawn external to and lie in the same plane as $\\triangle A B C$. Compute $Q T$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$2 \\sqrt{10}$"
} | {
"index": 2888,
"split": "test",
"question": "In triangle $A B C, A B=4, B C=6$, and $A C=8$. Squares $A B Q R$ and $B C S T$ are drawn external to and lie in the same plane as $\\triangle A B C$. Compute $Q T$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$, and the other focus is at $(d, 7)$. Compute $d$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{49}{3}$"
} | {
"index": 2890,
"split": "test",
"question": "An ellipse in the first quadrant is tangent to both the $x$-axis and $y$-axis. One focus is at $(3,7)$, and the other focus is at $(d, 7)$. Compute $d$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}$ be a regular octagon. Let $\\mathbf{u}$ be the vector from $A_{1}$ to $A_{2}$ and let $\\mathbf{v}$ be the vector from $A_{1}$ to $A_{8}$. The vector from $A_{1}$ to $A_{4}$ can be written as $a \\mathbf{u}+b \\mathbf{v}$ fo... | math | {
"style": "rule",
"ground_truth": "$\\quad(2+\\sqrt{2}, 1+\\sqrt{2})$"
} | {
"index": 2891,
"split": "test",
"question": "Let $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6} A_{7} A_{8}$ be a regular octagon. Let $\\mathbf{u}$ be the vector from $A_{1}$ to $A_{2}$ and let $\\mathbf{v}$ be the vector from $A_{1}$ to $A_{8}$. The vector from $A_{1}$ to $A_{4}$ can be written as $a \\mathbf{u}+b \\ma... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the integer $n$ such that $2009<n<3009$ and the sum of the odd positive divisors of $n$ is 1024 .\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2604"
} | {
"index": 2892,
"split": "test",
"question": "Compute the integer $n$ such that $2009<n<3009$ and the sum of the odd positive divisors of $n$ is 1024 ."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Points $A, R, M$, and $L$ are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point $A$ are $(11,5)$. If points $R, M$, and $L$ are all lattice points, and $R$ is in Quadrant I, compute the number of possible ordered pairs $(x, y)$ of ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2893,
"split": "test",
"question": "Points $A, R, M$, and $L$ are consecutively the midpoints of the sides of a square whose area is 650. The coordinates of point $A$ are $(11,5)$. If points $R, M$, and $L$ are all lattice points, and $R$ is in Quadrant I, compute the number of possible ordered pairs $... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The taxicab distance between points $\\left(x_{1}, y_{1}, z_{1}\\right)$ and $\\left(x_{2}, y_{2}, z_{2}\\right)$ is given by\n\n$$\nd\\left(\\left(x_{1}, y_{1}, z_{1}\\right),\\left(x_{2}, y_{2}, z_{2}\\right)\\right)=\\left|x_{1}-x_{2}\\right|+\\left|y_{1}-y_{2}\\right|+\\lef... | math | {
"style": "rule",
"ground_truth": "$\\frac{179}{250}$"
} | {
"index": 2894,
"split": "test",
"question": "The taxicab distance between points $\\left(x_{1}, y_{1}, z_{1}\\right)$ and $\\left(x_{2}, y_{2}, z_{2}\\right)$ is given by\n\n$$\nd\\left(\\left(x_{1}, y_{1}, z_{1}\\right),\\left(x_{2}, y_{2}, z_{2}\\right)\\right)=\\left|x_{1}-x_{2}\\right|+\\left|y_{1}-y_{2}\\r... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\quad$ Let $a$ and $b$ be real numbers such that\n\n$$\na^{3}-15 a^{2}+20 a-50=0 \\quad \\text { and } \\quad 8 b^{3}-60 b^{2}-290 b+2575=0\n$$\n\nCompute $a+b$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{15}{2}$"
} | {
"index": 2895,
"split": "test",
"question": "$\\quad$ Let $a$ and $b$ be real numbers such that\n\n$$\na^{3}-15 a^{2}+20 a-50=0 \\quad \\text { and } \\quad 8 b^{3}-60 b^{2}-290 b+2575=0\n$$\n\nCompute $a+b$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "For a positive integer $n$, define $s(n)$ to be the sum of $n$ and its digits. For example, $s(2009)=2009+2+0+0+9=2020$. Compute the number of elements in the set $\\{s(0), s(1), s(2), \\ldots, s(9999)\\}$.\n\nLet's think step by step and output the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "9046"
} | {
"index": 2896,
"split": "test",
"question": "For a positive integer $n$, define $s(n)$ to be the sum of $n$ and its digits. For example, $s(2009)=2009+2+0+0+9=2020$. Compute the number of elements in the set $\\{s(0), s(1), s(2), \\ldots, s(9999)\\}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Quadrilateral $A R M L$ is a kite with $A R=R M=5, A M=8$, and $R L=11$. Compute $A L$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$4 \\sqrt{5}$"
} | {
"index": 2897,
"split": "test",
"question": "Quadrilateral $A R M L$ is a kite with $A R=R M=5, A M=8$, and $R L=11$. Compute $A L$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=4 \\sqrt{5}$. If $x y=\\sqrt{5}, y z=5$, and $x z=T$, compute the positive value of $x$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 2898,
"split": "test",
"question": "Let $T=4 \\sqrt{5}$. If $x y=\\sqrt{5}, y z=5$, and $x z=T$, compute the positive value of $x$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\quad$ Let $T=2$. In how many ways can $T$ boys and $T+1$ girls be arranged in a row if all the girls must be standing next to each other?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"index": 2899,
"split": "test",
"question": "$\\quad$ Let $T=2$. In how many ways can $T$ boys and $T+1$ girls be arranged in a row if all the girls must be standing next to each other?"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\triangle A B C$ is on a coordinate plane such that $A=(3,6)$, $B=(T, 0)$, and $C=(2 T-1,1-T)$. Let $\\ell$ be the line containing the altitude to $\\overline{B C}$. Compute the $y$-intercept of $\\ell$.\n\nLet's think step by step and output the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2900,
"split": "test",
"question": "$\\triangle A B C$ is on a coordinate plane such that $A=(3,6)$, $B=(T, 0)$, and $C=(2 T-1,1-T)$. Let $\\ell$ be the line containing the altitude to $\\overline{B C}$. Compute the $y$-intercept of $\\ell$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=3$. In triangle $A B C, A B=A C-2=T$, and $\\mathrm{m} \\angle A=60^{\\circ}$. Compute $B C^{2}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"index": 2901,
"split": "test",
"question": "Let $T=3$. In triangle $A B C, A B=A C-2=T$, and $\\mathrm{m} \\angle A=60^{\\circ}$. Compute $B C^{2}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=19$. Let $\\mathcal{S}_{1}$ denote the arithmetic sequence $0, \\frac{1}{4}, \\frac{1}{2}, \\ldots$, and let $\\mathcal{S}_{2}$ denote the arithmetic sequence $0, \\frac{1}{6}, \\frac{1}{3}, \\ldots$ Compute the $T^{\\text {th }}$ smallest number that occurs in both sequ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 2902,
"split": "test",
"question": "Let $T=19$. Let $\\mathcal{S}_{1}$ denote the arithmetic sequence $0, \\frac{1}{4}, \\frac{1}{2}, \\ldots$, and let $\\mathcal{S}_{2}$ denote the arithmetic sequence $0, \\frac{1}{6}, \\frac{1}{3}, \\ldots$ Compute the $T^{\\text {th }}$ smallest number that occurs i... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "$\\quad$ Let $T=9$. An integer $n$ is randomly selected from the set $\\{1,2,3, \\ldots, 2 T\\}$. Compute the probability that the integer $\\left|n^{3}-7 n^{2}+13 n-6\\right|$ is a prime number.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{9}$"
} | {
"index": 2903,
"split": "test",
"question": "$\\quad$ Let $T=9$. An integer $n$ is randomly selected from the set $\\{1,2,3, \\ldots, 2 T\\}$. Compute the probability that the integer $\\left|n^{3}-7 n^{2}+13 n-6\\right|$ is a prime number."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $A=\\frac{1}{9}$, and let $B=\\frac{1}{25}$. In $\\frac{1}{A}$ minutes, 20 frogs can eat 1800 flies. At this rate, in $\\frac{1}{B}$ minutes, how many flies will 15 frogs be able to eat?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3750"
} | {
"index": 2904,
"split": "test",
"question": "Let $A=\\frac{1}{9}$, and let $B=\\frac{1}{25}$. In $\\frac{1}{A}$ minutes, 20 frogs can eat 1800 flies. At this rate, in $\\frac{1}{B}$ minutes, how many flies will 15 frogs be able to eat?"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=5$. If $|T|-1+3 i=\\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{25}$"
} | {
"index": 2905,
"split": "test",
"question": "Let $T=5$. If $|T|-1+3 i=\\frac{1}{z}$, compute the sum of the real and imaginary parts of $z$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=10$. Ann spends 80 seconds climbing up a $T$ meter rope at a constant speed, and she spends 70 seconds climbing down the same rope at a constant speed (different from her upward speed). Ann begins climbing up and down the rope repeatedly, and she does not pause after cli... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 2907,
"split": "test",
"question": "Let $T=10$. Ann spends 80 seconds climbing up a $T$ meter rope at a constant speed, and she spends 70 seconds climbing down the same rope at a constant speed (different from her upward speed). Ann begins climbing up and down the rope repeatedly, and she does not paus... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=800$. Simplify $2^{\\log _{4} T} / 2^{\\log _{16} 64}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2908,
"split": "test",
"question": "Let $T=800$. Simplify $2^{\\log _{4} T} / 2^{\\log _{16} 64}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $P(x)=x^{2}+T x+800$, and let $r_{1}$ and $r_{2}$ be the roots of $P(x)$. The polynomial $Q(x)$ is quadratic, it has leading coefficient 1, and it has roots $r_{1}+1$ and $r_{2}+1$. Find the sum of the coefficients of $Q(x)$.\n\nLet's think step by step and output the final... | math | {
"style": "rule",
"ground_truth": "800"
} | {
"index": 2909,
"split": "test",
"question": "Let $P(x)=x^{2}+T x+800$, and let $r_{1}$ and $r_{2}$ be the roots of $P(x)$. The polynomial $Q(x)$ is quadratic, it has leading coefficient 1, and it has roots $r_{1}+1$ and $r_{2}+1$. Find the sum of the coefficients of $Q(x)$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=12$. Equilateral triangle $A B C$ is given with side length $T$. Points $D$ and $E$ are the midpoints of $\\overline{A B}$ and $\\overline{A C}$, respectively. Point $F$ lies in space such that $\\triangle D E F$ is equilateral and $\\triangle D E F$ lies in a plane perp... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"index": 2910,
"split": "test",
"question": "Let $T=12$. Equilateral triangle $A B C$ is given with side length $T$. Points $D$ and $E$ are the midpoints of $\\overline{A B}$ and $\\overline{A C}$, respectively. Point $F$ lies in space such that $\\triangle D E F$ is equilateral and $\\triangle D E F$ lies in a... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In triangle $A B C, A B=5, A C=6$, and $\\tan \\angle B A C=-\\frac{4}{3}$. Compute the area of $\\triangle A B C$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 2911,
"split": "test",
"question": "In triangle $A B C, A B=5, A C=6$, and $\\tan \\angle B A C=-\\frac{4}{3}$. Compute the area of $\\triangle A B C$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the number of positive integers less than 25 that cannot be written as the difference of two squares of integers.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2912,
"split": "test",
"question": "Compute the number of positive integers less than 25 that cannot be written as the difference of two squares of integers."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "For digits $A, B$, and $C,(\\underline{A} \\underline{B})^{2}+(\\underline{A} \\underline{C})^{2}=1313$. Compute $A+B+C$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 2913,
"split": "test",
"question": "For digits $A, B$, and $C,(\\underline{A} \\underline{B})^{2}+(\\underline{A} \\underline{C})^{2}=1313$. Compute $A+B+C$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Points $P, Q, R$, and $S$ lie in the interior of square $A B C D$ such that triangles $A B P, B C Q$, $C D R$, and $D A S$ are equilateral. If $A B=1$, compute the area of quadrilateral $P Q R S$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$2-\\sqrt{3}$"
} | {
"index": 2914,
"split": "test",
"question": "Points $P, Q, R$, and $S$ lie in the interior of square $A B C D$ such that triangles $A B P, B C Q$, $C D R$, and $D A S$ are equilateral. If $A B=1$, compute the area of quadrilateral $P Q R S$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "For real numbers $\\alpha, B$, and $C$, the zeros of $T(x)=x^{3}+x^{2}+B x+C \\operatorname{are~}^{2} \\alpha$, $\\cos ^{2} \\alpha$, and $-\\csc ^{2} \\alpha$. Compute $T(5)$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{567}{4}$"
} | {
"index": 2915,
"split": "test",
"question": "For real numbers $\\alpha, B$, and $C$, the zeros of $T(x)=x^{3}+x^{2}+B x+C \\operatorname{are~}^{2} \\alpha$, $\\cos ^{2} \\alpha$, and $-\\csc ^{2} \\alpha$. Compute $T(5)$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $\\mathcal{R}$ denote the circular region bounded by $x^{2}+y^{2}=36$. The lines $x=4$ and $y=3$ partition $\\mathcal{R}$ into four regions $\\mathcal{R}_{1}, \\mathcal{R}_{2}, \\mathcal{R}_{3}$, and $\\mathcal{R}_{4}$. $\\left[\\mathcal{R}_{i}\\right]$ denotes the area of ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 2916,
"split": "test",
"question": "Let $\\mathcal{R}$ denote the circular region bounded by $x^{2}+y^{2}=36$. The lines $x=4$ and $y=3$ partition $\\mathcal{R}$ into four regions $\\mathcal{R}_{1}, \\mathcal{R}_{2}, \\mathcal{R}_{3}$, and $\\mathcal{R}_{4}$. $\\left[\\mathcal{R}_{i}\\right]$ denotes t... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $x$ be a real number in the interval $[0,360]$ such that the four expressions $\\sin x^{\\circ}, \\cos x^{\\circ}$, $\\tan x^{\\circ}, \\cot x^{\\circ}$ take on exactly three distinct (finite) real values. Compute the sum of all possible values of $x$.\n\nLet's think step b... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"index": 2917,
"split": "test",
"question": "Let $x$ be a real number in the interval $[0,360]$ such that the four expressions $\\sin x^{\\circ}, \\cos x^{\\circ}$, $\\tan x^{\\circ}, \\cot x^{\\circ}$ take on exactly three distinct (finite) real values. Compute the sum of all possible values of $x$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $a_{1}, a_{2}, a_{3}, \\ldots$ be an arithmetic sequence, and let $b_{1}, b_{2}, b_{3}, \\ldots$ be a geometric sequence. The sequence $c_{1}, c_{2}, c_{3}, \\ldots$ has $c_{n}=a_{n}+b_{n}$ for each positive integer $n$. If $c_{1}=1, c_{2}=4, c_{3}=15$, and $c_{4}=2$, compu... | math | {
"style": "rule",
"ground_truth": "61"
} | {
"index": 2918,
"split": "test",
"question": "Let $a_{1}, a_{2}, a_{3}, \\ldots$ be an arithmetic sequence, and let $b_{1}, b_{2}, b_{3}, \\ldots$ be a geometric sequence. The sequence $c_{1}, c_{2}, c_{3}, \\ldots$ has $c_{n}=a_{n}+b_{n}$ for each positive integer $n$. If $c_{1}=1, c_{2}=4, c_{3}=15$, and $c_{4... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In square $A B C D$ with diagonal $1, E$ is on $\\overline{A B}$ and $F$ is on $\\overline{B C}$ with $\\mathrm{m} \\angle B C E=\\mathrm{m} \\angle B A F=$ $30^{\\circ}$. If $\\overline{C E}$ and $\\overline{A F}$ intersect at $G$, compute the distance between the incenters of... | math | {
"style": "rule",
"ground_truth": "$4-2 \\sqrt{3}$"
} | {
"index": 2919,
"split": "test",
"question": "In square $A B C D$ with diagonal $1, E$ is on $\\overline{A B}$ and $F$ is on $\\overline{B C}$ with $\\mathrm{m} \\angle B C E=\\mathrm{m} \\angle B A F=$ $30^{\\circ}$. If $\\overline{C E}$ and $\\overline{A F}$ intersect at $G$, compute the distance between the i... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $a, b, m, n$ be positive integers with $a m=b n=120$ and $a \\neq b$. In the coordinate plane, let $A=(a, m), B=(b, n)$, and $O=(0,0)$. If $X$ is a point in the plane such that $A O B X$ is a parallelogram, compute the minimum area of $A O B X$.\n\nLet's think step by step ... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"index": 2920,
"split": "test",
"question": "Let $a, b, m, n$ be positive integers with $a m=b n=120$ and $a \\neq b$. In the coordinate plane, let $A=(a, m), B=(b, n)$, and $O=(0,0)$. If $X$ is a point in the plane such that $A O B X$ is a parallelogram, compute the minimum area of $A O B X$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $\\mathcal{S}$ be the set of integers from 0 to 9999 inclusive whose base- 2 and base- 5 representations end in the same four digits. (Leading zeros are allowed, so $1=0001_{2}=0001_{5}$ is one such number.) Compute the remainder when the sum of the elements of $\\mathcal{S... | math | {
"style": "rule",
"ground_truth": "6248"
} | {
"index": 2921,
"split": "test",
"question": "Let $\\mathcal{S}$ be the set of integers from 0 to 9999 inclusive whose base- 2 and base- 5 representations end in the same four digits. (Leading zeros are allowed, so $1=0001_{2}=0001_{5}$ is one such number.) Compute the remainder when the sum of the elements of $... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "If $A, R, M$, and $L$ are positive integers such that $A^{2}+R^{2}=20$ and $M^{2}+L^{2}=10$, compute the product $A \\cdot R \\cdot M \\cdot L$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2922,
"split": "test",
"question": "If $A, R, M$, and $L$ are positive integers such that $A^{2}+R^{2}=20$ and $M^{2}+L^{2}=10$, compute the product $A \\cdot R \\cdot M \\cdot L$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=49$. Compute the last digit, in base 10, of the sum\n\n$$\nT^{2}+(2 T)^{2}+(3 T)^{2}+\\ldots+\\left(T^{2}\\right)^{2}\n$$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2924,
"split": "test",
"question": "Let $T=49$. Compute the last digit, in base 10, of the sum\n\n$$\nT^{2}+(2 T)^{2}+(3 T)^{2}+\\ldots+\\left(T^{2}\\right)^{2}\n$$"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "A fair coin is flipped $n$ times. Compute the smallest positive integer $n$ for which the probability that the coin has the same result every time is less than $10 \\%$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 2925,
"split": "test",
"question": "A fair coin is flipped $n$ times. Compute the smallest positive integer $n$ for which the probability that the coin has the same result every time is less than $10 \\%$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=5$. Compute the smallest positive integer $n$ such that there are at least $T$ positive integers in the domain of $f(x)=\\sqrt{-x^{2}-2 x+n}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 2926,
"split": "test",
"question": "Let $T=5$. Compute the smallest positive integer $n$ such that there are at least $T$ positive integers in the domain of $f(x)=\\sqrt{-x^{2}-2 x+n}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=35$. Compute the smallest positive real number $x$ such that $\\frac{\\lfloor x\\rfloor}{x-\\lfloor x\\rfloor}=T$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{36}{35}$"
} | {
"index": 2927,
"split": "test",
"question": "Let $T=35$. Compute the smallest positive real number $x$ such that $\\frac{\\lfloor x\\rfloor}{x-\\lfloor x\\rfloor}=T$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let set $S=\\{1,2,3,4,5,6\\}$, and let set $T$ be the set of all subsets of $S$ (including the empty set and $S$ itself). Let $t_{1}, t_{2}, t_{3}$ be elements of $T$, not necessarily distinct. The ordered triple $\\left(t_{1}, t_{2}, t_{3}\\right)$ is called satisfactory if ei... | math | {
"style": "rule",
"ground_truth": "31186"
} | {
"index": 2947,
"split": "test",
"question": "Let set $S=\\{1,2,3,4,5,6\\}$, and let set $T$ be the set of all subsets of $S$ (including the empty set and $S$ itself). Let $t_{1}, t_{2}, t_{3}$ be elements of $T$, not necessarily distinct. The ordered triple $\\left(t_{1}, t_{2}, t_{3}\\right)$ is called satisfa... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $A B C D$ be a parallelogram with $\\angle A B C$ obtuse. Let $\\overline{B E}$ be the altitude to side $\\overline{A D}$ of $\\triangle A B D$. Let $X$ be the point of intersection of $\\overline{A C}$ and $\\overline{B E}$, and let $F$ be the point of intersection of $\\o... | math | {
"style": "rule",
"ground_truth": "$\\frac{222}{13}$"
} | {
"index": 2948,
"split": "test",
"question": "Let $A B C D$ be a parallelogram with $\\angle A B C$ obtuse. Let $\\overline{B E}$ be the altitude to side $\\overline{A D}$ of $\\triangle A B D$. Let $X$ be the point of intersection of $\\overline{A C}$ and $\\overline{B E}$, and let $F$ be the point of intersect... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the sum of all positive two-digit factors of $2^{32}-1$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"index": 2949,
"split": "test",
"question": "Compute the sum of all positive two-digit factors of $2^{32}-1$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations:\n\n$$\nx^{2}+y^{2}=6 y-4 x+12 \\quad \\text { and } \\quad 4 y=x^{2}+4 x+12\n$$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$(-6,6)$, $(2,6)$"
} | {
"index": 2950,
"split": "test",
"question": "Compute all ordered pairs of real numbers $(x, y)$ that satisfy both of the equations:\n\n$$\nx^{2}+y^{2}=6 y-4 x+12 \\quad \\text { and } \\quad 4 y=x^{2}+4 x+12\n$$"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Define $\\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\\log ^{*}(1000)=2$ since $\\log 1000=3$ and $\\log (\\log 1000)=\\log 3=0.477 \\ldots \\leq 1$. Let $a$ be the ... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 2951,
"split": "test",
"question": "Define $\\log ^{*}(n)$ to be the smallest number of times the log function must be iteratively applied to $n$ to get a result less than or equal to 1 . For example, $\\log ^{*}(1000)=2$ since $\\log 1000=3$ and $\\log (\\log 1000)=\\log 3=0.477 \\ldots \\leq 1$. Let ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "An integer $N$ is worth 1 point for each pair of digits it contains that forms a prime in its original order. For example, 6733 is worth 3 points (for 67,73 , and 73 again), and 20304 is worth 2 points (for 23 and 03). Compute the smallest positive integer that is worth exactly... | math | {
"style": "rule",
"ground_truth": "100337"
} | {
"index": 2952,
"split": "test",
"question": "An integer $N$ is worth 1 point for each pair of digits it contains that forms a prime in its original order. For example, 6733 is worth 3 points (for 67,73 , and 73 again), and 20304 is worth 2 points (for 23 and 03). Compute the smallest positive integer that is wo... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The six sides of convex hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings such that every triangle $A_{i} A_{j} A_{k}$ has at least one red side.\n\nLet's think step... | math | {
"style": "rule",
"ground_truth": "392"
} | {
"index": 2953,
"split": "test",
"question": "The six sides of convex hexagon $A_{1} A_{2} A_{3} A_{4} A_{5} A_{6}$ are colored red. Each of the diagonals of the hexagon is colored either red or blue. Compute the number of colorings such that every triangle $A_{i} A_{j} A_{k}$ has at least one red side."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the smallest positive integer $n$ such that $n^{n}$ has at least 1,000,000 positive divisors.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"index": 2954,
"split": "test",
"question": "Compute the smallest positive integer $n$ such that $n^{n}$ has at least 1,000,000 positive divisors."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Given an arbitrary finite sequence of letters (represented as a word), a subsequence is a sequence of one or more letters that appear in the same order as in the original sequence. For example, $N, C T, O T T$, and CONTEST are subsequences of the word CONTEST, but NOT, ONSET, a... | math | {
"style": "rule",
"ground_truth": "3851"
} | {
"index": 2955,
"split": "test",
"question": "Given an arbitrary finite sequence of letters (represented as a word), a subsequence is a sequence of one or more letters that appear in the same order as in the original sequence. For example, $N, C T, O T T$, and CONTEST are subsequences of the word CONTEST, but NO... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon $\\mathcal{H}$ with side length 1 , and so that the vertices not lying in the plane of $\\mathcal{H}$ (the \"top\" vertices) are themselves coplanar. A spherical ball of radius ... | math | {
"style": "rule",
"ground_truth": "$\\frac{\\sqrt{2}}{3}$"
} | {
"index": 2956,
"split": "test",
"question": "Six solid regular tetrahedra are placed on a flat surface so that their bases form a regular hexagon $\\mathcal{H}$ with side length 1 , and so that the vertices not lying in the plane of $\\mathcal{H}$ (the \"top\" vertices) are themselves coplanar. A spherical ball... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Derek starts at the point $(0,0)$, facing the point $(0,1)$, and he wants to get to the point $(1,1)$. He takes unit steps parallel to the coordinate axes. A move consists of either a step forward, or a $90^{\\circ}$ right (clockwise) turn followed by a step forward, so that hi... | math | {
"style": "rule",
"ground_truth": "529"
} | {
"index": 2957,
"split": "test",
"question": "Derek starts at the point $(0,0)$, facing the point $(0,1)$, and he wants to get to the point $(1,1)$. He takes unit steps parallel to the coordinate axes. A move consists of either a step forward, or a $90^{\\circ}$ right (clockwise) turn followed by a step forward,... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "The equations $x^{3}+A x+10=0$ and $x^{3}+B x^{2}+50=0$ have two roots in common. Compute the product of these common roots.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$5 \\sqrt[3]{4}$"
} | {
"index": 2958,
"split": "test",
"question": "The equations $x^{3}+A x+10=0$ and $x^{3}+B x^{2}+50=0$ have two roots in common. Compute the product of these common roots."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $N$ be a perfect square between 100 and 400 , inclusive. What is the only digit that cannot appear in $N$ ?\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 2960,
"split": "test",
"question": "Let $N$ be a perfect square between 100 and 400 , inclusive. What is the only digit that cannot appear in $N$ ?"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=7$. Let $A$ and $B$ be distinct digits in base $T$, and let $N$ be the largest number of the form $\\underline{A} \\underline{B} \\underline{A}_{T}$. Compute the value of $N$ in base 10 .\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "335"
} | {
"index": 2961,
"split": "test",
"question": "Let $T=7$. Let $A$ and $B$ be distinct digits in base $T$, and let $N$ be the largest number of the form $\\underline{A} \\underline{B} \\underline{A}_{T}$. Compute the value of $N$ in base 10 ."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let T be an integer. Given a nonzero integer $n$, let $f(n)$ denote the sum of all numbers of the form $i^{d}$, where $i=\\sqrt{-1}$, and $d$ is a divisor (positive or negative) of $n$. Compute $f(2 T+1)$.\n\nLet's think step by step and output the final answer within \\boxed{}... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"index": 2962,
"split": "test",
"question": "Let T be an integer. Given a nonzero integer $n$, let $f(n)$ denote the sum of all numbers of the form $i^{d}$, where $i=\\sqrt{-1}$, and $d$ is a divisor (positive or negative) of $n$. Compute $f(2 T+1)$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=0$. Compute the real value of $x$ for which there exists a solution to the system of equations\n\n$$\n\\begin{aligned}\nx+y & =0 \\\\\nx^{3}-y^{3} & =54+T .\n\\end{aligned}\n$$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2963,
"split": "test",
"question": "Let $T=0$. Compute the real value of $x$ for which there exists a solution to the system of equations\n\n$$\n\\begin{aligned}\nx+y & =0 \\\\\nx^{3}-y^{3} & =54+T .\n\\end{aligned}\n$$"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=3$. In $\\triangle A B C, A C=T^{2}, \\mathrm{~m} \\angle A B C=45^{\\circ}$, and $\\sin \\angle A C B=\\frac{8}{9}$. Compute $A B$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$8 \\sqrt{2}$"
} | {
"index": 2964,
"split": "test",
"question": "Let $T=3$. In $\\triangle A B C, A C=T^{2}, \\mathrm{~m} \\angle A B C=45^{\\circ}$, and $\\sin \\angle A C B=\\frac{8}{9}$. Compute $A B$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=9$. The sequence $a_{1}, a_{2}, a_{3}, \\ldots$ is an arithmetic progression, $d$ is the common difference, $a_{T}=10$, and $a_{K}=2010$, where $K>T$. If $d$ is an integer, compute the value of $K$ such that $|K-d|$ is minimal.\n\nLet's think step by step and output the ... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"index": 2966,
"split": "test",
"question": "Let $T=9$. The sequence $a_{1}, a_{2}, a_{3}, \\ldots$ is an arithmetic progression, $d$ is the common difference, $a_{T}=10$, and $a_{K}=2010$, where $K>T$. If $d$ is an integer, compute the value of $K$ such that $|K-d|$ is minimal."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $A$ be the number you will receive from position 7 , and let $B$ be the number you will receive from position 9 . There are exactly two ordered pairs of real numbers $\\left(x_{1}, y_{1}\\right),\\left(x_{2}, y_{2}\\right)$ that satisfy both $|x+y|=6(\\sqrt{A}-5)$ and $x^{2... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2967,
"split": "test",
"question": "Let $A$ be the number you will receive from position 7 , and let $B$ be the number you will receive from position 9 . There are exactly two ordered pairs of real numbers $\\left(x_{1}, y_{1}\\right),\\left(x_{2}, y_{2}\\right)$ that satisfy both $|x+y|=6(\\sqrt{A}-5)... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=23$. In triangle $A B C$, the altitude from $A$ to $\\overline{B C}$ has length $\\sqrt{T}, A B=A C$, and $B C=T-K$, where $K$ is the real root of the equation $x^{3}-8 x^{2}-8 x-9=0$. Compute the length $A B$.\n\nLet's think step by step and output the final answer with... | math | {
"style": "rule",
"ground_truth": "$6 \\sqrt{2}$"
} | {
"index": 2968,
"split": "test",
"question": "Let $T=23$. In triangle $A B C$, the altitude from $A$ to $\\overline{B C}$ has length $\\sqrt{T}, A B=A C$, and $B C=T-K$, where $K$ is the real root of the equation $x^{3}-8 x^{2}-8 x-9=0$. Compute the length $A B$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=8$. A cube has volume $T-2$. The cube's surface area equals one-eighth the surface area of a $2 \\times 2 \\times n$ rectangular prism. Compute $n$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"index": 2969,
"split": "test",
"question": "Let $T=8$. A cube has volume $T-2$. The cube's surface area equals one-eighth the surface area of a $2 \\times 2 \\times n$ rectangular prism. Compute $n$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=98721$, and let $K$ be the sum of the digits of $T$. Let $A_{n}$ be the number of ways to tile a $1 \\times n$ rectangle using $1 \\times 3$ and $1 \\times 1$ tiles that do not overlap. Tiles of both types need not be used; for example, $A_{3}=2$ because a $1 \\times 3$ ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 2970,
"split": "test",
"question": "Let $T=98721$, and let $K$ be the sum of the digits of $T$. Let $A_{n}$ be the number of ways to tile a $1 \\times n$ rectangle using $1 \\times 3$ and $1 \\times 1$ tiles that do not overlap. Tiles of both types need not be used; for example, $A_{3}=2$ because a $1 ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=3$, and let $K=T+2$. Compute the largest $K$-digit number which has distinct digits and is a multiple of 63.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98721"
} | {
"index": 2971,
"split": "test",
"question": "Let $T=3$, and let $K=T+2$. Compute the largest $K$-digit number which has distinct digits and is a multiple of 63."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T\\neq 0$. Suppose that $a, b, c$, and $d$ are real numbers so that $\\log _{a} c=\\log _{b} d=T$. Compute\n\n$$\n\\frac{\\log _{\\sqrt{a b}}(c d)^{3}}{\\log _{a} c+\\log _{b} d}\n$$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2972,
"split": "test",
"question": "Let $T\\neq 0$. Suppose that $a, b, c$, and $d$ are real numbers so that $\\log _{a} c=\\log _{b} d=T$. Compute\n\n$$\n\\frac{\\log _{\\sqrt{a b}}(c d)^{3}}{\\log _{a} c+\\log _{b} d}\n$$"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=2030$. Given that $\\mathrm{A}, \\mathrm{D}, \\mathrm{E}, \\mathrm{H}, \\mathrm{S}$, and $\\mathrm{W}$ are distinct digits, and that $\\underline{\\mathrm{W}} \\underline{\\mathrm{A}} \\underline{\\mathrm{D}} \\underline{\\mathrm{E}}+\\underline{\\mathrm{A}} \\underline{... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"index": 2973,
"split": "test",
"question": "Let $T=2030$. Given that $\\mathrm{A}, \\mathrm{D}, \\mathrm{E}, \\mathrm{H}, \\mathrm{S}$, and $\\mathrm{W}$ are distinct digits, and that $\\underline{\\mathrm{W}} \\underline{\\mathrm{A}} \\underline{\\mathrm{D}} \\underline{\\mathrm{E}}+\\underline{\\mathrm{A}} \... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $f(x)=2^{x}+x^{2}$. Compute the smallest integer $n>10$ such that $f(n)$ and $f(10)$ have the same units digit.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"index": 2974,
"split": "test",
"question": "Let $f(x)=2^{x}+x^{2}$. Compute the smallest integer $n>10$ such that $f(n)$ and $f(10)$ have the same units digit."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In rectangle $P A U L$, point $D$ is the midpoint of $\\overline{U L}$ and points $E$ and $F$ lie on $\\overline{P L}$ and $\\overline{P A}$, respectively such that $\\frac{P E}{E L}=\\frac{3}{2}$ and $\\frac{P F}{F A}=2$. Given that $P A=36$ and $P L=25$, compute the area of p... | math | {
"style": "rule",
"ground_truth": "630"
} | {
"index": 2975,
"split": "test",
"question": "In rectangle $P A U L$, point $D$ is the midpoint of $\\overline{U L}$ and points $E$ and $F$ lie on $\\overline{P L}$ and $\\overline{P A}$, respectively such that $\\frac{P E}{E L}=\\frac{3}{2}$ and $\\frac{P F}{F A}=2$. Given that $P A=36$ and $P L=25$, compute th... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Rectangle $A R M L$ has length 125 and width 8. The rectangle is divided into 1000 squares of area 1 by drawing in gridlines parallel to the sides of $A R M L$. Diagonal $\\overline{A M}$ passes through the interior of exactly $n$ of the 1000 unit squares. Compute $n$.\n\nLet's... | math | {
"style": "rule",
"ground_truth": "132"
} | {
"index": 2976,
"split": "test",
"question": "Rectangle $A R M L$ has length 125 and width 8. The rectangle is divided into 1000 squares of area 1 by drawing in gridlines parallel to the sides of $A R M L$. Diagonal $\\overline{A M}$ passes through the interior of exactly $n$ of the 1000 unit squares. Compute $n... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the least integer $n>1$ such that the product of all positive divisors of $n$ equals $n^{4}$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 2977,
"split": "test",
"question": "Compute the least integer $n>1$ such that the product of all positive divisors of $n$ equals $n^{4}$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Each of the six faces of a cube is randomly colored red or blue with equal probability. Compute the probability that no three faces of the same color share a common vertex.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{9}{32}$"
} | {
"index": 2978,
"split": "test",
"question": "Each of the six faces of a cube is randomly colored red or blue with equal probability. Compute the probability that no three faces of the same color share a common vertex."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Scalene triangle $A B C$ has perimeter 2019 and integer side lengths. The angle bisector from $C$ meets $\\overline{A B}$ at $D$ such that $A D=229$. Given that $A C$ and $A D$ are relatively prime, compute $B C$.\n\nLet's think step by step and output the final answer within \... | math | {
"style": "rule",
"ground_truth": "888"
} | {
"index": 2979,
"split": "test",
"question": "Scalene triangle $A B C$ has perimeter 2019 and integer side lengths. The angle bisector from $C$ meets $\\overline{A B}$ at $D$ such that $A D=229$. Given that $A C$ and $A D$ are relatively prime, compute $B C$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Given that $a$ and $b$ are positive and\n\n$$\n\\lfloor 20-a\\rfloor=\\lfloor 19-b\\rfloor=\\lfloor a b\\rfloor,\n$$\n\ncompute the least upper bound of the set of possible values of $a+b$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$\\frac{41}{5}$"
} | {
"index": 2980,
"split": "test",
"question": "Given that $a$ and $b$ are positive and\n\n$$\n\\lfloor 20-a\\rfloor=\\lfloor 19-b\\rfloor=\\lfloor a b\\rfloor,\n$$\n\ncompute the least upper bound of the set of possible values of $a+b$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the number of five-digit integers $\\underline{M} \\underline{A} \\underline{R} \\underline{T} \\underline{Y}$, with all digits distinct, such that $M>A>R$ and $R<T<Y$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1512"
} | {
"index": 2981,
"split": "test",
"question": "Compute the number of five-digit integers $\\underline{M} \\underline{A} \\underline{R} \\underline{T} \\underline{Y}$, with all digits distinct, such that $M>A>R$ and $R<T<Y$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "In parallelogram $A R M L$, points $P$ and $Q$ are the midpoints of sides $\\overline{R M}$ and $\\overline{A L}$, respectively. Point $X$ lies on segment $\\overline{P Q}$, and $P X=3, R X=4$, and $P R=5$. Point $I$ lies on segment $\\overline{R X}$ such that $I A=I L$. Comput... | math | {
"style": "rule",
"ground_truth": "$\\frac{4}{3}$"
} | {
"index": 2982,
"split": "test",
"question": "In parallelogram $A R M L$, points $P$ and $Q$ are the midpoints of sides $\\overline{R M}$ and $\\overline{A L}$, respectively. Point $X$ lies on segment $\\overline{P Q}$, and $P X=3, R X=4$, and $P R=5$. Point $I$ lies on segment $\\overline{R X}$ such that $I A=I... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Given that $a, b, c$, and $d$ are positive integers such that\n\n$$\na ! \\cdot b ! \\cdot c !=d ! \\quad \\text { and } \\quad a+b+c+d=37\n$$\n\ncompute the product $a b c d$.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2240"
} | {
"index": 2983,
"split": "test",
"question": "Given that $a, b, c$, and $d$ are positive integers such that\n\n$$\na ! \\cdot b ! \\cdot c !=d ! \\quad \\text { and } \\quad a+b+c+d=37\n$$\n\ncompute the product $a b c d$."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Compute the value of\n\n$$\n\\sin \\left(6^{\\circ}\\right) \\cdot \\sin \\left(12^{\\circ}\\right) \\cdot \\sin \\left(24^{\\circ}\\right) \\cdot \\sin \\left(42^{\\circ}\\right)+\\sin \\left(12^{\\circ}\\right) \\cdot \\sin \\left(24^{\\circ}\\right) \\cdot \\sin \\left(42^{\... | math | {
"style": "rule",
"ground_truth": "$\\frac{1}{16}$"
} | {
"index": 2984,
"split": "test",
"question": "Compute the value of\n\n$$\n\\sin \\left(6^{\\circ}\\right) \\cdot \\sin \\left(12^{\\circ}\\right) \\cdot \\sin \\left(24^{\\circ}\\right) \\cdot \\sin \\left(42^{\\circ}\\right)+\\sin \\left(12^{\\circ}\\right) \\cdot \\sin \\left(24^{\\circ}\\right) \\cdot \\sin \... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $a=19, b=20$, and $c=21$. Compute\n\n$$\n\\frac{a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a}{a+b+c}\n$$\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 2985,
"split": "test",
"question": "Let $a=19, b=20$, and $c=21$. Compute\n\n$$\n\\frac{a^{2}+b^{2}+c^{2}+2 a b+2 b c+2 c a}{a+b+c}\n$$"
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=60$ . Lydia is a professional swimmer and can swim one-fifth of a lap of a pool in an impressive 20.19 seconds, and she swims at a constant rate. Rounded to the nearest integer, compute the number of minutes required for Lydia to swim $T$ laps.\n\nLet's think step by ste... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 2986,
"split": "test",
"question": "Let $T=60$ . Lydia is a professional swimmer and can swim one-fifth of a lap of a pool in an impressive 20.19 seconds, and she swims at a constant rate. Rounded to the nearest integer, compute the number of minutes required for Lydia to swim $T$ laps."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=101$. In $\\triangle A B C, \\mathrm{~m} \\angle C=90^{\\circ}$ and $A C=B C=\\sqrt{T-3}$. Circles $O$ and $P$ each have radius $r$ and lie inside $\\triangle A B C$. Circle $O$ is tangent to $\\overline{A C}$ and $\\overline{B C}$. Circle $P$ is externally tangent to ci... | math | {
"style": "rule",
"ground_truth": "$3-\\sqrt{2}$"
} | {
"index": 2987,
"split": "test",
"question": "Let $T=101$. In $\\triangle A B C, \\mathrm{~m} \\angle C=90^{\\circ}$ and $A C=B C=\\sqrt{T-3}$. Circles $O$ and $P$ each have radius $r$ and lie inside $\\triangle A B C$. Circle $O$ is tangent to $\\overline{A C}$ and $\\overline{B C}$. Circle $P$ is externally ta... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Given that $p=6.6 \\times 10^{-27}$, then $\\sqrt{p}=a \\times 10^{b}$, where $1 \\leq a<10$ and $b$ is an integer. Compute $10 a+b$ rounded to the nearest integer.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 2988,
"split": "test",
"question": "Given that $p=6.6 \\times 10^{-27}$, then $\\sqrt{p}=a \\times 10^{b}$, where $1 \\leq a<10$ and $b$ is an integer. Compute $10 a+b$ rounded to the nearest integer."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=67$. A group of children and adults go to a rodeo. A child's admission ticket costs $\\$ 5$, and an adult's admission ticket costs more than $\\$ 5$. The total admission cost for the group is $\\$ 10 \\cdot T$. If the number of adults in the group were to increase by $20... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 2989,
"split": "test",
"question": "Let $T=67$. A group of children and adults go to a rodeo. A child's admission ticket costs $\\$ 5$, and an adult's admission ticket costs more than $\\$ 5$. The total admission cost for the group is $\\$ 10 \\cdot T$. If the number of adults in the group were to incr... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Let $T=67$. Rectangles $F A K E$ and $F U N K$ lie in the same plane. Given that $E F=T$, $A F=\\frac{4 T}{3}$, and $U F=\\frac{12}{5}$, compute the area of the intersection of the two rectangles.\n\nLet's think step by step and output the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "262"
} | {
"index": 2990,
"split": "test",
"question": "Let $T=67$. Rectangles $F A K E$ and $F U N K$ lie in the same plane. Given that $E F=T$, $A F=\\frac{4 T}{3}$, and $U F=\\frac{12}{5}$, compute the area of the intersection of the two rectangles."
} |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a switch mean... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 2991,
"split": "test",
"question": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a switch mean... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2992,
"split": "test",
"question": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a switch mean... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2993,
"split": "test",
"question": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a switch mean... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2994,
"split": "test",
"question": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a switch mean... | math | {
"style": "rule",
"ground_truth": "$\\frac{n}{2}$"
} | {
"index": 2995,
"split": "test",
"question": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a ... |
math-ai/olympiadbench | [
{
"role": "user",
"content": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a switch mean... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 2996,
"split": "test",
"question": "Elizabeth is in an \"escape room\" puzzle. She is in a room with one door which is locked at the start of the puzzle. The room contains $n$ light switches, each of which is initially off. Each minute, she must flip exactly $k$ different light switches (to \"flip\" a ... |
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