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": "There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$ , inclus... | math | {
"ground_truth": "5",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a positive arithmetic sequence $\\{a_{n}\\}$ with the sum of the first $n$ terms denoted as $S_{n}$, where $S_{3}=12$. If $a_{2}$ is the geometric mean of $2a_{1}$ and $a_{... | math | {
"ground_truth": "T_{n} = \\frac{21}{4} + \\frac{6n - 7}{4} \\cdot 3^{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": "Given that an equilateral triangle \\( \\triangle ABC \\) has sides \\( AB \\) and \\( AC \\) that are tangent to circle \\( \\odot O \\) at points \\( B \\) and \\( C \\) respec... | math | {
"ground_truth": "\\frac{4}{3} - \\frac{4\\sqrt{3}\\pi}{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": "Inside an equilateral triangle ABC lies a point O. It is known that $\\angle AOB = 113^\\circ$ and $\\angle BOC = 123^\\circ$. Find the angles of the triangle, where the sides ar... | math | {
"ground_truth": "53^\\circ, 63^\\circ \\text{ and } 64^\\circ",
"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 $\\triangle ABC$ , $X, Y$ , are points on BC such that $BX=XY=YC$ , $M , N$ are points on $AC$ such that $AM=MN=NC$ . $BM$ and $BN$ intersect $AY$ at $S$ and ... | math | {
"ground_truth": "\\frac{7}{48}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "**\nLet $M$ be the number of ordered pairs of nonempty sets $\\mathcal{C}$ and $\\mathcal{D}$ that have the following properties:\n$\\mathcal{C} \\cup \\mathcal{D} = \\{1,2,3,4,5... | math | {
"ground_truth": "3172",
"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 circle \\( C_{2} \\) of radius \\( 2R \\) rolls along the inside of another, fixed circle \\( C_{1} \\) of radius \\( nR \\), where \\( n \\) is an integer greater than 2. An a... | math | {
"ground_truth": "n - 1",
"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 equation of the line that passes through the point $(1,2)$ and has equal intercepts on both coordinate axes.",
"role": "user"
}
] | math | {
"ground_truth": "2x - y = 0 \\text{ or } x + y - 3 = 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": "A customer enters a supermarket. The probability that the customer buys bread is .60, the probability that the customer buys milk is .50, and the probability that the customer bu... | math | {
"ground_truth": "0.80",
"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": "Five marbles of various sizes are placed in a conical funnel. Each marble is in contact with the adjacent marble(s). Also, each marble is in contact all around the funnel wall. 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": "Given that Mona's age is a perfect square and at least half of the guesses $16, 25, 27, 32, 36, 40, 42, 49, 64, 81$ are too low, calculate Mona's age.",
"role": "user"
}
] | math | {
"ground_truth": "49",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given real numbers $a$ and $b$ satisfy $a^2 + b^2 + a^2b^2 = 4ab - 1$, then the value of $a + b$ is.",
"role": "user"
}
] | math | {
"ground_truth": "\\pm 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": "Given the function $f(x)=2\\cos \\left( \\frac {\\pi}{2}-x\\right)\\sin x+(\\sin x+\\cos x)^{2}$. \n$(1)$ Find the intervals of monotonic increase for the function $f(x)$. \n$(... | 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": "For a given positive integer $n >2$ , let $C_{1},C_{2},C_{3}$ be the boundaries of three convex $n-$ gons in the plane , such that \r\n $C_{1}\\cap C_{2}, C_{2}\\cap C_{3... | math | {
"ground_truth": "\\frac{3n}{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": "Find all integers $ n\\ge 2$ having the following property: for any $ k$ integers $ a_{1},a_{2},\\cdots,a_{k}$ which aren't congruent to each other (modulo $ n$ ), there e... | math | {
"ground_truth": "n = 2, 4, p",
"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 a circle with a radius of 5 units, \\( CD \\) and \\( AB \\) are mutually perpendicular diameters. A chord \\( CH \\) intersects \\( AB \\) at \\( K \\) and has a length of 8 ... | math | {
"ground_truth": "8.75",
"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 $\\sin 2\\theta = \\frac{2}{3}$, calculate the value of $\\tan\\theta + \\frac{1}{\\tan\\theta}$.",
"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": "8. You have been kidnapped by a witch and are stuck in the *Terrifying Tower*, which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The w... | 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": "The area of the ground plane of a truncated cone $K$ is four times as large as the surface of the top surface. A sphere $B$ is circumscribed in $K$ , that is to say that $B... | math | {
"ground_truth": "\\frac{9}{14}",
"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 units digit of $2^{1007} 6^{1008} 14^{1009}$.",
"role": "user"
}
] | math | {
"ground_truth": "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": "Compute the smallest positive angle $x,$ in degrees, such that:\n\\[\\tan 6x = \\frac{\\cos 2x - \\sin 2x}{\\cos 2x + \\sin 2x}.\\]",
"role": "user"
}
] | math | {
"ground_truth": "5.625^\\circ",
"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 $a<b<c<d<e$ are [consecutive](https://artofproblemsolving.com/wiki/index.php/Consecutive) [positive](https://artofproblemsolving.com/wiki/index.php/Positive) [integers](https:... | math | {
"ground_truth": "675",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given that thirty percent of the objects in the urn are beads, and fifty percent of the coins are silver, determine the percentage of all objects in the urn that are gold coins."... | math | {
"ground_truth": "35\\%",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $a$ and $b$ satisfy $ab = 6$ and $a+b = 7$. What is $|a-b|$?",
"role": "user"
}
] | math | {
"ground_truth": "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": "Points $R$ and $S$ lie on segment $CD$ such that $R$ divides $CD$ in the ratio $3:5$, and $S$ divides $CD$ in the ratio $4:7$. If the length $RS = 5$, find the length of the segm... | math | {
"ground_truth": "40",
"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 painting 18\" X 24\" is to be placed into a wooden frame with the longer dimension vertical. The wood at the top and bottom is twice as wide as the wood on the sides. If the fr... | math | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \\le P(x) < 120$ for all $x \\in \\{0,1,2,3,4,5\\}$ ?",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{125!}{5! \\cdot 120!}",
"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": "An elephant writes a sequence of numbers on a board starting with 1. Each minute, it doubles the sum of all the numbers on the board so far, and without erasing anything, writes ... | math | {
"ground_truth": "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": "Solve for real numbers $x$ and $y$ ,\r\n\\begin{eqnarray*} xy^2 &=& 15x^2 + 17xy +15y^2 ; x^2y &=& 20x^2 + 3y^2. \\end{eqnarray*}",
"role": "user"
}
] | math | {
"ground_truth": "(x, y) = (0, 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": "Five friends have a total of 10 identical markers, and each one has at least one marker. In how many ways can this happen?",
"role": "user"
}
] | math | {
"ground_truth": "126",
"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 what frequency does the digit 7 appear in the decimal representation of natural numbers? That is, find the limit of the ratio of the number of occurrences of the digit 7 in th... | math | {
"ground_truth": "\\frac{1}{10}",
"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 $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.",
"role": "user"
}
] | 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": "How many distinct prime factors does the sum of the positive divisors of $450$ have?",
"role": "user"
}
] | math | {
"ground_truth": "3",
"style": "rule"
} | {
"id": "numina_synthetic_math-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_math"
} | synthetic_math |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "The ratio of sports cars to sedans sold is 5:8, and 35 sports cars are anticipated to be sold. Determine the number of sedans anticipated to be sold.",
"role": "user"
}
] | math | {
"ground_truth": "56",
"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 area of the ring between two concentric circles is $12\\tfrac{1}{2}\\pi$ square inches. Calculate the length of a chord of the larger circle tangent to the smaller circle.",
... | math | {
"ground_truth": "5\\sqrt{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": "Find all sets of four positive real numbers \\((a, b, c, d)\\) satisfying\n\\[\n\\begin{array}{l}\na + b + c + d = 1, \\\\\n\\text{and} \\\\\n\\max \\left\\{\\frac{a^{2}}{b}, \\f... | math | {
"ground_truth": "\\left(\\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}, \\frac{1}{4}\\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": "Let \\( f, g: \\mathbf{Z}^{+} \\rightarrow \\mathbf{Z}^{+} \\) be strictly increasing sequences such that \\( f\\left(\\mathbf{Z}^{+}\\right) \\cap g\\left(\\mathbf{Z}^{+}\\right... | math | {
"ground_truth": "f(2m) = m + \\left\\lfloor \\sqrt{5}m \\right\\rfloor",
"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 number must be added to \\(4 - 8i\\) to obtain \\(1 + 10i\\)? Here, \\(i^2 = -1\\).",
"role": "user"
}
] | math | {
"ground_truth": "-3 + 18i",
"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 the height of a cylinder is tripled and its radius is doubled, by what factor is the cylinder's volume increased?",
"role": "user"
}
] | math | {
"ground_truth": "12",
"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 $\\left\\lceil -\\sqrt{\\frac{36}{9}} \\right\\rceil$.",
"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": "Find the number of different structures for a Boolean function device with 3 Boolean variables \\( x_{1}, x_{2}, x_{3} \\).",
"role": "user"
}
] | math | {
"ground_truth": "80",
"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 integer \\(n\\), \\(0 \\le n \\le 9\\), that satisfies \\[n \\equiv -1345 \\pmod{10}.\\]",
"role": "user"
}
] | math | {
"ground_truth": "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": "$$ Problem 4: $$ The sum of $5$ positive numbers equals $2$ . Let $S_k$ be the sum of the $k-th$ powers of\nthese numbers. Determine which of the numbers $2,S_2,S_3,S_4$... | math | {
"ground_truth": "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": "Let \\(Q\\) be the plane passing through the origin with normal vector \\(\\begin{pmatrix} 2 \\\\ -1 \\\\ 2 \\end{pmatrix}\\). Find the matrix \\(\\mathbf{Q}\\) such that for any... | math | {
"ground_truth": "\\begin{pmatrix} \\frac{5}{9} & \\frac{2}{9} & -\\frac{4}{9} \\\\ \\frac{2}{9} & \\frac{8}{9} & \\frac{2}{9} \\\\ -\\frac{4}{9} & \\frac{2}{9} & \\frac{5}{9} \\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": "Let $ABC$ be a triangle. The midpoints of the sides $BC$, $AC$, and $AB$ are denoted by $D$, $E$, and $F$, respectively.\n\nThe two medians $AD$ and $BE$ are perpendicular to eac... | math | {
"ground_truth": "22.5",
"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 sets \\( A \\), \\( B \\), \\( C \\) are subsets of \\(\\{1, 2, \\ldots, 2020\\}\\), and \\( A \\subseteq C \\), \\( B \\subseteq C \\), find the number of ordered tri... | math | {
"ground_truth": "5^{2020}",
"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 $mn \\ge 0$ and $m^3 + n^3 + 99mn = 33^3$, calculate the number of ordered pairs of integers $(m,n)$.",
"role": "user"
}
] | math | {
"ground_truth": "35",
"style": "rule"
} | {
"id": "numina_amc_aime-0",
"rating": null,
"split": "confidence_train",
"sub_source": "amc_aime"
} | amc_aime |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let \\( A, B, C, D \\) be four points in space that are not coplanar. Each pair of points is connected by an edge with a probability of \\( \\frac{1}{2} \\), and whether or not t... | math | {
"ground_truth": "\\frac{3}{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": "Let $p(x)$ be the monic cubic polynomial with roots $\\sin^2(1^{\\circ})$ , $\\sin^2(3^{\\circ})$ , and $\\sin^2(9^{\\circ})$ . Suppose that $p\\left(\\frac{1}{4}\\right)=\... | math | {
"ground_truth": "216",
"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 power function $f(x)=(m-1)^{2}x^{m^{2}-4m+2}$ is monotonically increasing on $(0, +\\infty)$, and the function $g(x)=2^{x}-k$,\n(Ⅰ) Find the value of $m$;\n(Ⅱ) For $x \... | math | {
"ground_truth": "0 \\leq k \\leq 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": "Find all integer pairs $(x, y)$ such that $$ y^2 = x^3 + 2x^2 + 2x + 1. $$",
"role": "user"
}
] | math | {
"ground_truth": "(x, y) \\in \\{(-1, 0), (0, -1), (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": "A sphere is cut into three equal wedges. The circumference of the sphere is $18\\pi$ inches. What is the volume of the intersection between one wedge and the top half of the sphe... | math | {
"ground_truth": "162\\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 the number of terms in an arithmetic progression (A.P.) is even, the sum of the odd-numbered terms is 36, the sum of the even-numbered terms is 45, and the last term exceed... | 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 that point P (3, -1) and Q (-1, 2) are on the same side of the line $ax+2y=1$, find the range of the real number $a$.",
"role": "user"
}
] | math | {
"ground_truth": "(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": "For which quadratic polynomials $f(x)$ does there exist a quadratic polynomial $g(x)$ such that the equations $g(f(x)) = 0$ and $f(x)g(x) = 0$ have the same roots, which ... | math | {
"ground_truth": "f(x) = \\frac{3}{2a}(x - a)(x - 2a)",
"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 positive odd multiples of $5$ less than $5000$ are perfect squares?",
"role": "user"
}
] | math | {
"ground_truth": "7",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Five friends did gardening for their local community and earned $15, $22, $28, $35, and $50 respectively. They decide to share their total earnings equally. How much money must t... | 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": "Find the number of ordered pairs of positive integers $(a,b)$ with $a+b$ prime, $1\\leq a, b \\leq 100$ , and $\\frac{ab+1}{a+b}$ is an integer.\n\n*Author: Alex Zhu*",
... | math | {
"ground_truth": "91",
"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 $C > A > 0$ and $C$ is 20% more than $A$, and $A$ is $y\\%$ less than $C$, calculate the value of $y$.",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{50}{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": "In polar coordinates, what is the equivalent point in the standard polar coordinate representation for the point $\\left( -5, \\frac{5 \\pi}{7} \\right)$? Enter your answer in th... | math | {
"ground_truth": "\\left( 5, \\frac{12 \\pi}{7} \\right)",
"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 hyperbola $C$: $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $F$ be its right focus. A line perpendicular to one asymptote of $C$ is drawn through... | math | {
"ground_truth": "\\frac{2\\sqrt{3}}{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": "Given that $x=-1$ is a root of the quadratic equation $x^{2}+mx=3$, determine the value of $m$.",
"role": "user"
}
] | math | {
"ground_truth": "-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": "Let $n$ be a positive integer and $a,b$ be invertible integers modulo $n$ such that $a \\equiv 2b^{-1} \\pmod{n}$. What is the remainder when $ab$ is divided by $n$?",
"role"... | 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": "Given that the perimeter of square $A$ is 20 and the perimeter of square $B$ is 40, and each side of square $C$ is equal to the sum of the side lengths of square $A$ and square $... | math | {
"ground_truth": "60",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A frog trainer places one frog at each vertex of an equilateral triangle $ABC$ of unit sidelength. The trainer can make one frog jump over another along the line joining the tw... | math | {
"ground_truth": "\\frac{\\left\\lfloor \\frac{\\ell + 2}{2} \\right\\rfloor \\left\\lfloor \\frac{\\ell + 4}{2} \\right\\rfloor \\left(\\left\\lfloor \\frac{\\ell + 1}{2} \\right\\rfloor \\left\\lfloor \\frac{\\ell + 3}{2} \\right\\rfloor\\right)^2}{8}",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "A special $24$-hour digital clock operates with a fault: every time it should display the digit $2$, it instead shows an $8$. For instance, when the correct time is 2:30 AM, the ... | math | {
"ground_truth": "\\frac{5}{8}",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given 3 segments are randomly selected from 5 segments of length 2, 4, 6, 8, and 10, calculate the probability that these 3 segments can form a triangle.",
"role": "user"
}... | math | {
"ground_truth": "\\frac{3}{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": "Let the function $f(x) = (\\sqrt{3}\\sin\\omega x + \\cos\\omega x)\\cos\\omega x - \\frac{1}{2}$ ($\\omega > 0$) have a minimum positive period of $4\\pi$.\n\n(Ⅰ) Find the inter... | math | {
"ground_truth": "(\\frac{1}{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": "Let $n$ be a fixed positive integer. Oriol has $n$ cards, each of them with a $0$ written on one side and $1$ on the other. We place these cards in line, some face up and... | math | {
"ground_truth": "\\lfloor \\frac{n}{2} \\rfloor \\text{ and } \\lceil \\frac{n}{2} \\rceil",
"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": "Derive formulas expressing $\\cos 5 \\alpha$ in terms of $\\cos \\alpha$ and $\\sin 5 \\alpha$ in terms of $\\sin \\alpha$.",
"role": "user"
}
] | math | {
"ground_truth": "\\sin 5\\alpha = 16 \\sin^5 \\alpha - 20 \\sin^3 \\alpha + 5 \\sin \\alpha",
"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 $M$ be the maximum possible value of $x_1x_2+x_2x_3+\\cdots +x_5x_1$ where $x_1, x_2, \\dots, x_5$ is a permutation of $(1,2,3,4,6)$ and let $N$ be the number of permutations... | math | {
"ground_truth": "67",
"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 fraction $\\frac{1}{x-7}$ is meaningful, then the range of real number $x$ is ____.",
"role": "user"
}
] | math | {
"ground_truth": "x \\neq 7",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Find the derivative.\n\n$$\ny=e^{a x}\\left(\\frac{1}{2 a}+\\frac{a \\cdot \\cos 2 b x+2 b \\cdot \\sin 2 b x}{2\\left(a^{2}+4 b^{2}\\right)}\\right)\n$$",
"role": "user"
}... | math | {
"ground_truth": "e^{ax} \\cos^2(bx)",
"style": "rule"
} | {
"id": "numina_olympiads-0",
"rating": null,
"split": "confidence_train",
"sub_source": "olympiads"
} | olympiads |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Evaluate the expression $1 + \\frac {1}{1 + \\frac {1}{1 + \\frac{1}{2}}}$.\nA) $\\frac{7}{5}$\nB) $\\frac{8}{5}$\nC) $\\frac{9}{5}$\nD) $\\frac{10}{5}$",
"role": "user"
}
... | math | {
"ground_truth": "\\frac{8}{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 $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$. It is known that $a_1 = \\frac{4}{5}$ and $a_{n+1}$ is defined as follows:\n$$\na_{n+1} =\n\\be... | math | {
"ground_truth": "\\frac{5047}{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 common foci of the ellipse $\\frac{x^{2}}{6} + \\frac{y^{2}}{2} = 1$ and the hyperbola $\\frac{x^{2}}{3} - y^2 = 1$ are $F_1$ and $F_2$. If P is a point of intersection of th... | 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": "Let $r_1$ , $r_2$ , $\\ldots$ , $r_m$ be a given set of $m$ positive rational numbers such that $\\sum_{k=1}^m r_k = 1$ . Define the function $f$ by $f(n)= n-\\sum_{k... | math | {
"ground_truth": "0 \\text{ and } m-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": "Write the equation of the plane passing through the point $A$ and perpendicular to the vector $\\overrightarrow{B C}$.\n\n$A(-3, -1, 7)$\n\n$B(0, 2, -6)$\n\n$C(2, 3, -5)$",
"... | math | {
"ground_truth": "2x + y + z = 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": "Given four different non-zero digits, it is possible to form 24 different four-digit numbers containing each of these four digits. What is the largest prime factor of the sum of ... | math | {
"ground_truth": "101",
"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 are given a natural number $d$ . Find all open intervals of maximum length $I \\subseteq R$ such that for all real numbers $a_0,a_1,...,a_{2d-1}$ inside interval $I$ , w... | math | {
"ground_truth": "\\left(1, 1 + \\frac{1}{d}\\right)",
"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 *$n$ -trønder walk* is a walk starting at $(0, 0)$ , ending at $(2n, 0)$ with no self intersection and not leaving the first quadrant, where every step is one of the vectors... | math | {
"ground_truth": "(2n-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": "Inside an angle of \\(30^\\circ\\) with vertex \\(A\\), a point \\(K\\) is chosen such that its distances to the sides of the angle are 1 and 2. Through point \\(K\\), all possib... | math | {
"ground_truth": "4(2 + \\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": "In triangle \\( ABC \\), \\( AB = 3 \\cdot AC \\). Points \\( D \\) and \\( E \\) are placed on \\( AB \\) and \\( BC \\), respectively, such that \\( \\angle BAE = \\angle ACD \... | math | {
"ground_truth": "The final answer given the choices is \\( \\textbf{B)}\\ 60^\\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": "For all $x,y,z$ positive real numbers, find the all $c$ positive real numbers that providing $$ \\frac{x^3y+y^3z+z^3x}{x+y+z}+\\frac{4c}{xyz}\\ge2c+2 $$",
"role": "user"... | math | {
"ground_truth": "c = 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": "The numbers 1447, 1005, and 1231 have something in common: each is a four-digit number beginning with 1 that has exactly two identical digits. How many such numbers are there?",... | math | {
"ground_truth": "432",
"style": "rule"
} | {
"id": "numina_aops_forum-0",
"rating": null,
"split": "confidence_train",
"sub_source": "aops_forum"
} | aops_forum |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 5.$ Find the minimum value of\n\\[\\frac{1}{x} + \\frac{4}{y} + \\frac{9}{z}.\\]",
"role": "user"
}
] | math | {
"ground_truth": "\\frac{36}{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": "Determine with proof the number of positive integers $n$ such that a convex regular polygon with $n$ sides has interior angles whose measures, in degrees, are integers.",
... | math | {
"ground_truth": "22",
"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 the set \\\\(A=\\\\{x| \\dfrac{1}{32}\\leqslant {2}^{-x}\\leqslant 4\\\\}\\\\), \\\\(B=\\\\{x｜x^{2}+2mx-3m^{2} < 0\\\\}\\\\) where \\\\(m > 0\\\\).\n\n\\\\((1)\\\\) If \\\\(m... | math | {
"ground_truth": "(0, \\dfrac{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": "Given Circle $C_1$ of radius $12$ units touches Circle $C_2$ externally at a point $P$, and a line through $P$ intersects $C_1$ at $Q$ and $C_2$ at $R$ such that $Q$ is closer to... | math | {
"ground_truth": "10",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "From a school of 2100 students, a sample of 30 students is randomly selected. The time (in minutes) each student spends on homework outside of class is as follows: \n75, 80, 85,... | math | {
"ground_truth": "630",
"style": "rule"
} | {
"id": "numina_cn_k12-0",
"rating": null,
"split": "confidence_train",
"sub_source": "cn_k12"
} | cn_k12 |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Each side of a unit square (side length 1) is also one side of an equilateral triangle that lies in the square. Compute the area of the intersection of (the interiors of) all fo... | math | {
"ground_truth": "\\frac{9 - 5\\sqrt{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": "Solve for $y$: $\\sqrt[3]{30y + \\sqrt[3]{30y + 26}} = 26.$",
"role": "user"
}
] | math | {
"ground_truth": "585",
"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 consisting of 0s and 1s of length 21, and with the conditions that it begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no four consec... | math | {
"ground_truth": "\\text{The valid number of sequences of length 21 that meet the given conditions is 151.}",
"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 least positive integer $n$ such that $15$ divides the product\n\\[a_1a_2\\dots a_{15}\\left (a_1^n+a_2^n+\\dots+a_{15}^n \\right )\\]\n, for every positive integers ... | 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": "Given the equation of the ellipse $\\dfrac {x^{2}}{a^{2}}+ \\dfrac {y^{2}}{b^{2}}=1(a > b > 0)$, with an eccentricity of $\\dfrac { \\sqrt {3}}{2}$, find the value of $\\dfrac {a... | math | {
"ground_truth": "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": "Calculate the remainder when the sum $2150 + 2151 + 2152 + 2153 + 2154 + 2155$ is divided by 12.",
"role": "user"
}
] | math | {
"ground_truth": "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": "each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of w... | math | {
"ground_truth": "1009",
"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 arithmetic mean of five numbers is 90. If the largest of these numbers is 130, then the mean of the remaining four numbers is?\nA) 78.0\nB) 79.0\nC) 80.0\nD) 81.0\nE) 82.0",
... | math | {
"ground_truth": "C) 80.0",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
math_box | [
{
"content": "Please reason step by step, and put your final answer within \\boxed{}.",
"role": "system"
},
{
"content": "Given a sequence $\\{a\\_n\\}$ that satisfies: there exists a positive integer $T$ such that for any positive integer $n$, $a_{n+T}=a_{n}$ holds. The sequence $\\{a\\_n\\}$ is ca... | math | {
"ground_truth": "②③",
"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": "Simplify the following expression:\n\\[\n\\frac{\\left(5^{2010}\\right)^2-\\left(5^{2008}\\right)^2}{\\left(5^{2009}\\right)^2-\\left(5^{2007}\\right)^2}.\n\\]\nA) $15$\nB) $20$\... | math | {
"ground_truth": "25",
"style": "rule"
} | {
"id": "numina_synthetic_amc-0",
"rating": null,
"split": "confidence_train",
"sub_source": "synthetic_amc"
} | synthetic_amc |
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