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": "Let $f(n)$ be the number of ones that occur in the decimal representations of all the numbers from 1 to $n$ . For example, this gives $f(8)=1$ , $f(9)=1$ , $f(10)=2$ , $f... | math | {
"ground_truth": "10^{101} + 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 that $ABC$ is a triangle with $AB = 6, BC = 12$ , and $\\angle B = 90^{\\circ}$ . Point $D$ lies on side $BC$ , and point $E$ is constructed on $AC$ such that ... | math | {
"ground_truth": "76",
"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 a new operation \"$\\oplus$\" defined as follows: when $a \\gt b$, $a\\oplus b=ab+b$; when $a \\lt b$, $a\\oplus b=ab-b$. If $3\\oplus \\left(x+2\\right) \\gt 0$, solve for... | math | {
"ground_truth": "(-2 < x < 1) \\text{ or } (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": "Given that $x + \\cos y = 1005$ and $x + 1005 \\sin y = 1003$, where $\\pi \\leq y \\leq \\frac{3\\pi}{2}$, find the value of $x+y$.",
"role": "user"
}
] | math | {
"ground_truth": "1005 + \\frac{3\\pi}{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": "Given that $\\sin 43^\\circ = a$, determine the relationship between $a$ and $\\frac{\\sqrt{2}}{2}$ (fill in the blank with either $<$ or $>$), and express $\\sin 73^\\circ$ in t... | math | {
"ground_truth": "\\sin 73^\\circ = \\frac{1}{2}\\sqrt{1 - a^2} + \\frac{\\sqrt{3}}{2}a",
"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 length of the arc of the parabola \\(2y = x^2\\) from the point \\((0,0)\\) to the point \\(\\left(x_0, \\frac{x_0^2}{2}\\right)\\).",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{x_0 \\sqrt{1 + x_0^2}}{2} + \\frac{1}{2} \\ln \\left( x_0 + \\sqrt{1 + x_0^2} \\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": "The ex-radii of a triangle are $10\\frac{1}{2}, 12$ and $14$ . If the sides of the triangle are the roots of the cubic $x^3-px^2+qx-r=0$ , where $p, q,r $ are integers , fi... | math | {
"ground_truth": "43",
"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": "Two right circular cones with vertices facing down contain the same amount of liquid. The radii of the tops of the liquid surfaces are \\(5\\) cm and \\(10\\) cm, respectively. A... | math | {
"ground_truth": "C) 4:1",
"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 $P(x,y)$ be a polynomial such that $\\deg_x(P), \\deg_y(P)\\le 2020$ and \\[P(i,j)=\\binom{i+j}{i}\\] over all $2021^2$ ordered pairs $(i,j)$ with $0\\leq i,j\\leq 2... | math | {
"ground_truth": "1555",
"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": "<u>Round 5</u>**p13.** Find the number of six-digit positive integers that satisfy all of the following conditions:\n(i) Each digit does not exceed $3$ .\n(ii) The number $1$ ... | math | {
"ground_truth": "128",
"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 the expression $[a - (b - (c+d))] - [(a+b) - (c-d)]$.\nA) $2b + 2d$\nB) $2a - 2c$\nC) $-2b + 2c$\nD) $2c - 2b$\nE) $0$",
"role": "user"
}
] | math | {
"ground_truth": "\\textbf{(C)} -2b + 2c",
"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 all the prime numbers between 10 and 20?",
"role": "user"
}
] | math | {
"ground_truth": "60",
"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 $\\left(20-\\left(2010-201\\right)\\right)+\\left(2010-\\left(201-20\\right)\\right)$, evaluate the expression.",
"role": "user"
}
] | math | {
"ground_truth": "40",
"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 positive real numbers $x$ that satisfy\n\\[x \\sqrt{20 - x} + \\sqrt{20x - x^3} \\ge 20.\\]",
"role": "user"
}
] | math | {
"ground_truth": "20",
"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 finite set \\( A \\), there exists a function \\( f: \\mathbf{N}^{*} \\rightarrow A \\) with the following property: if \\( i, j \\in \\mathbf{N}^{*} \\) and \\( |i-j| \\... | 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": "Divide a circle into $n$ sectors $(n \\geqslant 2)$, labeled as $s_{1}, s_{2}, \\cdots, s_{n}$. Each sector can be colored with one of three different colors: red, white, or blue... | math | {
"ground_truth": "a_n = 2(-1)^n + 2^n",
"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": "An engineering project can be completed by person A alone in 20 days, and by person B alone in 25 days. Person A works alone for 2 days, then both person A and person B work toge... | math | {
"ground_truth": "10",
"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 the smallest positive integer \\( n \\) such that no arithmetic progression of 1999 reals contains just \\( n \\) integers.",
"role": "user"
}
] | math | {
"ground_truth": "70",
"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 that the hyperbola C and the ellipse $\\frac{y^2}{5}+x^2=1$ share a common focus, and the distance from the focus to the asymptote of the hyperbola C is 1, determine the eq... | math | {
"ground_truth": "\\frac{y^2}{3}-x^2=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": "Determine the value of $\\theta$ where the line $y=x$ is perpendicular to the line $\\begin{cases} x=1+t\\cos \\theta \\\\ y=t\\sin \\theta \\end{cases}$, where $t$ is a paramete... | math | {
"ground_truth": "\\dfrac {3\\pi}{4}",
"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": "Julia and James pick a random integer between $1$ and $10$ , inclusive. The probability they pick the same number can be written in the form $m/n$ , where $m$ and $n$ ar... | math | {
"ground_truth": "11",
"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 a number $a$ is exactly the square of another integer, then mathematically, this number $a$ is called a square number. It is known that the digit in the tens place of a square... | math | {
"ground_truth": "6",
"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 height in centimeters of the water in the cylinder when the water from an inverted cone with base radius $15 \\mathrm{cm}$ and height $20 \\mathrm{cm}$ is poured into... | math | {
"ground_truth": "1.\\overline{6}",
"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": "Evaluate $y$ for a given expression where:\n\\[ y = \\sqrt{x^2 - 4x + 4} + \\sqrt{x^2 + 6x + 9} - 2 \\]\nA) $|x-2| + |x+3|$\nB) $|x+3| - |x-2|$\nC) $|x-2| - |x+3|$ \nD) $|x-2| + ... | math | {
"ground_truth": "|x-2| + |x+3| - 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": "Given vectors $\\overrightarrow{a}=(-3,1)$, $\\overrightarrow{b}=(1,-2)$, $\\overrightarrow{m}=\\overrightarrow{a}+k\\overrightarrow{b}$ ($k\\in \\mathbb{R}$). \n$(1)$ If $\\ove... | math | {
"ground_truth": "-\\frac{1}{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": "If 100 students are selected using systematic sampling from 1000 students evenly divided into 100 groups, and the number drawn from the first group is 6, determine the number dra... | math | {
"ground_truth": "96",
"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": "Assume the sum of two numbers is \\( S \\). If \\( 5 \\) is subtracted from each number and then each resulting number is tripled, what is the sum of these newly transformed numb... | math | {
"ground_truth": "B) \\( 3S - 30 \\)",
"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 Sidorov family, there are 3 people: the father, mother, and son. The father works as a programmer with an hourly rate of 1500 rubles. The mother is a home-based hairdresse... | math | {
"ground_truth": "19350 \\text{ rubles}",
"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": "Express $.\\overline{46}$ as a common fraction.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{46}{99}",
"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 square with side length s, a rectangle is constructed on one diagonal of the square such that its area equals the area of the square. If the length of the rectangle is th... | math | {
"ground_truth": "\\frac{s}{\\sqrt{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": "Given the complex number $z= \\frac {2i}{1+i}$, calculate the imaginary part of its conjugate complex number.",
"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": "Rationalize the denominator of $\\displaystyle \\frac{1}{\\sqrt{3} + \\sqrt{5} + \\sqrt{11}}$ and write your answer in the form\n\\[\n\\frac{A\\sqrt{3} + B\\sqrt{5} + C\\sqrt{11}... | math | {
"ground_truth": "97",
"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": "Peppy the Pigeon and Freddy the Frog are going up the same staircase. Peppy can fly up three steps per jump, and Freddy can hop four steps per jump. However, both may take fewer ... | math | {
"ground_truth": "B) 19",
"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": "If $a^{2}-4a-12=0$, then calculate the value of $-2a^{2}+8a+40$.",
"role": "user"
}
] | math | {
"ground_truth": "16",
"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 triangle $ABC$ with orthocenter $H$ is inscribed in a circle with center $K$ and radius $1$ , where the angles at $B$ and $C$ are non-obtuse. If the lines $HK$ and... | math | {
"ground_truth": "\\frac{\\sqrt{3}}{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": "Marie repeatedly flips a fair coin and stops after she gets tails for the second time. What is the expected number of times Marie flips the coin?",
"role": "user"
}
] | 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": "The towns \\( A \\), \\( B \\), and \\( C \\) are connected by straight roads. The distance from town \\( A \\) to the road connecting towns \\( B \\) and \\( C \\) is 100 km, an... | math | {
"ground_truth": "30.7 \\text{ liters}",
"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 \\( O \\) and \\( H \\) be the circumcenter and orthocenter of triangle \\( ABC \\), respectively. Let \\( a, b, \\) and \\( c \\) be the side lengths, and let \\( R \\) deno... | math | {
"ground_truth": "850",
"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 $X=\\{1,2,\\ldots,2001\\}$ . Find the least positive integer $m$ such that for each subset $W\\subset X$ with $m$ elements, there exist $u,v\\in W$ (not necessarily ... | math | {
"ground_truth": "999",
"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 triples of integers for which the product is four times the sum, and one of them is twice the sum of the other two.",
"role": "user"
}
] | math | {
"ground_truth": "(1, 6, 14), (-1, -6, -14), (2, 3, 10), (-2, -3, -10), (a, -a, 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": "Let $P_1$ be a regular $n$ -gon, where $n\\in\\mathbb{N}$ . We construct $P_2$ as the regular $n$ -gon whose vertices are the midpoints of the edges of $P_1$ . Continuing... | math | {
"ground_truth": "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": "Determine the maximal size of the set $S$ such that:\ni) all elements of $S$ are natural numbers not exceeding $100$ ;\nii) for any two elements $a,b$ in $S$ , there exis... | math | {
"ground_truth": "72",
"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 numerical value of a particular regular hexagon's area is equal to the numerical value of its perimeter. What is the length of a side of the hexagon?",
"role": "user"
}... | math | {
"ground_truth": "\\frac{4\\sqrt{3}}{3} \\text{ units}",
"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": "Evaluate $$125^{1/3}\\cdot81^{-1/4}\\cdot32^{1/5}\\div8^{1/3}.$$",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{5}{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": "Given the table with the middle cell and the cell in the top left corner filled, determine the sum $A+B$.",
"role": "user"
}
] | math | {
"ground_truth": "6",
"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 proposition: \"There exists $x \\in [1,2]$ such that $x^{2}+2x+a \\geqslant 0$\" is true, the range of values for $a$ is _______ .",
"role": "user"
}
] | math | {
"ground_truth": "a \\geqslant -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 a cube of side length $10$ and balls of clay of radius $3$, calculate the maximum number of balls of clay that can be packed completely inside the cube.",
"role": "user... | 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": "Our school's boys soccer team consists of 18 players, including a set of twins: Bob and Bill. The coach needs to choose a starting lineup of 8 players. Every starting lineup must... | math | {
"ground_truth": "3602",
"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 four-element set $\\{a, b, c, d\\}$ of positive integers is called *good* if there are two of them such that their product is a mutiple of the greatest common divisor of the ... | math | {
"ground_truth": "230",
"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 dimensions of an \"L\"-shaped lawn consisting of a $120$-foot by $50$-foot rectangular section and a $70$-foot by $50$-foot rectangular section, both sharing one $50$-f... | math | {
"ground_truth": "0.95",
"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": "Calculate the value of $(5^{-1} - 4^{-1})^{-1}$.",
"role": "user"
}
] | 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": "There are seven red cubes, three blue cubes, and nine green cubes. Ten cubes were placed into a gift bag. In how many different ways could this have been done?",
"role": "use... | math | {
"ground_truth": "31",
"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": "If the complex number $z = (m-1)i + m^2 - 1$ is purely imaginary, then find the value of the real number $m$.",
"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": "A point in space $(x, y, z)$ is randomly selected such that $-2 \\leq x \\leq 2$, $-2 \\leq y \\leq 2$, $-2 \\leq z \\leq 2$. What is the probability that $x^2 + y^2 + z^2 \\leq ... | math | {
"ground_truth": "\\frac{\\pi}{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": "Compute $1 - 2 + 3 - 4 + \\dots + 197 - 198 + 199$.",
"role": "user"
}
] | math | {
"ground_truth": "100",
"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 skew lines $e$ and $f$ are perpendicular to each other. The segment $AB$ of unit length moves such that $A$ is always on $e$ and $B$ is always on $f$. Determine the locus of ... | math | {
"ground_truth": "\\text{A circle perpendicular to } EF \\text{ with radius } \\sqrt{\\frac{1}{4} - \\left(\\frac{EF}{2}\\right)^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 $\\triangle XYZ$, point $W$ is placed on side $XZ$ such that $XW=9$ and $WZ=15$. What is the ratio of the area of $\\triangle XYW$ to the area of $\\triangle WYZ$? Write your ... | math | {
"ground_truth": "3:5",
"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": "Suppose a function $f(x)$ is defined on the domain $[-12,6]$. If we define a new function $g(x)$ by $$g(x) = f(3x),$$ what is the domain of $g(x)$? Express your answer in interva... | math | {
"ground_truth": "[-4, 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 trapezoid $ABCD$, where $AB$ is parallel to $CD$, the height from $B$ to $AD$ is 5 units, and the bases $AB$ and $CD$ have unknown equal lengths. If $AD = 20$ units and $BC = ... | math | {
"ground_truth": "2.5 \\times (20 + \\sqrt{41})",
"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": "Find the number of solution-pairs in positive integers of the equation \\(4x + 7y = 588\\).",
"role": "user"
}
] | math | {
"ground_truth": "21",
"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": "Simplify $\\sin 15^\\circ - \\sin 75^\\circ$.",
"role": "user"
}
] | math | {
"ground_truth": "-\\frac{\\sqrt{6}}{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": "How many solutions does the equation $\\tan(2x)=\\cos(\\tfrac{x}{2})$ have on the interval $[0,2\\pi]$?",
"role": "user"
}
] | math | {
"ground_truth": "5",
"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": "Four balls of radius $1$ are placed in space so that each of them touches the other three. What is the radius of the smallest sphere containing all of them?",
"role": "user... | math | {
"ground_truth": "\\frac{\\sqrt{6} + 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": "Given an ellipse (C) with the equation $\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$ and a focal length of $2\\sqrt{3}$, and the ellipse (C) passes through point $A(1, \\... | 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": "Choose a number from $-\\frac{1}{2}$, $-1$, $1$, $2$, and $5$ as $a$, the probability that the parabola $y=ax^{2}+bx+c$ opens upwards is ____.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{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": "The escalator of the station **champion butcher** has this property that if $m$ persons are on it, then it's speed is $m^{-\\alpha}$ where $\\alpha$ is a fixed positive rea... | math | {
"ground_truth": "l \\cdot n^{\\min(\\alpha, 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": "A semicircle with diameter length $16$ contains a circle radius $3$ tangent both to the inside of the semicircle and its diameter as shown. A second larger circle is tangent ... | math | {
"ground_truth": "361",
"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 set \\( A = \\{1, 2, 3, \\cdots, 2011\\} \\), the number of elements whose last digit is 1 is ______.",
"role": "user"
}
] | math | {
"ground_truth": "202",
"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 an isosceles triangle \\(ABC\\), it is known that \\(\\angle A = \\alpha > 90^\\circ\\) and \\(BC = a\\). Find the distance between the orthocenter and the circumcenter.",
... | math | {
"ground_truth": "\\frac{a}{2}\\left(\\operatorname{tg} \\frac{\\alpha}{2}-\\operatorname{ctg} \\alpha\\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": "Given that the center of circle $C$ lies on the line $x-2y=0$.\n\n1. If circle $C$ is tangent to the positive semi-axis of the $y$-axis, and the chord intercepted by the circle o... | math | {
"ground_truth": "b= \\frac {5± \\sqrt {15}}{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": "Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five?",
"role": "user"
}
] | math | {
"ground_truth": "72000",
"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 line $(2m+1)x+(m+1)y-7m-4=0$ passes through a fixed point $(x,y)$. Solve for $(x,y)$.",
"role": "user"
}
] | math | {
"ground_truth": "(3,1)",
"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 circular spinner for a game has a radius of 15 cm. The probability of winning on one spin of this spinner has two possibilities with each winning sector having an equal chance,... | math | {
"ground_truth": "75\\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": "If $R$ and $S$ are two rectangles with integer sides such that the perimeter of $R$ equals the area of $S$ and the perimeter of $S$ equals the area of $R$ , then we ca... | math | {
"ground_truth": "\\{\\{22, 5\\}, \\{54, 1\\}\\}, \\{\\{13, 6\\}, \\{38, 1\\}\\}, \\{\\{10, 7\\}, \\{34, 1\\}\\}, \\{\\{10, 3\\}, \\{13, 2\\}\\}, \\{\\{6, 4\\}, \\{10, 2\\}\\}, \\{\\{6, 3\\}, \\{6, 3\\}\\}, \\{\\{4, 4\\}, \\{4, 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": "A pentagon is inscribed in a circle. If an angle is inscribed into each of the five segments outside the pentagon, the sum of these five angles, expressed in degrees, is:\nA) 540... | math | {
"ground_truth": "720^\\circ",
"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": "Suppose $\\alpha$ is inversely proportional to $\\beta$, and directly proportional to $\\gamma$. If $\\alpha = 4$ when $\\beta = 27$ and $\\gamma = 3$, find $\\alpha$ when $\\bet... | 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": "Ecological priority, green development, let the beautiful earth add more \"Chinese green\". A certain community purchased two types of seedlings, A and B, to focus on \"park gree... | math | {
"ground_truth": "38",
"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 marked price of the jackets was reduced by a discount of $\\frac{1}{4}$, and the cost of the jackets was $\\frac{4}{5}$ of the price at which they were actually sold. Determi... | math | {
"ground_truth": "\\frac{3}{5}",
"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 sequence \\(a_{1}, a_{2}, a_{3}, \\cdots\\) such that \\(a_{1} = 1\\) and \\(a_{n+1} = \\frac{1}{16}\\left(1 + 4a_{n} + \\sqrt{1 + 24a_{n}}\\right)\\) for \\(n = 1, 2, \\... | math | {
"ground_truth": "\\frac{1}{3} + 6 * (\\frac{1}{2})^n + 4 * (\\frac{1}{2})^{2n-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": "Suppose triangle $ABC$ is a right triangle with $\\angle B = 90^\\circ$, $\\angle A = 45^\\circ$, and the length of hypotenuse $\\overline{BC} = 30$ units. Calculate the area of ... | math | {
"ground_truth": "225",
"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": "Compute $\\begin{pmatrix} 2 & -2 \\\\ 2 & -1 \\end{pmatrix}^4.$",
"role": "user"
}
] | math | {
"ground_truth": "\\begin{pmatrix} -8 & 8 \\\\ 0 & 3 \\end{pmatrix}",
"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 that two-thirds of the people in a room are seated in three-fourths of the chairs, and there are 6 empty chairs, find the total number of people in the room.",
"role": ... | math | {
"ground_truth": "27",
"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 $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether $$ \\exp \\left( \\sum_{n=1}^{\\infty} \\frac{ B(n)}{n(n+1)} \\ri... | 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": "Find the equation of the line that passes through point P (1, 0) and is parallel to the line $2x+y-5=0$.",
"role": "user"
}
] | math | {
"ground_truth": "2x+y-2=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": "Consider lattice points of a $6*7$ grid.We start with two points $A,B$ .We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from ... | math | {
"ground_truth": "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": "Given the radius (4) of an inscribed circle in a triangle and segments 6 and 8, into which the point of tangency divides one side of the triangle. Find the other two sides.",
... | math | {
"ground_truth": "BC = 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": "Given the functions $f(x)=|x-1|$ and $g(x)=2|x-a|$, where $a\\in\\mathbb{R}$.\n1. If $a=2$, find the solution set of the inequality $f(x)-g(x)\\leq x-3$.\n2. If for all $m > 1$, ... | math | {
"ground_truth": "[-2-2 \\sqrt {6}, 2 \\sqrt {6}+4]",
"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 lame king is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. A lame ki... | math | {
"ground_truth": "43",
"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 farmer said: \"I have $N$ rabbits. Exactly 13 of them have long ears. Exactly 17 of them can jump far.\"\n\nThe traveler rightly noted: \"Therefore, among your rabbits, at le... | math | {
"ground_truth": "27",
"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 the smallest natural number that ends with the digit 6 such that moving this digit to the front increases the number exactly fourfold.",
"role": "user"
}
] | math | {
"ground_truth": "153846",
"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 $\\sin \\theta +\\cos \\theta =\\frac{1}{5}$, and $\\frac{\\pi }{2}\\leqslant \\theta \\leqslant \\frac{3\\pi }{4}$, then the value of $\\cos 2\\theta$ is __________.",
... | math | {
"ground_truth": "-\\frac{7}{25}",
"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 \\( ABC \\) be an acute triangle, and let \\( M \\) be the midpoint of \\( AC \\). A circle \\(\\omega\\) passing through \\( B \\) and \\( M \\) meets the sides \\( AB \\) a... | math | {
"ground_truth": "\\sqrt{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": "Find the area of the overlap, assumed to be a rhombus, where two strips overlap at an angle of $\\beta = \\frac{\\alpha}{2}$, where $\\alpha$ is a given acute angle.",
"role"... | math | {
"ground_truth": "\\frac{1}{\\sin \\frac{\\alpha}{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": "Independent random variables \\( X \\) and \\( Y \\) are given by the following probability density functions:\n\n\\[\nf_{1}(x) = e^{-x} \\quad (0 \\leq x < \\infty), \\quad f_{2... | math | {
"ground_truth": "g(z) = e^{-z / 2}\\left(1 - e^{-z / 2}\\right), \\text{for } z \\geqslant 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": "The distance from $P(3,-4)$ to the $x$-axis is ____.",
"role": "user"
}
] | math | {
"ground_truth": "4",
"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 parabola $C: y^2 = 4x$ and the point $M(0, 2)$, a line passing through the focus of $C$ with a slope of $k$ intersects $C$ at points $A$ and $B$. If $\\overrightarrow{M... | 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": "The centers of three spheres, with radii 3, 4, and 6, are located at the vertices of an equilateral triangle with a side length of 11. How many planes exist that simultaneously t... | math | {
"ground_truth": "6",
"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 cart rolls down a hill, traveling 5 inches in the first second. It accelerates so that during each successive 1-second interval, it travels 9 inches more than during the previo... | math | {
"ground_truth": "7220",
"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": "Rectangle $EFGH$ has area $2457$. An ellipse with area $2457\\pi$ passes through $E$ and $G$ and has foci at $F$ and $H$. What is the perimeter of the rectangle?",
"role": "u... | math | {
"ground_truth": "8\\sqrt{1228.5}",
"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": "Given the function $y=\\begin{cases} x^{2}+1 & (x\\leqslant 0) \\\\ 2x & (x > 0) \\end{cases}$, find the value(s) of $a$ for which $f(a)=10$.",
"role": "user"
}
] | math | {
"ground_truth": "-3 or 5",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
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