data_source stringclasses 1
value | prompt listlengths 2 2 | ability stringclasses 1
value | reward_model dict | extra_info dict | sub_source stringclasses 6
values |
|---|---|---|---|---|---|
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the complex number $z=|\\frac{1-i}{1+i}|+2i$, determine the value of $z$.",
"role": "user"
}
] | math | {
"ground_truth": "1+2i",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $A,B,C$ be nodes of the lattice $Z\\times Z$ such that inside the triangle $ABC$ lies a unique node $P$ of the lattice. Denote $E = AP \\cap BC$ . Determine max $\\... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given three red beads, two white beads, and one blue bead are placed in line in random order, determine the probability that no two neighboring beads are the same color.",
"r... | math | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $\\{a_n\\}$ be a sequence of integers satisfying $(n-1)a_{n+1}=(n+1)a_n-2(n-1) \\forall n\\ge 1$ . If $2000|a_{1999}$ , find the smallest $n\\ge 2$ such that $2000|a_n... | math | {
"ground_truth": "249",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $N > 1$, then $\\sqrt[3]{N^2 \\sqrt[3]{N^3\\sqrt[3]{N^2}}}$ equals:\nA) $N^{\\frac{1}{9}}$\nB) $N^{\\frac{2}{9}}$\nC) $N^{\\frac{1}{3}}$\nD) $N^{\\frac{29}{27}}$\nE) $N^{\\fra... | math | {
"ground_truth": "N^{\\frac{29}{27}}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given Javier uses 5 pounds of meat to make 10 hamburgers for his family gathering, determine the amount of meat, in pounds, needed to make 40 hamburgers for a community event.",
... | math | {
"ground_truth": "20",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of $g(-2)$ if \\( g(x) = 2x^2 - 3x + 1 \\)?",
"role": "user"
}
] | math | {
"ground_truth": "15",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For $x \\in \\mathbb{R}$, the function $f(x)$ satisfies $f(-x) = f(x)$, and $f(x)$ is monotonically decreasing on $[0, +\\infty)$. Given $f(2) = 0$, determine the range of $x$ fo... | math | {
"ground_truth": "(-\\infty, -2) \\cup (2, +\\infty)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Each face of a regular tetrahedron is painted either red, white, or blue. Two colorings are considered indistinguishable if two congruent tetrahedra with those colorings can be r... | math | {
"ground_truth": "15",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider rectangle $ABCD$ with $AB = 6$ and $BC = 8$ . Pick points $E, F, G, H$ such that the angles $\\angle AEB, \\angle BFC, \\angle CGD, \\angle AHD$ are all right.... | math | {
"ground_truth": "98",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Food safety issues are increasingly attracting people's attention. The abuse of pesticides and chemical fertilizers poses certain health risks to the public. To provide consumers... | math | {
"ground_truth": "282",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The sequence \\(a_{n}\\) is constructed as follows: \\(a_{1}, a_{2}\\) are arbitrary real numbers, and for \\(n \\geq 3\\), the number \\(a_{n}\\) is the smallest of the numbers ... | math | {
"ground_truth": "21",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the functions $f(x)=x\\ln x$ and $g(x)=\\frac{x}{e^{x}}$.\n(I) Let $F(x)=f(x)-g(x)$. Determine the number of zeros of $F(x)$ in the interval $(1,2)$ and explain the reasoni... | math | {
"ground_truth": "x\\_1+x\\_2 > 2x\\_0",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given Ellen has a $3 \\times 7$ index card. If she shortens the length of one side of this card by $2$ inches and the other side by $1$ inch, the card would have an area of $10$ ... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let N be the third smallest positive integer that is divisible by every positive integer less than 8. Find the sum of the digits of N.",
"role": "user"
}
] | math | {
"ground_truth": "9",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( a_{i} \\) (where \\( i \\in \\mathbf{Z}_{+} \\) and \\( i \\leq 2020 \\)) be non-negative real numbers such that \\( \\sum_{i=1}^{2020} a_{i}=1 \\). Find the maximum valu... | math | {
"ground_truth": "\\frac{5}{11}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "We call $ A_1, A_2, \\ldots, A_n$ an $ n$ -division of $ A$ if\r\n\r\n(i) $ A_1 \\cap A_2 \\cap \\cdots \\cap A_n \\equal{} A$ ,\r\n(ii) $ A_i \\cap A_j \\neq \\emptyset$ ... | math | {
"ground_truth": "56",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the expression $\\[(3^2-3)-(4^2-4)+(5^2-5)-(6^2-6)\\], calculate its value.",
"role": "user"
}
] | math | {
"ground_truth": "-16",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a music club consisting of 80 members, 45 members play guitar, 30 members play piano, and 18 members play both instruments. How many members do not play either guitar or piano... | math | {
"ground_truth": "23",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the angles of a right triangle as \\(a^\\circ\\), \\(b^\\circ\\), and \\(90^\\circ\\). If \\(a\\) and \\(b\\) are both unique prime numbers such that \\(a > b+2\\), what... | math | {
"ground_truth": "D) 7",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a right triangle $ABC$ with $\\angle BAC = 90^\\circ$ and $\\angle ABC = 60^\\circ$. If the hypotenuse $BC$ is 30 units, find the area of triangle $ABC$.",
"role": "use... | math | {
"ground_truth": "112.5\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "If $-2$ degrees Celsius is denoted as $-2^{\\circ}\\mathrm{C}$, then $2$ degrees Celsius above zero is denoted as $^{\\circ}\\mathrm{C}$.",
"role": "user"
}
] | math | {
"ground_truth": "+2^{\\circ}\\mathrm{C}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A circle has a radius \\(3\\) units. Line segments of length \\(2\\) units each are tangent to the circle at one endpoint. Find the area of the region consisting of all such line... | math | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given two lines in a coordinate plane, where the first line passes through points (0, 5) and (10, 2) and the second line passes through points (2, 6) and (8, 1), find the area of... | math | {
"ground_truth": "C) 13.33",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( g(x) \\) be a polynomial with leading coefficient \\( 1 \\), whose roots are the reciprocals of the two roots of \\( f(x) = x^2 + px + q \\), where \\( p < q \\). Determi... | math | {
"ground_truth": "**C)** \\( \\frac{1+p+q}{q} \\)",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A $(2^n - 1) \\times (2^n +1)$ board is to be divided into rectangles with sides parallel to the sides of the board and integer side lengths such that the area of each rectangl... | math | {
"ground_truth": "2n",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( X = \\{1, 2, 3, \\ldots, 17\\} \\). Find the number of subsets \\( Y \\) of \\( X \\) with odd cardinalities.",
"role": "user"
}
] | math | {
"ground_truth": "65536",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "fantasticbobob is proctoring a room for the SiSiEyMB with $841$ seats arranged in $29$ rows and $29$ columns. The contestants sit down, take part $1$ of the contest, go o... | math | {
"ground_truth": "421",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x \\geq 0$, $f(x) = x\\ln(x+1)$. Find the expression for $f(x)$ when $x < 0$.",
"role": ... | math | {
"ground_truth": "f(x) = x\\ln(-x+1)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A cylinder and a cone entirely fit inside it are given. Calculate the ratio of the volume of the cone to the volume of the cylinder. The cylinder has a height of 10 cm and radius... | math | {
"ground_truth": "\\frac{1}{6}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "George and Henry start swimming from opposite ends of a 50-meter pool. After 2 minutes, they meet for the first time 20 meters from one of the ends. If both swimmers turn instant... | math | {
"ground_truth": "\\textbf{(C)} \\quad 4",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that convex quadrilateral ABCD has AB = 9 and CD = 12, diagoals AC and BD intersect at E, AC = 14, and triangle AED and triangle BEC have equal areas, determine the length ... | math | {
"ground_truth": "6",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\\pi$.",
"role"... | math | {
"ground_truth": "49.25\\pi",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a complex number $z$ satisfies $z\\cdot \\left(1+i\\right)=1-2i$, determine the imaginary part of the complex number $\\overline{z}$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a triangle $ABC, L$ and $K$ are the points of intersections of the angle bisectors of $\\angle ABC$ and $\\angle BAC$ with the segments $AC$ and $BC$ , respectively... | math | {
"ground_truth": "120^\\circ",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Collinear points $A$ , $B$ , and $C$ are given in the Cartesian plane such that $A= (a, 0)$ lies along the x-axis, $B$ lies along the line $y=x$ , $C$ lies along the l... | math | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the value of $\\sqrt{4^3 + 4^3 + 4^3}$?",
"role": "user"
}
] | math | {
"ground_truth": "8\\sqrt{3}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $w$, $x$, $y$, and $z$ be whole numbers. If $2^w \\cdot 3^x \\cdot 5^y \\cdot 7^z = 588$, then what does $2w + 3x + 5y + 7z$ equal?",
"role": "user"
}
] | math | {
"ground_truth": "21",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In a photo lab, there are 130 photographs, of which 35 are of the fourth grade, 30 of the fifth grade, 25 of the sixth grade, 20 of the seventh grade, and the rest are from the f... | math | {
"ground_truth": "77",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Suppose that $x,$ $y,$ and $z$ are three positive numbers that satisfy the equations $xyz=1,$ $x+\\frac{1}{z}=5,$ and $y+\\frac{1}{x}=29.$ Then $z+\\frac{1}{y}=\\fra... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all pairs of positive integers $ (m,n)$ such that $ mn \\minus{} 1$ divides $ (n^2 \\minus{} n \\plus{} 1)^2$ .\r\n\r\n*Aaron Pixton.*",
"role": "user"
}
] | math | {
"ground_truth": "(m, n) = (k^2 + 1, k)",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "At Pine Lake Summer Camp, $70\\%$ of the children play soccer, $50\\%$ of the children swim, and $30\\%$ of the soccer players swim. Find the percent of the non-swimmers that pla... | math | {
"ground_truth": "98\\%",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $P(x)$ be a real quadratic trinomial, so that for all $x\\in \\mathbb{R}$ the inequality $P(x^3+x)\\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$ .\n\n*Prop... | math | {
"ground_truth": "4",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Square corners, each 7 units on each side, are removed from a 25 unit by 35 unit rectangular sheet of cardboard. The sides of the remaining material are then folded up to form an... | math | {
"ground_truth": "C) 679 square units",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is $\\frac{2+4+6}{1+3+5} - \\frac{1+3+5}{2+4+6}$",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{7}{12}",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $-5$ is a solution to $x^2 + bx - 45 = 0$, what is the value of $b$?",
"role": "user"
}
] | math | {
"ground_truth": "-4",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $T_1$ and $T_2$ be the points of tangency of the excircles of a triangle $ABC$ with its sides $BC$ and $AC$ respectively. It is known that the reflection of the inc... | math | {
"ground_truth": "90^\\circ",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $x = \\frac{2}{9}$ is a solution to the equation $81x^2 + 220 = 196x - 15,$ what is the other value of $x$ that will solve the equation? Express your answer as a commo... | math | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Walking down Jane Street, Ralph passed four houses in a row, each painted a different color. He passed the orange house before the red house, and he passed the blue house before ... | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A square \\( S_1 \\) has an area of 25 square units. Each side is bisected, and a smaller square \\( S_2 \\) is constructed using these midpoints as vertices. This process is rep... | math | {
"ground_truth": "B) 3.125",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $n$ blue lines, no two of which are parallel and no three concurrent, be drawn on a plane. An intersection of two blue lines is called a blue point. Through any two blue po... | math | {
"ground_truth": "\\frac{n(n-1)(n-2)(n-3)(n-4)}{8}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A chef bakes a $24$-inch by $15$-inch pan of brownies. Each brownie is cut into pieces that measure $3$ inches by $2$ inches. Calculate the number of pieces of brownie the pan co... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Alex and Katy play a game on an \\(8 \\times 8\\) grid made of 64 unit cells. They take turns to play, with Alex going first. On Alex's turn, he writes 'A' in an empty cell. On K... | math | {
"ground_truth": "32",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The domain of the function $f(x)=\\ln (1-5^{x})$ is $(-\\infty, 0)$.",
"role": "user"
}
] | math | {
"ground_truth": "(-\\infty,0)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $p$ and $q$ be nonzero real numbers such that\n\\[(3 - 4i)(p + qi)\\] is pure imaginary. Find $\\frac{p}{q}$.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The graph of the function $y=\\sin x$ is translated according to the vector $\\overrightarrow{a}=(-\\frac{\\pi}{2}, 2)$ and then coincides with the graph of the function $g(x)$. ... | math | {
"ground_truth": "\\cos x+2",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For a positive integer $n$ , and a non empty subset $A$ of $\\{1,2,...,2n\\}$ , call $A$ good if the set $\\{u\\pm v|u,v\\in A\\}$ does not contain the set $\\{1,2,...,n... | math | {
"ground_truth": "\\frac{6}{5}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider polynomials $P$ of degree $2015$ , all of whose coefficients are in the set $\\{0,1,\\dots,2010\\}$ . Call such a polynomial *good* if for every integer $m$ , one o... | math | {
"ground_truth": "460",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the maximum value of real number $A$ such that $$ 3x^2 + y^2 + 1 \\geq A(x^2 + xy + x) $$ for all positive integers $x, y.$",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that two tangents to a circle are drawn from a point $A$, the points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2:3$. Find the degree mea... | math | {
"ground_truth": "36^\\circ",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "At a conference, 85% of the delegates know English, and 75% know Spanish. What proportion of the delegates know both languages?",
"role": "user"
}
] | math | {
"ground_truth": "60\\%",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the largest positive integer $n$ for which there is a unique integer $k$ such that $\\frac{8}{15} < \\frac{n}{n + k} < \\frac{7}{13}$ ?",
"role": "user"
}
] | math | {
"ground_truth": "112",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The smallest sum one can get by adding three different numbers from the set { 7, 25, -1, 12, -3 } is",
"role": "user"
}
] | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find all integer solutions to the equation $y^k = x^2 + x$ , where $k$ is a natural number greater than $1$ .",
"role": "user"
}
] | math | {
"ground_truth": "x = 0",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The sum of the first $n$ terms of the arithmetic sequences ${a_n}$ and ${b_n}$ are $S_n$ and $T_n$ respectively. If $$\\frac {S_{n}}{T_{n}}= \\frac {2n+1}{3n+2}$$, find the value... | math | {
"ground_truth": "\\frac{129}{130}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the points $(x_1, y_1), (x_2, y_2), (x_3, y_3), (x_4, y_4)$ where $x_1 < x_2 < x_3 < x_4$ and $x_4 - x_3 = x_3 - x_2 = x_2 - x_1 = d$, calculate the slope of the line that ... | math | {
"ground_truth": "\\frac{y_4 - y_1}{x_4 - x_1}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"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 base $7$ representation of $999_{10}$?",
"role": "user"
}
] | math | {
"ground_truth": "15",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region... | math | {
"ground_truth": "n = 108",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Vertex $A$ of $\\triangle ABC$ lies on the parabola defined by $y = 4x^2$ at the point $(2, 16)$. Points $B$ and $C$ are symmetric about the $y$-axis, and $\\overline{BC}$ is hor... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $U=\\{0,1,2,3\\}$, $A=\\{x\\in U|x^{2}+mx=0\\}$, and if $C_{U}A=\\{1,2\\}$, then the real number $m=$ _______",
"role": "user"
}
] | math | {
"ground_truth": "m=-3",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For how many $n$ in $\\{1, 2, 3, ..., 50 \\}$ is the tens digit of $n^2$ odd when $n$ ends in 3 or 7?",
"role": "user"
}
] | math | {
"ground_truth": "0",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function f has the property that for each real number x in its domain, 1/x is also in its domain and f(x) + f(1/x) = x. Determine the largest set of real numbers that c... | math | {
"ground_truth": "\\{-1, 1\\}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "For each value of $x$, $g(x)$ is defined to be the minimum value of the three numbers $3x + 3$, $\\frac{1}{3}x + 1$, and $-\\frac{2}{3}x + 8$. Find the maximum value of $g(x)$.",... | math | {
"ground_truth": "\\frac{10}{3}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "All divisors of a natural number \\( n \\) are listed in ascending order from 1 to \\( n \\). It turns out that the second-to-last number in the list is 101 times greater than th... | math | {
"ground_truth": "101^3",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "What is the maximum number of checkers that can be placed on a $6 \\times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) are on the sam... | math | {
"ground_truth": "12",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given sets $A$, $B$, and $C$, with $A \\subseteq B$ and $A \\subseteq C$, if $B = \\{0, 1, 2, 3, 4\\}$ and $C = \\{0, 2, 4, 8\\}$, then the number of sets $A$ that satisfy these ... | math | {
"ground_truth": "8",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that the functions $f(x)$ and $g(x)$ defined on $\\mathbb{R}$ are an odd function and an even function, respectively, and $f(x) + g(x) = x^{2} - 2x$, then $f(2) + g(1) = \\... | math | {
"ground_truth": "-3",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Monica produces a sequence of numbers beginning with a positive integer. She uses applicable rules based on the nature of each number:\n- **Rule 1**: If a number is less than 10,... | math | {
"ground_truth": "C) 115",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given real numbers $b_0,b_1,\\ldots, b_{2019}$ with $b_{2019}\\neq 0$ , let $z_1,z_2,\\ldots, z_{2019}$ be the roots in the complex plane of the polynomial\n\\[\nP(z) = \\su... | math | {
"ground_truth": "2019^{-2019^{-1}}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Consider the series $3 + \\frac{3}{3} + \\frac{1}{3} + \\frac{1}{9} + \\cdots$. Identify from the following statements which are correct:\n\n(1) The sum approaches a finite value... | math | {
"ground_truth": "A) Only 1",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A rhombus $\\mathcal R$ has short diagonal of length $1$ and long diagonal of length $2023$ . Let $\\mathcal R'$ be the rotation of $\\mathcal R$ by $90^\\circ$ about ... | math | {
"ground_truth": "\\frac{1}{2023}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given $S_n$ is the sum of the first $n$ terms of the sequence $\\{a_n\\}$, and the vectors $\\overrightarrow{a}=(a_{n}-1,-2)$, $\\overrightarrow{b}=(4,S_{n})$ satisfy $\\overrigh... | math | {
"ground_truth": "\\frac{31}{7}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The area of the plane region represented by the set \n$$\nA=\\left\\{\\left.(x, y)\\left|(x-t)^{2}+y^{2} \\leq\\left(1-\\frac{t}{2}\\right)^{2},\\right| t \\right\\rvert\\, \\leq... | math | {
"ground_truth": "4 \\sqrt{3} + \\frac{8 \\pi}{3}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x_0, a, b$ be reals given such that $b > 0$ and $x_0 \\ne 0$ . For every nonnegative integer $n$ a real value $x_{n+1}$ is chosen that satisfies $$ x^2_{n+1}= ax_... | math | {
"ground_truth": "x_0 \\frac{\\left( a + \\sqrt{a^2 + 4b} \\right)^{n+1} - \\left( a - \\sqrt{a^2 + 4b} \\right)^{n+1}}{2^{n+2} \\sqrt{a^2 + 4b}}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Real numbers $x,y,z$ satisfy the inequalities $$ x^2\\le y+z,\\qquad y^2\\le z+x\\qquad z^2\\le x+y. $$ Find the minimum and maximum possible values of $z$ .",
"role": "... | math | {
"ground_truth": "[-\\frac{1}{4}, 2]",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that $a \\geqslant 0$, the function $f(x)=( x^{2}-2a x) e^{x}$ is monotonically decreasing on the interval $[-1 , 1 ]$. Determine the range of values for $a$.",
"role":... | math | {
"ground_truth": "\\left[\\frac{3}{4},+\\infty \\right)",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Rani, James, and Fiona start a game with $$3$ each. A bell rings every $10$ seconds, and at each ring, each of the players who has more than $1 dollar will give $$1$ to one of th... | math | {
"ground_truth": "\\frac{2}{27}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the diagram below, $AB$ is 12 cm and $BC$ is 9 cm. Triangle $ABC$ is right-angled at $B$. In square centimeters, what is the area of triangle $ABC$?\n\n[asy]\ndefaultpen(linew... | math | {
"ground_truth": "54\\text{ cm}^2",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Does there exist a natural number $ n$ such that $ n>2$ and the sum of squares of\r\nsome $ n$ consecutive integers is a perfect square?",
"role": "user"
}
] | math | {
"ground_truth": "n = 2k+1",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Three circles with centers at $A, B,$ and $C$ have radii of $2, 3,$ and $4$ respectively. They all lie on the same side of line $m$ and are tangent to $m$ at points $A', B',$ and... | math | {
"ground_truth": "1",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Unconventional dice are to be designed such that the six faces are marked with numbers from $1$ to $6$ with $1$ and $2$ appearing on opposite faces. Further, each face is... | math | {
"ground_truth": "48",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the expression \\((ax+iby)(ax-iby)\\) where \\(a, b,x,y \\in \\mathbb{R}\\), simplify this expression.\nA) \\(a^2x^2 + b^2y^2\\)\nB) \\(-a^2x^2 - b^2y^2\\)\nC) \\(a^2x^2 - ... | math | {
"ground_truth": "C) \\(a^2x^2 - b^2y^2\\)",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "**T**wo ordered pairs $(a,b)$ and $(c,d)$ , where $a,b,c,d$ are real numbers, form a basis of the coordinate plane if $ad \\neq bc$ . Determine the number of ordered quadru... | math | {
"ground_truth": "63",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given the function $f(x)=2\\cos^2x-2\\sqrt{3}\\sin x\\cos x-1$ $(x\\in\\mathbb{R})$.\n\n$(1)$ Calculate the value of $f\\left(-\\dfrac{\\pi}{12}\\right)$;\n\n$(2)$ Find the small... | math | {
"ground_truth": "\\left[k\\pi-\\dfrac{2\\pi}{3},k\\pi-\\dfrac{\\pi}{6}\\right]",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Calculate the integral \\( I = \\iint_{\\sigma} \\left( x \\cos \\alpha + \\cos \\beta + x z^{2} \\cos \\gamma \\right) d \\sigma \\), where \\(\\sigma\\) is the part of the sphe... | math | {
"ground_truth": "\\frac{\\cos \\alpha \\pi}{6} + \\frac{\\cos \\beta \\pi}{4} + \\frac{2 \\cos \\gamma}{15}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate the expression: $\\dfrac{3+6+9+\\cdots + 150}{3+5+7+\\cdots+99}$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{153}{100}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Sequence $A$ is a geometric sequence. Sequence $B$ is an arithmetic sequence. Each sequence stops as soon as one of its terms is greater than $300$. What is the least positive di... | math | {
"ground_truth": "2",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the plane, the area of a triangle is $S$, and its perimeter is $C$. The radius of the incircle is $\\gamma= \\frac {2S}{C}$. In space, the volume of a triangular pyramid is $V... | math | {
"ground_truth": "\\frac {3V}{S}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \\cdot lcm(a -1, b)$ , where $lcm (a, b)$ denotes the smallest ... | math | {
"ground_truth": "(6, 15)",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate $\\left|\\frac{3}{4} - 3i\\right|$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{\\sqrt{153}}{4}",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
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