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": "If $5(3x + 7\\pi) = Q$, solve for the value of $10(6x + 14\\pi)$.",
"role": "user"
}
] | math | {
"ground_truth": "4Q",
"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 $f(x) = 2x - 3$, where $x \\in \\{x \\in \\mathbb{N} | 1 \\leq x \\leq \\frac{10}{3}\\}$, find the maximum value of the function.",
"role": "user"
}
] | 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": "A Senate subcommittee has 6 Democrats and 4 Republicans. Assuming all politicians are distinguishable, in how many ways can they sit around a circular table if no two Republicans... | math | {
"ground_truth": "43,\\!200",
"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 $\\tan \\alpha < 0$ and $\\sin \\alpha = -\\frac{\\sqrt{3}}{3}$, find the value of $\\sin 2\\alpha$.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{2\\sqrt{2}}{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": "Let the complex number $z = (3m-2) + (m-1)i$, where $m \\in \\mathbb{R}$.\n\n(1) For what value of $m$ is $z$ a purely imaginary number?\n\n(2) For what value of $m$ does the poi... | math | {
"ground_truth": "\\frac{2}{3} < m < 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 ball is thrown upward from a height \\( h \\) with an initial velocity \\( V_0 \\) and accelerates downward due to gravity \\( g \\). It reaches a final velocity \\( V \\) afte... | math | {
"ground_truth": "\\textbf{(B)}\\ \\frac{2(S - h)}{V + V_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": "In the coordinate plane \\( xOy \\), point \\( P \\) is a moving point on the ellipse \\( \\frac{x^{2}}{25}+\\frac{y^{2}}{16}=1 \\) in the third quadrant. From point \\( P \\), t... | math | {
"ground_truth": "\\frac{81}{20}",
"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": "Find $23\\cdot\\left(1\\frac{2}{3} + 2\\frac{1}{4}\\right) \\div \\left(1\\frac{1}{2} + 1\\frac{1}{5}\\right)$. Express your answer as a mixed number.",
"role": "user"
}
] | math | {
"ground_truth": "3\\frac{43}{108}",
"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": "It is given the function $f:\\mathbb{R} \\to \\mathbb{R}$ such that it holds $f(\\sin x)=\\sin (2011x)$ . Find the value of $f(\\cos x)$ .",
"role": "user"
}
] | math | {
"ground_truth": "\\cos(2011x)",
"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": "In the diagram, a network of roads between the homes of five friends is shown. The shortest distance by road from Asya to Galia is 12 km, from Galia to Borya is 10 km, from Asya ... | math | {
"ground_truth": "18 \\, \\text{km}",
"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 define the sequence \\(a_i\\) as follows:\n\n\\[a_{1}=1, \\quad a_{n}=\\frac{n+1}{n-1}\\left(a_{1}+\\cdots+a_{n-1}\\right)\\]\n\nDetermine the value of \\(a_{1998}\\).",
"... | math | {
"ground_truth": "a_{1998} = 1999 \\cdot 2^{1996}",
"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": "\\(\\left( \\frac{36}{49} \\right)^{-\\frac{1}{2}}-\\log_{2}\\sqrt[6]{2}=\\) ______.",
"role": "user"
}
] | math | {
"ground_truth": "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": "Expand the following product: $\\frac{3}{7}\\left(\\frac{7}{x^3} + 14x^5\\right)$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{3}{x^3} + 6x^5",
"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 $(a_1,a_2,\\ldots, a_{13})$ be a permutation of $(1, 2, \\ldots, 13)$ . Ayvak takes this permutation and makes a series of *moves*, each of which consists of choosing an i... | math | {
"ground_truth": "13703",
"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": "How many non-empty subsets \\(S\\) of \\(\\{1,2,3,\\ldots,10\\}\\) have the following two properties?\n1. No two consecutive integers belong to \\(S\\).\n2. If \\(S\\) contains \... | math | {
"ground_truth": "143",
"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": "When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $abc = 216$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac1{216}",
"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 pressure \\( P \\) on a roof varies jointly as the area \\( A \\) of the roof and the square of the velocity \\( V \\) of the wind. There is a pressure of 2 pounds on a squar... | math | {
"ground_truth": "The correct answer is \\( \\textbf{(C)}\\ 80\\text{ mph} \\).",
"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(z) \\) and \\( S(z) \\) be the unique polynomials such that \\[ z^{2023}+1 = (z^2+z+1)P(z) + S(z) \\] and the degree of \\( S \\) is less than 2. What is \\( S(z)? \\)\... | math | {
"ground_truth": "C) z + 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": "Let $f:\\mathbb{R}^+\\to \\mathbb{R}^+$ be a function such that for all $x,y \\in \\mathbb{R}+,\\, f(x)f(y)=f(xy)+f\\left(\\frac{x}{y}\\right)$ , where $\\mathbb{R}^+$ repre... | math | {
"ground_truth": "07",
"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 $M$ denote the number of permutations of the $16$-character string $AAAABBBCCCCDDDD$ such that:\n- None of the first five letters is an $A$.\n- None of the next four letters ... | math | {
"ground_truth": "140",
"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 sequence $\\{a\\_n\\}$, its first $n$ terms sum is $S\\_n$, and $a\\_1=1$, $S\\_n=n^2a\\_n$ $(n∈N^+)$, find the inductive guess for the expression of $S\\_n$.",
"role... | math | {
"ground_truth": "\\frac{2n}{n+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": "Let the universal set be $U=\\{1,2,3,4,5\\}$, set $M=\\{1,4\\}$, $N=\\{2,5\\}$. Determine the set $N\\cup \\complement _{U}M$.",
"role": "user"
}
] | math | {
"ground_truth": "\\{2,3,5\\}",
"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": "What is the remainder when $17^{49}$ is divided by 5?",
"role": "user"
}
] | 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": "Determine the smallest natural number $ n$ such that among any $ n$ integers one can choose $ 18$ integers whose sum is divisible by $ 18$ .",
"role": "user"
}
] | math | {
"ground_truth": "35",
"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": "How many 3-digit whole numbers, whose digit-sum is $27$ and whose middle digit is even, are even?\nA) 0\nB) 1\nC) 2\nD) 3\nE) 4",
"role": "user"
}
] | math | {
"ground_truth": "\\text{(A)}\\ 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": "A parabola has a maximum height of 16 inches and a span of 40 inches. What is the height, in inches, of the arc at a point 5 inches away from the center?",
"role": "user"
}... | math | {
"ground_truth": "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": "Points $A$, $B$, $C$, and $D$ are midpoints of the respective sides of a large square. Consider the triangle formed by points $A$, $B$, and $C$. If the larger square has area $12... | math | {
"ground_truth": "16",
"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 $T$ be the set of all positive integer divisors of $200,000.$ How many numbers are the product of two distinct elements of $T$?",
"role": "user"
}
] | math | {
"ground_truth": "117",
"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": "Solve the equations:\n(1) $(x-2)^2=4$;\n(2) $x^2+2x-1=0$ (solve by completing the square);\n(3) $25x^2-9(x-1)^2=0$.",
"role": "user"
}
] | math | {
"ground_truth": "x_2 = -\\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 order for the inequality $2x^{2}-9x+a<0$ (with a non-empty solution set) to be satisfied for every value of $x$ that satisfies at least one of the inequalities $x^{2}-4x+3<0$ ... | math | {
"ground_truth": "a < 4",
"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 Laila had a 10-hour work day and attended three different meetings during this period, the first meeting took 50 minutes, the second meeting twice as long as the first, and... | math | {
"ground_truth": "33.33\\%",
"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": "When the graph of $y = 2x^2 - x + 7$ is shifted six units to the right, we obtain the graph of $y = ax^2 + bx + c$. Find $a + b + c$.",
"role": "user"
}
] | math | {
"ground_truth": "62",
"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 the radius of a circle is increased $100\\%$, find the percentage increase in the area.",
"role": "user"
}
] | math | {
"ground_truth": "300\\%",
"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": "For an integer \\( m \\), let \\( p(m) \\) denote the largest prime factor of \\( m \\). For convenience, define \\( p(\\pm 1) = 1 \\) and \\( p(0) = \\infty \\). Find all intege... | math | {
"ground_truth": "f(x) = c\\left(4x - a_{1}^{2}\\right)\\left(4x - a_{2}^{2}\\right) \\cdots \\left(4x - a_{k}^{2}\\right)",
"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": "A rubber tire has an outer diameter of 25 inches. Calculate the approximate percentage increase in the number of rotations in one mile when the radius of the tire decreases by \\... | math | {
"ground_truth": "2\\%",
"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": "In how many ways is it possible to arrange the digits of 2022 to form a four-digit number that is a multiple of 2?",
"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": "Construct the graph of the equation \\(x^{2} - y^{4} = \\sqrt{18x - x^{2} - 81}\\), i.e., plot on the coordinate plane all points \\((x, y)\\) that satisfy this equation.",
"... | math | {
"ground_truth": "(9, \\sqrt{3}) \\text{ and } (9, -\\sqrt{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": "Daniel has 5 indistinguishable bronze coins and 3 indistinguishable copper coins. Each coin has an engraving of one face on one side. To stack these 8 coins such that no two adja... | math | {
"ground_truth": "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": "A newly designed car travels 4.2 kilometers further per liter of gasoline than an older model. The fuel consumption for the new car is 2 liters less per 100 kilometers. How many ... | math | {
"ground_truth": "5.97",
"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": "Calculate the indefinite integral:\n\n$$\n\\int \\frac{\\arccos ^{3} x-1}{\\sqrt{1-x^{2}}} \\, dx\n$$",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{(\\arccos x)^4}{4} - \\arccos x + C",
"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": "Parallel resistances. If we connect two conductors in parallel, whose resistances are respectively \\(x\\) and \\(y\\), the resistance \\(z\\) of such a section can be obtained f... | math | {
"ground_truth": "(6, 6, 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 \\( P \\) be any point inside triangle \\( \\triangle ABC \\), where the sides opposite to angles \\( A \\), \\( B \\), and \\( C \\) are \\( a \\), \\( b \\), and \\( c \\) ... | math | {
"ground_truth": "4",
"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 $i^2=-1$, simplify the expression $\\left(i-i^{-1}\\right)^{-1}$.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{i}{2}",
"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 cyclist planned to travel from point \\( A \\) to point \\( B \\) in 5 hours, moving at a constant speed. He rode at the planned speed until the halfway point and then increase... | math | {
"ground_truth": "4 \\text{ hours} 30 \\text{ minutes}",
"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": "In the same Cartesian coordinate system, there are infinite lines given by the equation \\( y = kx + b \\) (where \\( k \\) and \\( b \\) are real numbers and \\( k \\neq 0 \\)),... | math | {
"ground_truth": "7",
"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": "Convex quadrilateral $ABCD$ has $AB = 8$ and $CD = 10$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 15$, and $\\triangle AED$ and $\\triangle BEC$ have equal areas. What is $... | math | {
"ground_truth": "\\frac{20}{3}",
"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": "Find the range of the function \\( f(x) = \\operatorname{arctg} \\sqrt{6 \\log_{0.25}^{-1} \\left( \\frac{\\sin x}{\\sin x + 15} \\right)} \\).",
"role": "user"
}
] | math | {
"ground_truth": "(0, \\ \\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": "What is $\\frac{3^4 + 3^4}{3^{-4} + 3^{-4}}$?",
"role": "user"
}
] | math | {
"ground_truth": "6561",
"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": "Mike sells one enchilada and four tacos for $3.50, and he sells four enchiladas and one taco for $4.20. Assuming a fixed price per item, what is the cost, in dollars, of two ench... | math | {
"ground_truth": "\\$5.04",
"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": "What is the maximum value of $n$ for which there is a set of distinct positive integers $k_1, k_2, ... k_n$ for which\n\\[k^2_1 + k^2_2 + ... + k^2_n = 2010?\\]\nA) 14\nB) 15\nC)... | math | {
"ground_truth": "The final correct answer, given the choices, is (D) 17.",
"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 certain store is test-selling a type of clothing with a cost of 60 yuan per item. It is stipulated that during the test sale period, the selling price per item must not be lowe... | math | {
"ground_truth": "864",
"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": "Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\\dfrac{a^3b-1}{a+1}$ and $\\dfrac{b^3a+1}{b-1}$ are both positive integers.",
"role": "user"... | math | {
"ground_truth": "(1, 3), (2, 2), (3, 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\n\\[\n\\cos^2 \\alpha \\sin^2 \\beta + \\sin^2 \\alpha \\cos^2 \\beta = \\cos \\alpha \\sin \\alpha \\cos \\beta \\sin \\beta,\n\\]\nfind the value of\n\\[\n\\frac{\\s... | math | {
"ground_truth": "1",
"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 is a simple graph with $239$ vertices, such that it is not bipartite and each vertex has degree at least $3$ . Find the smallest $k$ , such that each odd cycle has leng... | math | {
"ground_truth": "k = 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": "Starting with ten billion, Barbara creates a sequence where she alternately divides by 5 and multiplies by 2. The sequence begins with 10,000,000,000. What is the last integer in... | math | {
"ground_truth": "(2^{16})(5^4)",
"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": "At a bus depot, there are 5 red, 6 blue, and 5 yellow buses. A lineup of 7 buses is randomly formed. What is the probability that the first bus in the lineup is red, and among th... | math | {
"ground_truth": "\\frac{75}{8080}",
"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": "Players $A$, $B$, $C$, and $D$ start a game with $16$, $15$, $14$, and $13$ tokens, respectively. The game rule is the same: in each round, the player(s) with the most tokens giv... | math | {
"ground_truth": "A) 36",
"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": "Vova has 19 grades in his journal for math, all of which are 2s and 3s, with the first four grades being 2s. It turns out that among Vova's consecutive four grades, all 16 possib... | math | {
"ground_truth": "3222",
"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": "Formulate the equation of the lowest degree with integer coefficients that has $\\sqrt{2} + \\sqrt{3}$ as one of its roots.",
"role": "user"
}
] | math | {
"ground_truth": "x^4 - 10x^2 + 1 = 0",
"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": "Vasya drew several circles on a plane and drew all possible common tangents to each pair of these circles. It turned out that the drawn lines contain all the sides of some regula... | math | {
"ground_truth": "504",
"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": "Consider a 10 x 10 checkerboard made up of 100 squares. How many squares are there that can be formed by the union of one or more squares on the checkerboard?",
"role": "user... | math | {
"ground_truth": "385",
"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": "How many ways are there to color the cells of a $10 \\times 10$ board in blue and green such that each $2 \\times 2$ square has two blue and two green cells?",
"role": "user"... | math | {
"ground_truth": "2046",
"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 function $f(n) =\\left\\{\\begin{matrix}\\log_{4}{n}, &\\text{if }\\log_{4}{n}\\text{ is rational,}\\\\ 0, &\\text{otherwise,}\\end{matrix}\\right.$ evaluate $\\sum_{n ... | math | {
"ground_truth": "39",
"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": "Define a function $f(x)$ on $\\mathbb{R}$, for any $x$, it satisfies $f(x+4) \\leq f(x) + 4$ and $f(x+3) \\geq f(x) + 3$, and $f(1) = 2$, $f(0) = 1$. Find $f(2009)$.",
"role"... | math | {
"ground_truth": "2010",
"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 $L$ be a positive constant. For a point $P(t,\\ 0)$ on the positive part of the $x$ axis on the coordinate plane, denote $Q(u(t),\\ v(t))$ the point at which the poin... | math | {
"ground_truth": "-\\frac{L}{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": "The common sum of the numbers in each row, the numbers in each column, and the numbers along each of the main diagonals of a 5-by-5 square formed by arranging the 25 integers fro... | math | {
"ground_truth": "10",
"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 all possible rectangles that can be drawn on a $8 \\times 8$ chessboard, covering only whole cells. Calculate the sum of their areas.\n\nWhat formula is obtained if “ ... | math | {
"ground_truth": "\\sum_{k=1}^{a} \\sum_{l=1}^{b} k \\times l \\times (a+1-k) \\times (b+1-l)",
"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": "In the product\n\n\\[ \n24^{a} \\cdot 25^{b} \\cdot 26^{c} \\cdot 27^{d} \\cdot 28^{e} \\cdot 29^{f} \\cdot 30^{g} \n\\]\n\nseven numbers \\(1, 2, 3, 5, 8, 10, 11\\) were assigne... | 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": "Let $\\alpha ,\\ \\beta$ be the distinct positive roots of the equation of $2x=\\tan x$ .\r\n\r\nEvaluate the following definite integral.\r\n\\[\\int_{0}^{1}\\sin \\alpha x\\... | math | {
"ground_truth": "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": "Consider a square quilt block made from sixteen smaller squares, where each smaller square is identical in size. Four of these squares are divided into two right-angled triangles... | math | {
"ground_truth": "\\frac{1}{8}",
"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 following drawing, \\(ABCD\\) is a square and points \\(E\\) and \\(F\\) are on sides \\(BC\\) and \\(CD\\) such that \\(AEF\\) forms a right triangle, \\(AE = 4\\) and \\... | math | {
"ground_truth": "\\frac{256}{17}",
"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": "Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$ , $b - d$ , an... | math | {
"ground_truth": "(1, 2, 0, 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": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$, and satisfies $f(x+2)=-f(x)$, and when $0 \\leq x \\leq 1$, $f(x) = \\frac {1}{2}x$, determine the val... | math | {
"ground_truth": "4n-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": "The vertices of a $6 - 8 - 10$ right triangle are the centers of three mutually externally tangent circles. Calculate the sum of the areas of these circles.",
"role": "user"
... | math | {
"ground_truth": "56\\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 right triangle \\( \\triangle A B C \\) with \\( A \\) as the right-angle vertex and \\( \\angle B = 60^\\circ \\), \\( A D \\perp B C \\), and \\( A E \\) bisects \\( \\... | math | {
"ground_truth": "( \\sqrt{3} - \\frac{3}{2}) a",
"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": "In $\\triangle ABC$, if $\\overrightarrow{AB} \\cdot \\overrightarrow{AC} = 7$ and $\\left|\\overrightarrow{AB} - \\overrightarrow{AC}\\right| = 6$, then the maximum area of $\\t... | 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": "There are 4 containers weighing 3 tons each, 5 containers weighing 2.5 tons each, 14 containers weighing 1.5 tons each, and 7 containers weighing 1 ton each. What is the minimum ... | 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 an isosceles triangle $ABC$ with sides $AB = AC = 4\\sqrt{6}$, and a circle with radius $6\\sqrt{2}$ tangent to line $AB$ at $B$ and line $AC$ at $C$, find the area of the ... | math | {
"ground_truth": "48\\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": "Let $P(x,\\ y)\\ (x>0,\\ y>0)$ be a point on the curve $C: x^2-y^2=1$ . If $x=\\frac{e^u+e^{-u}}{2}\\ (u\\geq 0)$ , then find the area bounded by the line $OP$ , the $x$ a... | math | {
"ground_truth": "\\frac{1}{2} u",
"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 $P$ be the foot of the perpendicular dropped from vertex $C$ of the smaller base $BC$ of an isosceles trapezoid $ABCD$ to its larger base $AD$. Find $DP$ and $AP$ if the base... | math | {
"ground_truth": "\\frac{a - b}{2}, \\frac{a + b}{2}",
"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 chords \\(AB\\) and \\(CD\\) of a circle, find a point \\(X\\) on the circle such that chords \\(AX\\) and \\(BX\\) cut off a segment \\(EF\\) on \\(CD\\) of a given length... | math | {
"ground_truth": "2",
"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": "Determine the largest value the expression $$ \\sum_{1\\le i<j\\le 4} \\left( x_i+x_j \\right)\\sqrt{x_ix_j} $$ may achieve, as $ x_1,x_2,x_3,x_4 $ run through the non-neg... | math | {
"ground_truth": "\\frac{3}{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": "In the coordinate plane consider the set $ S$ of all points with integer coordinates. For a positive integer $ k$ , two distinct points $A$ , $ B\\in S$ will be called $ k... | math | {
"ground_truth": "180180",
"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 $n\\geq 2$ an integer. Find the least value of $\\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...... | math | {
"ground_truth": "\\frac{1}{2(n-1)^{n-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": "Suppose a non-identically zero function $f$ satisfies $f\\left(x\\right)f\\left(y\\right)=f\\left(\\sqrt{x^2+y^2}\\right)$ for all $x$ and $y$ . Compute $$ f\\left(1\\ri... | math | {
"ground_truth": "-1",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Pairwise distinct points $P_1,P_2,\\ldots, P_{16}$ lie on the perimeter of a square with side length $4$ centered at $O$ such that $\\lvert P_iP_{i+1} \\rvert = 1$ for $... | math | {
"ground_truth": "64",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In triangle \\(ABC\\), the side \\(BC\\) is 19 cm. The perpendicular \\(DF\\), drawn to side \\(AB\\) through its midpoint \\(D\\), intersects side \\(BC\\) at point \\(F\\). Fin... | math | {
"ground_truth": "29 \\, \\text{cm}",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many three-digit numbers are composed of three distinct non-consecutive digits, such that one digit is the average of the other two?",
"role": "user"
}
] | math | {
"ground_truth": "66",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of line $l$ is $$\\begin{cases} x=t\\cos\\alpha \\\\ y=-2+t\\sin\\alpha \\end{cases}$$ (where $t$ is the paramet... | math | {
"ground_truth": "D(0,2)",
"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": "LeRoy and Bernardo went on a week-long trip and agreed to split some of their expenses based on the ratio of the amounts they could afford. Bernardo agreed to pay twice as much a... | math | {
"ground_truth": "\\frac{B-A}{2} - 75",
"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 rational number that is equal to its reciprocal is ______.",
"role": "user"
}
] | math | {
"ground_truth": "\\pm 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": "Let \\( A \\) be a set of \\( m \\) real numbers with the sum of its elements equal to \\( a \\); \\( B \\) be a set of \\( n \\) real numbers with the sum of its elements equal ... | math | {
"ground_truth": "n^2 a^2 + m^2 b^2",
"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 $\\log_k{x} \\cdot \\log_7{k} = 6$, find the value of $x$.",
"role": "user"
}
] | math | {
"ground_truth": "117649",
"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": "The region is bounded by two cones with a common base, whose height is 4 and base radius is 3. Three spheres are placed in the region, touching each other externally. Two of the ... | math | {
"ground_truth": "\\frac{27}{35}",
"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": "A line $l$ passes through the point $A(\\sqrt{3},1)$ and intersects the circle $x^{2}+y^{2}=1$ at some point. The range of the slope angle of line $l$ is __________.",
"role"... | math | {
"ground_truth": "\\left[0, \\dfrac {\\pi}{3}\\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": "A sequence of positive integers with $n$ terms satisfies $\\sum_{i=1}^{n} a_i=2007$ . Find the least positive integer $n$ such that there exist some consecutive terms in the... | math | {
"ground_truth": "1019",
"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 $c$ and $d$ be positive real numbers, with $c > 2d$. Compute the sum of the infinite series:\n\\[\n\\frac{1}{cd} + \\frac{1}{c(3c - 2d)} + \\frac{1}{(3c - 2d)(5c - 4d)} + \\f... | math | {
"ground_truth": "\\frac{1}{(2c - 2d)d}",
"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 the function $f(x)=x^{2}-kx+(2k-3)$.\n(1) If $k= \\frac {3}{2}$, solve the inequality $f(x) > 0$;\n(2) If $f(x) > 0$ holds for any $x∈R$, find the range of values for the r... | math | {
"ground_truth": "(6, \\frac {13}{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": "$1991$ students sit around a circle and play the following game. Starting from some student $A$ and counting clockwise, each student on turn says a number. The numbers are $1... | math | {
"ground_truth": "1093",
"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 $y=\\log_{a}(x-3)+2$ $(a > 0, a\\neq 1)$, its graph passes through a fixed point $P$, and the terminal side of angle $\\alpha$ passes through point $P$. Determ... | math | {
"ground_truth": "\\frac{7}{5}",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
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