data_source stringclasses 8 values | prompt listlengths 2 2 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. Compute $2^{\\log _{T} 8}-8^{\\log _{T} 2}$.",
"role": "us... | math | {
"ground_truth": "0",
"style": "rule"
} | {
"index": 400,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. At some point during a given week, a law enforcement officer h... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"index": 401,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "$\\quad$ Let $T=T N Y W R$. In parallelogram $A R M L$, points $P$ and $Q$ trisec... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 402,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. Compute the number of positive perfect cubes that are divisors... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 403,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. The graph of $y=x^{2}+2 x-T$ intersects the $x$-axis at points... | math | {
"ground_truth": "74",
"style": "rule"
} | {
"index": 404,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $S$ be the set of prime factors of the numbers you receive from positions 7 a... | math | {
"ground_truth": "$\\frac{67 \\pi}{3}$",
"style": "rule"
} | {
"index": 405,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. A group of $n$ friends goes camping; two of them are selected ... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 406,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. The parabola $y=x^{2}+T x$ is tangent to the parabola $y=-(x-2... | math | {
"ground_truth": "184",
"style": "rule"
} | {
"index": 407,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. The first two terms of a sequence are $a_{1}=3 / 5$ and $a_{2}... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 408,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. A regular $n$-gon has exactly $T$ more diagonals than a regula... | math | {
"ground_truth": "19",
"style": "rule"
} | {
"index": 409,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. The sequence $a_{1}, a_{2}, a_{3}, \\ldots$, is arithmetic wit... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 410,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=T N Y W R$. A rectangular prism has a length of 1 , a width of 3 , a heigh... | math | {
"ground_truth": "$\\frac{27}{2}$",
"style": "rule"
} | {
"index": 411,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The zeros of $x^{2}+b x+93$ are $r$ and $s$. If the zeros of $x^{2}-22 x+c$ are $... | math | {
"ground_truth": "114",
"style": "rule"
} | {
"index": 412,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $N=888,888 \\times 9,999,999$. Compute the sum of the digits of $N$.",
"r... | math | {
"ground_truth": "63",
"style": "rule"
} | {
"index": 413,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Five equilateral triangles are drawn in the plane so that no two sides of any of ... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 414,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "$\\quad$ Let $S$ be the set of four-digit positive integers for which the sum of ... | math | {
"ground_truth": "2302",
"style": "rule"
} | {
"index": 415,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $E U C L I D$ be a hexagon inscribed in a circle of radius 5 . Given that $E ... | math | {
"ground_truth": "$\\frac{14}{5}$",
"style": "rule"
} | {
"index": 416,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The ARMLLexicon consists of 10 letters: $\\{A, R, M, L, e, x, i, c, o, n\\}$. A p... | math | {
"ground_truth": "99956",
"style": "rule"
} | {
"index": 417,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $10^{y}$ be the product of all real numbers $x$ such that $\\log x=\\frac{3+\... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 418,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The solutions to the equation $x^{2}-180 x+8=0$ are $r_{1}$ and $r_{2}$. Compute\... | math | {
"ground_truth": "508",
"style": "rule"
} | {
"index": 419,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Circle $\\omega$ is tangent to parallel lines $\\ell_{1}$ and $\\ell_{2}$ at $A$ ... | math | {
"ground_truth": "$5 \\sqrt{10}$",
"style": "rule"
} | {
"index": 420,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Given quadrilateral $A R M L$ with $A R=20, R M=23, M L=25$, and $A M=32$, comput... | math | {
"ground_truth": "49",
"style": "rule"
} | {
"index": 421,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $\\mathcal{S}$ denote the set of all real polynomials $A(x)$ with leading coe... | math | {
"ground_truth": "46760",
"style": "rule"
} | {
"index": 422,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=688$. Let $a$ be the least nonzero digit in $T$, and let $b$ be the greate... | math | {
"ground_truth": "36",
"style": "rule"
} | {
"index": 423,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=36$. Square $A B C D$ has area $T$. Points $M, N, O$, and $P$ lie on $\\ov... | math | {
"ground_truth": "$6 \\sqrt{2}-2$",
"style": "rule"
} | {
"index": 424,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In a game, a player chooses 2 of the 13 letters from the first half of the alphab... | math | {
"ground_truth": "$\\frac{1}{6084}$",
"style": "rule"
} | {
"index": 425,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=\\frac{1}{6084}$. Compute the least positive integer $n$ such that when a ... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 426,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=13$. Compute the least integer $n>2023$ such that the equation $x^{2}-T x-... | math | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": 427,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In a sequence of $n$ consecutive positive integers, where $n>1$, an element of th... | math | {
"ground_truth": "1,2",
"style": "rule"
} | {
"index": 428,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In a sequence of $n$ consecutive positive integers, where $n>1$, an element of th... | math | {
"ground_truth": "1,3",
"style": "rule"
} | {
"index": 429,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For an integer $n \\geq 4$, define $a_{n}$ to be the product of all real numbers ... | math | {
"ground_truth": "-2019",
"style": "rule"
} | {
"index": 430,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Suppose that $u$ and $v$ are distinct numbers chosen at random from the set $\\{1... | math | {
"ground_truth": "$\\frac{17}{145}$",
"style": "rule"
} | {
"index": 431,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The degree-measures of the interior angles of convex hexagon TIEBRK are all integ... | math | {
"ground_truth": "65",
"style": "rule"
} | {
"index": 432,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A six-digit natural number is \"sort-of-decreasing\" if its first three digits ar... | math | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": 433,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "For each positive integer $N$, let $P(N)$ denote the product of the digits of $N$... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"index": 434,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the least integer value of the function\n\n$$\nf(x)=\\frac{x^{4}-6 x^{3}+... | math | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": 435,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Suppose that noncongruent triangles $A B C$ and $X Y Z$ are given such that $A B=... | math | {
"ground_truth": "$25 \\sqrt{3}$",
"style": "rule"
} | {
"index": 436,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The mean, median, and unique mode of a list of positive integers are three consec... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"index": 437,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "David builds a circular table; he then carves one or more positive integers into ... | math | {
"ground_truth": "7711",
"style": "rule"
} | {
"index": 438,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In quadrilateral $A B C D, \\mathrm{~m} \\angle B+\\mathrm{m} \\angle D=270^{\\ci... | math | {
"ground_truth": "$\\sqrt{130}$",
"style": "rule"
} | {
"index": 439,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $i=\\sqrt{-1}$. The complex number $z=-142+333 \\sqrt{5} i$ can be expressed ... | math | {
"ground_truth": "17",
"style": "rule"
} | {
"index": 440,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Parallelogram $A B C D$ is rotated about $A$ in the plane, resulting in $A B^{\\p... | math | {
"ground_truth": "$\\sqrt{2}-1$,$\\frac{3-\\sqrt{2}}{7}$",
"style": "rule"
} | {
"index": 441,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the least integer greater than 2023 , the sum of whose digits is 17 .",
... | math | {
"ground_truth": "2069",
"style": "rule"
} | {
"index": 442,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T$ = 2069, and let $K$ be the sum of the digits of $T$. Let $r$ and $s$ be t... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 443,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ 7, and let $K=9 T$. Let $A_{1}=2$, and for $n \\geq 2$, let\n\n$$\nA_{n}... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"index": 444,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ 21. The number $20^{T} \\cdot 23^{T}$ has $K$ positive divisors. Compute... | math | {
"ground_truth": "43",
"style": "rule"
} | {
"index": 445,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=43$. Compute the positive integer $n \\neq 17$ for which $\\left(\\begin{a... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 446,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=23$ . Compute the units digit of $T^{2023}+T^{20}-T^{23}$.",
"role": "... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 447,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ 3. Suppose that $T$ fair coins are flipped. Compute the probability that... | math | {
"ground_truth": "$\\frac{7}{8}$",
"style": "rule"
} | {
"index": 448,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ $\\frac{7}{8}$. The number $T$ can be expressed as a reduced fraction $\... | math | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": 449,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ -8, and let $i=\\sqrt{-1}$. Compute the positive integer $k$ for which $... | math | {
"ground_truth": "16",
"style": "rule"
} | {
"index": 450,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ 16. Compute the value of $x$ that satisfies $\\log _{4} T=\\log _{2} x$.... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"index": 451,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ 4. Pyramid $L E O J S$ is a right square pyramid with base $E O J S$, wh... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 452,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=$ 7. Compute the units digit of $T^{2023}+(T-2)^{20}-(T+10)^{23}$.",
"... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 453,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $r=1$ and $R=5$. A circle with radius $r$ is centered at $A$, and a circle wi... | math | {
"ground_truth": "$\\sqrt{15}$",
"style": "rule"
} | {
"index": 454,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the largest prime divisor of $15 !-13$ !.",
"role": "user"
}
] | math | {
"ground_truth": "19",
"style": "rule"
} | {
"index": 455,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Three non-overlapping squares of positive integer side lengths each have one vert... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 456,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A circle with center $O$ and radius 1 contains chord $\\overline{A B}$ of length ... | math | {
"ground_truth": "$\\frac{\\sqrt{3}}{32}$",
"style": "rule"
} | {
"index": 457,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "$\\quad$ Suppose that $p$ and $q$ are two-digit prime numbers such that $p^{2}-q^... | math | {
"ground_truth": "162",
"style": "rule"
} | {
"index": 458,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The four zeros of the polynomial $x^{4}+j x^{2}+k x+225$ are distinct real number... | math | {
"ground_truth": "-50",
"style": "rule"
} | {
"index": 459,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the smallest positive integer $n$ such that\n\n$$\nn,\\lfloor\\sqrt{n}\\r... | math | {
"ground_truth": "4096",
"style": "rule"
} | {
"index": 460,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "If $n$ is a positive integer, then $n$ !! is defined to be $n(n-2)(n-4) \\cdots 2... | math | {
"ground_truth": "1510",
"style": "rule"
} | {
"index": 461,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "On the complex plane, the parallelogram formed by the points $0, z, \\frac{1}{z}$... | math | {
"ground_truth": "$\\frac{50}{37}$",
"style": "rule"
} | {
"index": 462,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "One face of a $2 \\times 2 \\times 2$ cube is painted (not the entire cube), and ... | math | {
"ground_truth": "$\\frac{1}{16}$",
"style": "rule"
} | {
"index": 463,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "In triangle $A B C, A B=B C$. A trisector of $\\angle B$ intersects $\\overline{A... | math | {
"ground_truth": "146",
"style": "rule"
} | {
"index": 464,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The rational number $r$ is the largest number less than 1 whose base-7 expansion ... | math | {
"ground_truth": "95",
"style": "rule"
} | {
"index": 465,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=95$. Triangle $A B C$ has $A B=A C$. Points $M$ and $N$ lie on $\\overline... | math | {
"ground_truth": "75",
"style": "rule"
} | {
"index": 466,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=75$. At Wash College of Higher Education (Wash Ed.), the entering class ha... | math | {
"ground_truth": "37",
"style": "rule"
} | {
"index": 467,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the least positive integer $n$ such that the set of angles\n\n$$\n\\left\... | math | {
"ground_truth": "11",
"style": "rule"
} | {
"index": 468,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=11$. In ARMLvania, license plates use only the digits 1-9, and each licens... | math | {
"ground_truth": "256",
"style": "rule"
} | {
"index": 469,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=256$. Let $\\mathcal{R}$ be the region in the plane defined by the inequal... | math | {
"ground_truth": "$1024-256 \\pi$",
"style": "rule"
} | {
"index": 470,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Triangle $A B C$ has $\\mathrm{m} \\angle A>\\mathrm{m} \\angle B>\\mathrm{m} \\a... | math | {
"ground_truth": "$44^{\\circ}$",
"style": "rule"
} | {
"index": 471,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the number of ordered pairs of integers $(b, c)$, with $-20 \\leq b \\leq... | math | {
"ground_truth": "81",
"style": "rule"
} | {
"index": 472,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "A seventeen-sided die has faces numbered 1 through 17, but it is not fair: 17 com... | math | {
"ground_truth": "$\\frac{7}{128}$",
"style": "rule"
} | {
"index": 473,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Compute the number of ordered pairs of integers $(a, b)$ such that $1<a \\leq 50,... | math | {
"ground_truth": "81",
"style": "rule"
} | {
"index": 474,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Suppose that 5-letter \"words\" are formed using only the letters A, R, M, and L.... | math | {
"ground_truth": "165",
"style": "rule"
} | {
"index": 475,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Positive integers $a_{1}, a_{2}, a_{3}, \\ldots$ form an arithmetic sequence. If ... | math | {
"ground_truth": "820",
"style": "rule"
} | {
"index": 476,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The graphs of $y=x^{2}-|x|-12$ and $y=|x|-k$ intersect at distinct points $A, B, ... | math | {
"ground_truth": "$10+2 \\sqrt{2}$",
"style": "rule"
} | {
"index": 477,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "The zeros of $f(x)=x^{6}+2 x^{5}+3 x^{4}+5 x^{3}+8 x^{2}+13 x+21$ are distinct co... | math | {
"ground_truth": "$-\\frac{23}{60}$",
"style": "rule"
} | {
"index": 478,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $N=\\left\\lfloor(3+\\sqrt{5})^{34}\\right\\rfloor$. Compute the remainder wh... | math | {
"ground_truth": "47",
"style": "rule"
} | {
"index": 479,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $A B C$ be a triangle with $\\mathrm{m} \\angle B=\\mathrm{m} \\angle C=80^{\... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 480,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "If $\\lceil u\\rceil$ denotes the least integer greater than or equal to $u$, and... | math | {
"ground_truth": "$\\frac{189 \\sqrt{11}}{11}$",
"style": "rule"
} | {
"index": 481,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "If $x, y$, and $z$ are positive integers such that $x y=20$ and $y z=12$, compute... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 482,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=8$. Let $A=(1,5)$ and $B=(T-1,17)$. Compute the value of $x$ such that $(x... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 483,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let T be a rational number. Let $N$ be the smallest positive $T$-digit number tha... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 484,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=15$. For complex $z$, define the function $f_{1}(z)=z$, and for $n>1, f_{n... | math | {
"ground_truth": "$\\sqrt{26}$",
"style": "rule"
} | {
"index": 485,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=\\sqrt{26}$. Compute the number of ordered pairs of positive integers $(a,... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"index": 486,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=32$. Given that $\\sin \\theta=\\frac{\\sqrt{T^{2}-64}}{T}$, compute the l... | math | {
"ground_truth": "$\\frac{1}{3}$",
"style": "rule"
} | {
"index": 487,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=\\frac{9}{17}$. When $T$ is expressed as a reduced fraction, let $m$ and $... | math | {
"ground_truth": "450",
"style": "rule"
} | {
"index": 488,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=-14$, and let $d=|T|$. A person whose birthday falls between July 23 and A... | math | {
"ground_truth": "$\\frac{9}{17}$",
"style": "rule"
} | {
"index": 489,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=-10$. Given that $\\log _{2} 4^{8 !}+\\log _{4} 2^{8 !}=6 ! \\cdot T \\cdo... | math | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": 490,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=20$. For some real constants $a$ and $b$, the solution sets of the equatio... | math | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": 491,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let T be a rational number, and let $K=T-2$. If $K$ workers can produce 9 widgets... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 492,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $T=2018$, and append the digits of $T$ to $\\underline{A} \\underline{A} \\un... | math | {
"ground_truth": "14",
"style": "rule"
} | {
"index": 493,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Given that April $1^{\\text {st }}, 2012$ fell on a Sunday, what is the next year... | math | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": 494,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Let $p$ be a prime number. If $p$ years ago, the ages of three children formed a ... | math | {
"ground_truth": "28",
"style": "rule"
} | {
"index": 495,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Define a reverse prime to be a positive integer $N$ such that when the digits of ... | math | {
"ground_truth": "79",
"style": "rule"
} | {
"index": 496,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Some students in a gym class are wearing blue jerseys, and the rest are wearing r... | math | {
"ground_truth": "7",
"style": "rule"
} | {
"index": 497,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "Point $P$ is on the hypotenuse $\\overline{E N}$ of right triangle $B E N$ such t... | math | {
"ground_truth": "$\\frac{1}{60}$",
"style": "rule"
} | {
"index": 498,
"split": "default"
} |
olympiad_bench | [
{
"content": "You are Qwen, created by Alibaba Cloud. You are a helpful assistant. Solve the following math problem step by step. Put your final answer within <answer></answer> tags.",
"role": "system"
},
{
"content": "$\\quad$ Compute all real values of $x$ such that $\\log _{2}\\left(\\log _{2} x\... | math | {
"ground_truth": "$\\sqrt{2}$",
"style": "rule"
} | {
"index": 499,
"split": "default"
} |
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