data_source stringclasses 1 value | answer stringlengths 1 204 | source_dataset stringclasses 4 values | prompt listlengths 2 2 | ability stringclasses 1 value | reward_model dict | extra_info dict |
|---|---|---|---|---|---|---|
numina_math | 3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "At present, the sum of the ages of a husband and wife, $P$, is six times the sum of the ages of their children, $C$. Two years ago, the sum of the ages of the husband and wife wa... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"index": 900,
"split": "test"
} |
numina_math | 9 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of $x$ such that $\\log _{2}\\left(\\log _{2}(2 x-2)\\right)=2$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "9",
"style": "rule"
} | {
"index": 901,
"split": "test"
} |
numina_math | 210 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(x)=2^{k x}+9$, where $k$ is a real number. If $f(3): f(6)=1: 3$, determine the value of $f(9)-f(3)$. Let's think step by step and output the final answer within \\boxed{}.... | math | {
"ground_truth": "210",
"style": "rule"
} | {
"index": 902,
"split": "test"
} |
numina_math | (-\infty,-5) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine, with justification, all values of $k$ for which $y=x^{2}-4$ and $y=2|x|+k$ do not intersect. Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "(-\\infty,-5)",
"style": "rule"
} | {
"index": 903,
"split": "test"
} |
numina_math | 13 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $2 \\leq x \\leq 5$ and $10 \\leq y \\leq 20$, what is the maximum value of $15-\\frac{y}{x}$ ? Let's think step by step and output the final answer within \\boxed{}.",
"r... | math | {
"ground_truth": "13",
"style": "rule"
} | {
"index": 904,
"split": "test"
} |
numina_math | -84 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The functions $f$ and $g$ satisfy\n\n$$\n\\begin{aligned}\n& f(x)+g(x)=3 x+5 \\\\\n& f(x)-g(x)=5 x+7\n\\end{aligned}\n$$\n\nfor all values of $x$. Determine the value of $2 f(2) ... | math | {
"ground_truth": "-84",
"style": "rule"
} | {
"index": 905,
"split": "test"
} |
numina_math | $\frac{2}{5}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three different numbers are chosen at random from the set $\\{1,2,3,4,5\\}$.\n\nThe numbers are arranged in increasing order.\n\nWhat is the probability that the resulting sequen... | math | {
"ground_truth": "$\\frac{2}{5}$",
"style": "rule"
} | {
"index": 906,
"split": "test"
} |
numina_math | 48 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the largest two-digit number that becomes $75 \\%$ greater when its digits are reversed? Let's think step by step and output the final answer within \\boxed{}.",
"rol... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 907,
"split": "test"
} |
numina_math | 45 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Serge likes to paddle his raft down the Speed River from point $A$ to point $B$. The speed of the current in the river is always the same. When Serge paddles, he always paddles a... | math | {
"ground_truth": "45",
"style": "rule"
} | {
"index": 908,
"split": "test"
} |
numina_math | $\frac{32}{9}$,$\frac{1}{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square $O P Q R$ has vertices $O(0,0), P(0,8), Q(8,8)$, and $R(8,0)$. The parabola with equation $y=a(x-2)(x-6)$ intersects the sides of the square $O P Q R$ at points $K, L, M$,... | math | {
"ground_truth": "$\\frac{32}{9}$,$\\frac{1}{2}$",
"style": "rule"
} | {
"index": 909,
"split": "test"
} |
numina_math | 62.5% | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A 75 year old person has a $50 \\%$ chance of living at least another 10 years.\n\nA 75 year old person has a $20 \\%$ chance of living at least another 15 years. An 80 year old ... | math | {
"ground_truth": "62.5%",
"style": "rule"
} | {
"index": 910,
"split": "test"
} |
numina_math | 1000 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all values of $x$ for which $2^{\\log _{10}\\left(x^{2}\\right)}=3\\left(2^{1+\\log _{10} x}\\right)+16$. Let's think step by step and output the final answer within \\... | math | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": 911,
"split": "test"
} |
numina_math | 4090 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Sieve of Sundaram uses the following infinite table of positive integers:\n\n| 4 | 7 | 10 | 13 | $\\cdots$ |\n| :---: | :---: | :---: | :---: | :---: |\n| 7 | 12 | 17 | 22 | ... | math | {
"ground_truth": "4090",
"style": "rule"
} | {
"index": 912,
"split": "test"
} |
numina_math | $2RC+R+C$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The Sieve of Sundaram uses the following infinite table of positive integers:\n\n| 4 | 7 | 10 | 13 | $\\cdots$ |\n| :---: | :---: | :---: | :---: | :---: |\n| 7 | 12 | 17 | 22 | ... | math | {
"ground_truth": "$2RC+R+C$",
"style": "rule"
} | {
"index": 913,
"split": "test"
} |
numina_math | 4085 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\lfloor x\\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\\lfloor 3.1\\rfloor=3$ and $\\lfloor-1.4\\rfloor=-2$.\n\nSuppose that $f(n)=2 n-\\l... | math | {
"ground_truth": "4085",
"style": "rule"
} | {
"index": 914,
"split": "test"
} |
numina_math | 55 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\lfloor x\\rfloor$ denote the greatest integer less than or equal to $x$. For example, $\\lfloor 3.1\\rfloor=3$ and $\\lfloor-1.4\\rfloor=-2$.\n\nSuppose that $f(n)=2 n-\\l... | math | {
"ground_truth": "55",
"style": "rule"
} | {
"index": 915,
"split": "test"
} |
numina_math | $\frac{14}{15}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Six tickets numbered 1 through 6 are placed in a box. Two tickets are randomly selected and removed together. What is the probability that the smaller of the two numbers on the t... | math | {
"ground_truth": "$\\frac{14}{15}$",
"style": "rule"
} | {
"index": 916,
"split": "test"
} |
numina_math | (6,5) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A goat starts at the origin $(0,0)$ and then makes several moves. On move 1 , it travels 1 unit up to $(0,1)$. On move 2 , it travels 2 units right to $(2,1)$. On move 3 , it tra... | math | {
"ground_truth": "(6,5)",
"style": "rule"
} | {
"index": 917,
"split": "test"
} |
numina_math | 1 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all possible values of $r$ such that the three term geometric sequence 4, $4 r, 4 r^{2}$ is also an arithmetic sequence.\n\n(An arithmetic sequence is a sequence in whi... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"index": 918,
"split": "test"
} |
numina_math | 5,1 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $f(x)=\\sin ^{2} x-2 \\sin x+2$, what are the minimum and maximum values of $f(x)$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user... | math | {
"ground_truth": "5,1",
"style": "rule"
} | {
"index": 919,
"split": "test"
} |
numina_math | 1002001 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the sum of the digits of the integer equal to $\\left(10^{3}+1\\right)^{2}$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "1002001",
"style": "rule"
} | {
"index": 920,
"split": "test"
} |
numina_math | $8 \%$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bakery sells small and large cookies. Before a price increase, the price of each small cookie is $\\$ 1.50$ and the price of each large cookie is $\\$ 2.00$. The price of each ... | math | {
"ground_truth": "$8 \\%$",
"style": "rule"
} | {
"index": 921,
"split": "test"
} |
numina_math | 7,14,18 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Qing is twice as old as Rayna. Qing is 4 years younger than Paolo. The average age of Paolo, Qing and Rayna is 13. Determine their ages. Let's think step by step and output the f... | math | {
"ground_truth": "7,14,18",
"style": "rule"
} | {
"index": 922,
"split": "test"
} |
numina_math | 48 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The parabola with equation $y=-2 x^{2}+4 x+c$ has vertex $V(1,18)$. The parabola intersects the $y$-axis at $D$ and the $x$-axis at $E$ and $F$. Determine the area of $\\triangle... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"index": 923,
"split": "test"
} |
numina_math | $\frac{58}{3}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $3\\left(8^{x}\\right)+5\\left(8^{x}\\right)=2^{61}$, what is the value of the real number $x$ ? Let's think step by step and output the final answer within \\boxed{}.",
"... | math | {
"ground_truth": "$\\frac{58}{3}$",
"style": "rule"
} | {
"index": 924,
"split": "test"
} |
numina_math | 1,-1,7,-7 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For some real numbers $m$ and $n$, the list $3 n^{2}, m^{2}, 2(n+1)^{2}$ consists of three consecutive integers written in increasing order. Determine all possible values of $m$.... | math | {
"ground_truth": "1,-1,7,-7",
"style": "rule"
} | {
"index": 925,
"split": "test"
} |
numina_math | (-7,-1) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Chinara starts with the point $(3,5)$, and applies the following three-step process, which we call $\\mathcal{P}$ :\n\nStep 1: Reflect the point in the $x$-axis.\n\nStep 2: Trans... | math | {
"ground_truth": "(-7,-1)",
"style": "rule"
} | {
"index": 926,
"split": "test"
} |
numina_math | 20 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $n>5$ and that the numbers $t_{1}, t_{2}, t_{3}, \\ldots, t_{n-2}, t_{n-1}, t_{n}$ form an arithmetic sequence with $n$ terms. If $t_{3}=5, t_{n-2}=95$, and the sum ... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 927,
"split": "test"
} |
numina_math | $a=2$, $a=-6-4 \sqrt{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $a$ and $r$ are real numbers. A geometric sequence with first term $a$ and common ratio $r$ has 4 terms. The sum of this geometric sequence is $6+6 \\sqrt{2}$. A sec... | math | {
"ground_truth": "$a=2$, $a=-6-4 \\sqrt{2}$",
"style": "rule"
} | {
"index": 928,
"split": "test"
} |
numina_math | $\frac{4}{7}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bag contains 3 green balls, 4 red balls, and no other balls. Victor removes balls randomly from the bag, one at a time, and places them on a table. Each ball in the bag is equa... | math | {
"ground_truth": "$\\frac{4}{7}$",
"style": "rule"
} | {
"index": 929,
"split": "test"
} |
numina_math | $\frac{1}{6} \pi, \frac{5}{6} \pi, \frac{1}{4} \pi, \frac{3}{4} \pi$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $f(a)=2 a^{2}-3 a+1$ for all real numbers $a$ and $g(b)=\\log _{\\frac{1}{2}} b$ for all $b>0$. Determine all $\\theta$ with $0 \\leq \\theta \\leq 2 \\pi$ for which... | math | {
"ground_truth": "$\\frac{1}{6} \\pi, \\frac{5}{6} \\pi, \\frac{1}{4} \\pi, \\frac{3}{4} \\pi$",
"style": "rule"
} | {
"index": 930,
"split": "test"
} |
numina_math | $(3,2),(-3,2),(3,-2),(-3,-2)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $a=5$ and $b=4$. Determine all pairs of integers $(K, L)$ for which $K^{2}+3 L^{2}=a^{2}+b^{2}-a b$. Let's think step by step and output the final answer within \\bo... | math | {
"ground_truth": "$(3,2),(-3,2),(3,-2),(-3,-2)$",
"style": "rule"
} | {
"index": 931,
"split": "test"
} |
numina_math | $(-\sqrt{11},-2)\cup (\sqrt{11},9)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all values of $x$ for which $0<\\frac{x^{2}-11}{x+1}<7$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "$(-\\sqrt{11},-2)\\cup (\\sqrt{11},9)$",
"style": "rule"
} | {
"index": 932,
"split": "test"
} |
numina_math | 34 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The numbers $a_{1}, a_{2}, a_{3}, \\ldots$ form an arithmetic sequence with $a_{1} \\neq a_{2}$. The three numbers $a_{1}, a_{2}, a_{6}$ form a geometric sequence in that order. ... | math | {
"ground_truth": "34",
"style": "rule"
} | {
"index": 933,
"split": "test"
} |
numina_math | 1,2,5,8,9 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For some positive integers $k$, the parabola with equation $y=\\frac{x^{2}}{k}-5$ intersects the circle with equation $x^{2}+y^{2}=25$ at exactly three distinct points $A, B$ and... | math | {
"ground_truth": "1,2,5,8,9",
"style": "rule"
} | {
"index": 934,
"split": "test"
} |
numina_math | $(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the following system of equations in which all logarithms have base 10:\n\n$$\n\\begin{aligned}\n(\\log x)(\\log y)-3 \\log 5 y-\\log 8 x & =a \\\\\n(\\log y)(\\log z)-4... | math | {
"ground_truth": "$(10^{4}, 10^{3}, 10^{10}),(10^{2}, 10^{-1}, 10^{-2})$",
"style": "rule"
} | {
"index": 935,
"split": "test"
} |
numina_math | $\frac{11}{36}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Two fair dice, each having six faces numbered 1 to 6 , are thrown. What is the probability that the product of the two numbers on the top faces is divisible by 5 ? Let's think st... | math | {
"ground_truth": "$\\frac{11}{36}$",
"style": "rule"
} | {
"index": 936,
"split": "test"
} |
numina_math | $(3,0),(-3,1)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $f(x)=x^{2}-x+2, g(x)=a x+b$, and $f(g(x))=9 x^{2}-3 x+2$, determine all possible ordered pairs $(a, b)$ which satisfy this relationship. Let's think step by step and output t... | math | {
"ground_truth": "$(3,0),(-3,1)$",
"style": "rule"
} | {
"index": 937,
"split": "test"
} |
numina_math | 60 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Digital images consist of a very large number of equally spaced dots called pixels The resolution of an image is the number of pixels/cm in each of the horizontal and vertical di... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"index": 938,
"split": "test"
} |
numina_math | -3,0 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $T=x^{2}+\\frac{1}{x^{2}}$, determine the values of $b$ and $c$ so that $x^{6}+\\frac{1}{x^{6}}=T^{3}+b T+c$ for all non-zero real numbers $x$. Let's think step by step and ou... | math | {
"ground_truth": "-3,0",
"style": "rule"
} | {
"index": 939,
"split": "test"
} |
numina_math | (4,2,3,2,4,3,1,1),(1,1,3,4,2,3,2,4),(4,1,1,3,4,2,3,2),(2,3,2,4,3,1,1,4),(3,4,2,3,2,4,1,1),(1,1,4,2,3,2,4,3) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A Skolem sequence of order $n$ is a sequence $\\left(s_{1}, s_{2}, \\ldots, s_{2 n}\\right)$ of $2 n$ integers satisfying the conditions:\n\ni) for every $k$ in $\\{1,2,3, \\ldot... | math | {
"ground_truth": "(4,2,3,2,4,3,1,1),(1,1,3,4,2,3,2,4),(4,1,1,3,4,2,3,2),(2,3,2,4,3,1,1,4),(3,4,2,3,2,4,1,1),(1,1,4,2,3,2,4,3)",
"style": "rule"
} | {
"index": 940,
"split": "test"
} |
numina_math | (7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A Skolem sequence of order $n$ is a sequence $\\left(s_{1}, s_{2}, \\ldots, s_{2 n}\\right)$ of $2 n$ integers satisfying the conditions:\n\ni) for every $k$ in $\\{1,2,3, \\ldot... | math | {
"ground_truth": "(7,5,1,1,9,3,5,7,3,8,6,4,2,9,2,4,6,8)",
"style": "rule"
} | {
"index": 941,
"split": "test"
} |
numina_math | 623 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The three-digit positive integer $m$ is odd and has three distinct digits. If the hundreds digit of $m$ equals the product of the tens digit and ones (units) digit of $m$, what i... | math | {
"ground_truth": "623",
"style": "rule"
} | {
"index": 942,
"split": "test"
} |
numina_math | 40 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to... | math | {
"ground_truth": "40",
"style": "rule"
} | {
"index": 943,
"split": "test"
} |
numina_math | 1, 3, 5, 15 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $n$ is a positive integer and that the value of $\\frac{n^{2}+n+15}{n}$ is an integer. Determine all possible values of $n$. Let's think step by step and output the ... | math | {
"ground_truth": "1, 3, 5, 15",
"style": "rule"
} | {
"index": 944,
"split": "test"
} |
numina_math | 340 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Ada starts with $x=10$ and $y=2$, and applies the following process:\n\nStep 1: Add $x$ and $y$. Let $x$ equal the result. The value of $y$ does not change. Step 2: Multiply $x$ ... | math | {
"ground_truth": "340",
"style": "rule"
} | {
"index": 945,
"split": "test"
} |
numina_math | -2,-1,1,2 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all integers $k$, with $k \\neq 0$, for which the parabola with equation $y=k x^{2}+6 x+k$ has two distinct $x$-intercepts. Let's think step by step and output the fina... | math | {
"ground_truth": "-2,-1,1,2",
"style": "rule"
} | {
"index": 946,
"split": "test"
} |
numina_math | $\frac{19}{34}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The positive integers $a$ and $b$ have no common divisor larger than 1 . If the difference between $b$ and $a$ is 15 and $\\frac{5}{9}<\\frac{a}{b}<\\frac{4}{7}$, what is the val... | math | {
"ground_truth": "$\\frac{19}{34}$",
"style": "rule"
} | {
"index": 947,
"split": "test"
} |
numina_math | $-\frac{30}{17}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A geometric sequence has first term 10 and common ratio $\\frac{1}{2}$.\n\nAn arithmetic sequence has first term 10 and common difference $d$.\n\nThe ratio of the 6th term in the... | math | {
"ground_truth": "$-\\frac{30}{17}$",
"style": "rule"
} | {
"index": 948,
"split": "test"
} |
numina_math | 5 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive real number $x$, define $f(x)$ to be the number of prime numbers $p$ that satisfy $x \\leq p \\leq x+10$. What is the value of $f(f(20))$ ? Let's think step by ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 949,
"split": "test"
} |
numina_math | (1,-4,-2),(3,2,3),(13,2,-2) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all triples $(x, y, z)$ of real numbers that satisfy the following system of equations:\n\n$$\n\\begin{aligned}\n(x-1)(y-2) & =0 \\\\\n(x-3)(z+2) & =0 \\\\\nx+y z & =9\... | math | {
"ground_truth": "(1,-4,-2),(3,2,3),(13,2,-2)",
"style": "rule"
} | {
"index": 950,
"split": "test"
} |
numina_math | $4 \pi^{2}-1$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that the function $g$ satisfies $g(x)=2 x-4$ for all real numbers $x$ and that $g^{-1}$ is the inverse function of $g$. Suppose that the function $f$ satisfies $g\\left(f... | math | {
"ground_truth": "$4 \\pi^{2}-1$",
"style": "rule"
} | {
"index": 951,
"split": "test"
} |
numina_math | $(45^{\circ}, 60^{\circ}),(135^{\circ}, 60^{\circ})$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all pairs of angles $(x, y)$ with $0^{\\circ} \\leq x<180^{\\circ}$ and $0^{\\circ} \\leq y<180^{\\circ}$ that satisfy the following system of equations:\n\n$$\n\\begin... | math | {
"ground_truth": "$(45^{\\circ}, 60^{\\circ}),(135^{\\circ}, 60^{\\circ})$",
"style": "rule"
} | {
"index": 952,
"split": "test"
} |
numina_math | $\frac{1-p^{2}}{3}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Four tennis players Alain, Bianca, Chen, and Dave take part in a tournament in which a total of three matches are played. First, two players are chosen randomly to play each othe... | math | {
"ground_truth": "$\\frac{1-p^{2}}{3}$",
"style": "rule"
} | {
"index": 953,
"split": "test"
} |
numina_math | $\frac{41}{12}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three microphones $A, B$ and $C$ are placed on a line such that $A$ is $1 \\mathrm{~km}$ west of $B$ and $C$ is $2 \\mathrm{~km}$ east of $B$. A large explosion occurs at a point... | math | {
"ground_truth": "$\\frac{41}{12}$",
"style": "rule"
} | {
"index": 954,
"split": "test"
} |
numina_math | (1,7,103, 105), (3, 5, 101, 107) | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Kerry has a list of $n$ integers $a_{1}, a_{2}, \\ldots, a_{n}$ satisfying $a_{1} \\leq a_{2} \\leq \\ldots \\leq a_{n}$. Kerry calculates the pairwise sums of all $m=\\frac{1}{2... | math | {
"ground_truth": "(1,7,103, 105), (3, 5, 101, 107)",
"style": "rule"
} | {
"index": 955,
"split": "test"
} |
numina_math | $-1$,$2$,$-2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all values of $x$ for which $\\frac{x^{2}+x+4}{2 x+1}=\\frac{4}{x}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "$-1$,$2$,$-2$",
"style": "rule"
} | {
"index": 956,
"split": "test"
} |
numina_math | 8 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the number of positive divisors of 900, including 1 and 900, that are perfect squares. (A positive divisor of 900 is a positive integer that divides exactly into 900.) ... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"index": 957,
"split": "test"
} |
numina_math | $8$,$4$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Points $A(k, 3), B(3,1)$ and $C(6, k)$ form an isosceles triangle. If $\\angle A B C=\\angle A C B$, determine all possible values of $k$. Let's think step by step and output the... | math | {
"ground_truth": "$8$,$4$",
"style": "rule"
} | {
"index": 958,
"split": "test"
} |
numina_math | 17.5% | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A chemist has three bottles, each containing a mixture of acid and water:\n\n- bottle A contains $40 \\mathrm{~g}$ of which $10 \\%$ is acid,\n- bottle B contains $50 \\mathrm{~g... | math | {
"ground_truth": "17.5%",
"style": "rule"
} | {
"index": 959,
"split": "test"
} |
numina_math | 10 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $x$ and $y$ are real numbers with $3 x+4 y=10$. Determine the minimum possible value of $x^{2}+16 y^{2}$. Let's think step by step and output the final answer within... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"index": 960,
"split": "test"
} |
numina_math | 26 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bag contains 40 balls, each of which is black or gold. Feridun reaches into the bag and randomly removes two balls. Each ball in the bag is equally likely to be removed. If the... | math | {
"ground_truth": "26",
"style": "rule"
} | {
"index": 961,
"split": "test"
} |
numina_math | 20 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The geometric sequence with $n$ terms $t_{1}, t_{2}, \\ldots, t_{n-1}, t_{n}$ has $t_{1} t_{n}=3$. Also, the product of all $n$ terms equals 59049 (that is, $t_{1} t_{2} \\cdots ... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"index": 962,
"split": "test"
} |
numina_math | 4027 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $\\frac{(x-2013)(y-2014)}{(x-2013)^{2}+(y-2014)^{2}}=-\\frac{1}{2}$, what is the value of $x+y$ ? Let's think step by step and output the final answer within \\boxed{}.",
... | math | {
"ground_truth": "4027",
"style": "rule"
} | {
"index": 963,
"split": "test"
} |
numina_math | $10^{100}$,$10^{1 / 100}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all real numbers $x$ for which\n\n$$\n\\left(\\log _{10} x\\right)^{\\log _{10}\\left(\\log _{10} x\\right)}=10000\n$$ Let's think step by step and output the final ans... | math | {
"ground_truth": "$10^{100}$,$10^{1 / 100}$",
"style": "rule"
} | {
"index": 964,
"split": "test"
} |
numina_math | $221,$8$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Without using a calculator, determine positive integers $m$ and $n$ for which\n\n$$\n\\sin ^{6} 1^{\\circ}+\\sin ^{6} 2^{\\circ}+\\sin ^{6} 3^{\\circ}+\\cdots+\\sin ^{6} 87^{\\ci... | math | {
"ground_truth": "$221,$8$",
"style": "rule"
} | {
"index": 965,
"split": "test"
} |
numina_math | 202 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $f(n)$ be the number of positive integers that have exactly $n$ digits and whose digits have a sum of 5. Determine, with proof, how many of the 2014 integers $f(1), f(2), \\l... | math | {
"ground_truth": "202",
"style": "rule"
} | {
"index": 966,
"split": "test"
} |
numina_math | 1000 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $\\log _{10} x=3+\\log _{10} y$, what is the value of $\\frac{x}{y}$ ? Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "1000",
"style": "rule"
} | {
"index": 967,
"split": "test"
} |
numina_math | $\frac{97}{36}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $x+\\frac{1}{x}=\\frac{13}{6}$, determine all values of $x^{2}+\\frac{1}{x^{2}}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
... | math | {
"ground_truth": "$\\frac{97}{36}$",
"style": "rule"
} | {
"index": 968,
"split": "test"
} |
numina_math | $\frac{2}{9}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A die, with the numbers $1,2,3,4,6$, and 8 on its six faces, is rolled. After this roll, if an odd number appears on the top face, all odd numbers on the die are doubled. If an e... | math | {
"ground_truth": "$\\frac{2}{9}$",
"style": "rule"
} | {
"index": 969,
"split": "test"
} |
numina_math | 16,3,1,1 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The table below gives the final standings for seven of the teams in the English Cricket League in 1998. At the end of the year, each team had played 17 matches and had obtained t... | math | {
"ground_truth": "16,3,1,1",
"style": "rule"
} | {
"index": 970,
"split": "test"
} |
numina_math | 4.25 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\lfloor x\\rfloor$ represent the greatest integer which is less than or equal to $x$. For example, $\\lfloor 3\\rfloor=3,\\lfloor 2.6\\rfloor=2$. If $x$ is positive and $x\... | math | {
"ground_truth": "4.25",
"style": "rule"
} | {
"index": 971,
"split": "test"
} |
numina_math | 6 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cube has edges of length $n$, where $n$ is an integer. Three faces, meeting at a corner, are painted red. The cube is then cut into $n^{3}$ smaller cubes of unit length. If exa... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 972,
"split": "test"
} |
numina_math | 7,4 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Thurka bought some stuffed goats and some toy helicopters. She paid a total of $\\$ 201$. She did not buy partial goats or partial helicopters. Each stuffed goat cost $\\$ 19$ an... | math | {
"ground_truth": "7,4",
"style": "rule"
} | {
"index": 973,
"split": "test"
} |
numina_math | -6,-8,-10 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all real values of $x$ for which $(x+8)^{4}=(2 x+16)^{2}$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "-6,-8,-10",
"style": "rule"
} | {
"index": 974,
"split": "test"
} |
numina_math | $g(x)=x^2-2x+2$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $f(x)=2 x+1$ and $g(f(x))=4 x^{2}+1$, determine an expression for $g(x)$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "$g(x)=x^2-2x+2$",
"style": "rule"
} | {
"index": 975,
"split": "test"
} |
numina_math | 5 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A geometric sequence has 20 terms.\n\nThe sum of its first two terms is 40 .\n\nThe sum of its first three terms is 76 .\n\nThe sum of its first four terms is 130 .\n\nDetermine ... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"index": 976,
"split": "test"
} |
numina_math | $1$,$\frac{3 + \sqrt{21}}{2}$,$\frac{3 - \sqrt{21}}{2}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all real values of $x$ for which $3^{(x-1)} 9^{\\frac{3}{2 x^{2}}}=27$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
... | math | {
"ground_truth": "$1$,$\\frac{3 + \\sqrt{21}}{2}$,$\\frac{3 - \\sqrt{21}}{2}$",
"style": "rule"
} | {
"index": 977,
"split": "test"
} |
numina_math | $(1,0),(\frac{1}{100},-8),(100,8)$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all points $(x, y)$ where the two curves $y=\\log _{10}\\left(x^{4}\\right)$ and $y=\\left(\\log _{10} x\\right)^{3}$ intersect. Let's think step by step and output the... | math | {
"ground_truth": "$(1,0),(\\frac{1}{100},-8),(100,8)$",
"style": "rule"
} | {
"index": 978,
"split": "test"
} |
numina_math | $\frac{27}{64}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Oi-Lam tosses three fair coins and removes all of the coins that come up heads. George then tosses the coins that remain, if any. Determine the probability that George tosses exa... | math | {
"ground_truth": "$\\frac{27}{64}$",
"style": "rule"
} | {
"index": 979,
"split": "test"
} |
numina_math | $4^{\circ}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Ross starts with an angle of measure $8^{\\circ}$ and doubles it 10 times until he obtains $8192^{\\circ}$. He then adds up the reciprocals of the sines of these 11 angles. That ... | math | {
"ground_truth": "$4^{\\circ}$",
"style": "rule"
} | {
"index": 980,
"split": "test"
} |
numina_math | 2,4,3 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triang... | math | {
"ground_truth": "2,4,3",
"style": "rule"
} | {
"index": 981,
"split": "test"
} |
numina_math | 309 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each positive integer $n$, let $T(n)$ be the number of triangles with integer side lengths, positive area, and perimeter $n$. For example, $T(6)=1$ since the only such triang... | math | {
"ground_truth": "309",
"style": "rule"
} | {
"index": 982,
"split": "test"
} |
numina_math | $\frac{3}{4}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose $0^{\\circ}<x<90^{\\circ}$ and $2 \\sin ^{2} x+\\cos ^{2} x=\\frac{25}{16}$. What is the value of $\\sin x$ ? Let's think step by step and output the final answer within ... | math | {
"ground_truth": "$\\frac{3}{4}$",
"style": "rule"
} | {
"index": 983,
"split": "test"
} |
numina_math | 153 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The first term of a sequence is 2007. Each term, starting with the second, is the sum of the cubes of the digits of the previous term. What is the 2007th term? Let's think step b... | math | {
"ground_truth": "153",
"style": "rule"
} | {
"index": 984,
"split": "test"
} |
numina_math | 5,15 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Sequence A has $n$th term $n^{2}-10 n+70$.\n\n(The first three terms of sequence $\\mathrm{A}$ are $61,54,49$. )\n\nSequence B is an arithmetic sequence with first term 5 and com... | math | {
"ground_truth": "5,15",
"style": "rule"
} | {
"index": 985,
"split": "test"
} |
numina_math | 6 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all values of $x$ for which $2+\\sqrt{x-2}=x-2$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "6",
"style": "rule"
} | {
"index": 986,
"split": "test"
} |
numina_math | $100,\frac{1}{100}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all values of $x$ for which $(\\sqrt{x})^{\\log _{10} x}=100$. Let's think step by step and output the final answer within \\boxed{}.",
"role": "user"
}
] | math | {
"ground_truth": "$100,\\frac{1}{100}$",
"style": "rule"
} | {
"index": 987,
"split": "test"
} |
numina_math | 23 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $f(x)=x^{2}+(2 n-1) x+\\left(n^{2}-22\\right)$ for some integer $n$. What is the smallest positive integer $n$ for which $f(x)$ has no real roots? Let's think step b... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"index": 988,
"split": "test"
} |
numina_math | $\frac{10}{21}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A bag contains 3 red marbles and 6 blue marbles. Akshan removes one marble at a time until the bag is empty. Each marble that they remove is chosen randomly from the remaining ma... | math | {
"ground_truth": "$\\frac{10}{21}$",
"style": "rule"
} | {
"index": 989,
"split": "test"
} |
numina_math | 417 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine the number of quadruples of positive integers $(a, b, c, d)$ with $a<b<c<d$ that satisfy both of the following system of equations:\n\n$$\n\\begin{aligned}\na c+a d+b c... | math | {
"ground_truth": "417",
"style": "rule"
} | {
"index": 990,
"split": "test"
} |
numina_math | 222 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $\\triangle A B C$ is right-angled at $B$ and has $A B=n(n+1)$ and $A C=(n+1)(n+4)$, where $n$ is a positive integer. Determine the number of positive integers $n<10... | math | {
"ground_truth": "222",
"style": "rule"
} | {
"index": 991,
"split": "test"
} |
numina_math | $[\frac{1}{2}, 8]$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all real values of $x$ for which\n\n$$\n\\sqrt{\\log _{2} x \\cdot \\log _{2}(4 x)+1}+\\sqrt{\\log _{2} x \\cdot \\log _{2}\\left(\\frac{x}{64}\\right)+9}=4\n$$ Let's t... | math | {
"ground_truth": "$[\\frac{1}{2}, 8]$",
"style": "rule"
} | {
"index": 992,
"split": "test"
} |
numina_math | 590 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For every real number $x$, define $\\lfloor x\\rfloor$ to be equal to the greatest integer less than or equal to $x$. (We call this the \"floor\" of $x$.) For example, $\\lfloor ... | math | {
"ground_truth": "590",
"style": "rule"
} | {
"index": 993,
"split": "test"
} |
numina_math | $p(x)=\frac{(x-1)(x-2)}{6}$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For every real number $x$, define $\\lfloor x\\rfloor$ to be equal to the greatest integer less than or equal to $x$. (We call this the \"floor\" of $x$.) For example, $\\lfloor ... | math | {
"ground_truth": "$p(x)=\\frac{(x-1)(x-2)}{6}$",
"style": "rule"
} | {
"index": 994,
"split": "test"
} |
numina_math | $166$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "One of the faces of a rectangular prism has area $27 \\mathrm{~cm}^{2}$. Another face has area $32 \\mathrm{~cm}^{2}$. If the volume of the prism is $144 \\mathrm{~cm}^{3}$, dete... | math | {
"ground_truth": "$166$",
"style": "rule"
} | {
"index": 995,
"split": "test"
} |
numina_math | $2,-1,-18$ | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The equations $y=a(x-2)(x+4)$ and $y=2(x-h)^{2}+k$ represent the same parabola. What are the values of $a, h$ and $k$ ? Let's think step by step and output the final answer withi... | math | {
"ground_truth": "$2,-1,-18$",
"style": "rule"
} | {
"index": 996,
"split": "test"
} |
numina_math | -5,7 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In an arithmetic sequence with 5 terms, the sum of the squares of the first 3 terms equals the sum of the squares of the last 2 terms. If the first term is 5 , determine all poss... | math | {
"ground_truth": "-5,7",
"style": "rule"
} | {
"index": 997,
"split": "test"
} |
numina_math | 1469 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Dan was born in a year between 1300 and 1400. Steve was born in a year between 1400 and 1500. Each was born on April 6 in a year that is a perfect square. Each lived for 110 year... | math | {
"ground_truth": "1469",
"style": "rule"
} | {
"index": 998,
"split": "test"
} |
numina_math | 1,3,9,11 | weqweasdas/olympiadbench | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all values of $k$ for which the points $A(1,2), B(11,2)$ and $C(k, 6)$ form a right-angled triangle. Let's think step by step and output the final answer within \\boxed... | math | {
"ground_truth": "1,3,9,11",
"style": "rule"
} | {
"index": 999,
"split": "test"
} |
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