data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Two circles are drawn in a 12-inch by 14-inch rectangle. Each circle has a diameter of 6 inches. If the circles do not extend beyond the rectangular region, what is the greatest possible distance (in inches) between the centers of the two circles? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "10\\text{ inches}",
"style": "rule"
} | {
"index": "38000",
"split": "train"
} |
math | [
{
"content": "On a spherical surface with an area of $60\\pi$, there are four points $S$, $A$, $B$, and $C$, and $\\triangle ABC$ is an equilateral triangle. The distance from the center $O$ of the sphere to the plane $ABC$ is $\\sqrt{3}$. If the plane $SAB$ is perpendicular to the plane $ABC$, then the maximum... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38001",
"split": "train"
} |
math | [
{
"content": "In the second year of junior high school this semester, a basketball game was held. In order to better train the participating athletes, the sports department plans to purchase two brands of basketballs, brand A and brand B. It is known that the unit price of brand A basketball is $40$ yuan lower ... | {
"ground_truth": "m = 20",
"style": "rule"
} | {
"index": "38002",
"split": "train"
} |
math | [
{
"content": "In a game of Fish, R2 and R3 are each holding a positive number of cards so that they are collectively holding a total of 24 cards. Each player gives an integer estimate for the number of cards he is holding, such that each estimate is an integer between $80 \\%$ of his actual number of cards and ... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "38003",
"split": "train"
} |
math | [
{
"content": "Suppose a point P(m, n) is formed by using the numbers m and n obtained from rolling a dice twice as its horizontal and vertical coordinates, respectively. The probability that point P(m, n) falls below the line x+y=4 is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "38004",
"split": "train"
} |
math | [
{
"content": "Place the arithmetic operation signs and parentheses between the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ so that the resulting expression equals 100. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38005",
"split": "train"
} |
math | [
{
"content": "Compute the number of positive real numbers $x$ that satisfy $\\left(3 \\cdot 2^{\\left\\lfloor\\log _{2} x\\right\\rfloor}-x\\right)^{16}=2022 x^{13}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38006",
"split": "train"
} |
math | [
{
"content": "What is the average of all the integer values of $M$ such that $\\frac{M}{56}$ is strictly between $\\frac{3}{7}$ and $\\frac{1}{4}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "38007",
"split": "train"
} |
math | [
{
"content": "Given a rectangular grid of dots with 5 rows and 6 columns, determine how many different squares can be formed using these dots as vertices. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "38008",
"split": "train"
} |
math | [
{
"content": "Given the complex number $z$ satisfies $|z+3-\\sqrt{3} i|=\\sqrt{3}$, what is the minimum value of $\\arg z$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{5}{6} \\pi$",
"style": "rule"
} | {
"index": "38009",
"split": "train"
} |
math | [
{
"content": "In a certain business district parking lot, the temporary parking fee is charged by time period. The fee is 6 yuan for each car if the parking duration does not exceed 1 hour, and each additional hour (or a fraction thereof is rounded up to a full hour) costs 8 yuan. It is known that two individua... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "38010",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$\n\\[\n\\sqrt{xyz} + d |x^2 - y^2 + z^2| \\ge \\frac{x + y + z}{3}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38011",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of parts into which the coordinate plane \\(xOy\\) can be divided by the graphs of 100 quadratic polynomials of the form \n\\[ y = a_{n} x^{2} + b_{n} x + c_{n} \\quad (n=1, 2, \\ldots, 100) ? \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "10001",
"style": "rule"
} | {
"index": "38012",
"split": "train"
} |
math | [
{
"content": "The value of $999 + 999$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1998",
"style": "rule"
} | {
"index": "38013",
"split": "train"
} |
math | [
{
"content": "Given $$\\overrightarrow {m} = (\\sin \\omega x + \\cos \\omega x, \\sqrt {3} \\cos \\omega x)$$, $$\\overrightarrow {n} = (\\cos \\omega x - \\sin \\omega x, 2\\sin \\omega x)$$ ($\\omega > 0$), and the function $f(x) = \\overrightarrow {m} \\cdot \\overrightarrow {n}$, if the distance between tw... | {
"ground_truth": "\\sqrt {3}",
"style": "rule"
} | {
"index": "38014",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a cyclic quadrilateral. The side lengths of $ABCD$ are distinct integers less than $15$ such that $BC \\cdot CD = AB \\cdot DA$. What is the largest possible value of $BD$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\dfrac{425}{2}}",
"style": "rule"
} | {
"index": "38015",
"split": "train"
} |
math | [
{
"content": "A phone number \\( d_{1} d_{2} d_{3}-d_{4} d_{5} d_{6} d_{7} \\) is called \"legal\" if the number \\( d_{1} d_{2} d_{3} \\) is equal to \\( d_{4} d_{5} d_{6} \\) or to \\( d_{5} d_{6} d_{7} \\). For example, \\( 234-2347 \\) is a legal phone number. Assume each \\( d_{i} \\) can be any digit from... | {
"ground_truth": "19990",
"style": "rule"
} | {
"index": "38016",
"split": "train"
} |
math | [
{
"content": "Arrange the $n$ consecutive positive integers from 1 to $n$ (where $n > 1$) in a sequence such that the sum of each pair of adjacent terms is a perfect square. Find the minimum value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38017",
"split": "train"
} |
math | [
{
"content": "There are $10$ birds on the ground. For any $5$ of them, there are at least $4$ birds on a circle. Determine the least possible number of birds on the circle with the most birds. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "38018",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of all points with coordinates $(x,y,z)$, where $x$, $y$, and $z$ are each chosen from the set $\\{0,1,2\\}$. How many equilateral triangles have all their vertices in $S$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "38019",
"split": "train"
} |
math | [
{
"content": "A sequence of ten $0$s and/or $1$s is randomly generated. If the probability that the sequence does not contain two consecutive $1$s can be written in the form $\\dfrac{m}{n}$, where $m,n$ are relatively prime positive integers, find $m+n$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "73",
"style": "rule"
} | {
"index": "38020",
"split": "train"
} |
math | [
{
"content": "Suppose $a,$ $b,$ and $c$ are real numbers such that\n\\[\\frac{ac}{a + b} + \\frac{ba}{b + c} + \\frac{cb}{c + a} = -9\\]and\n\\[\\frac{bc}{a + b} + \\frac{ca}{b + c} + \\frac{ab}{c + a} = 10.\\]Compute the value of\n\\[\\frac{b}{a + b} + \\frac{c}{b + c} + \\frac{a}{c + a}.\\] Please output the ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "38021",
"split": "train"
} |
math | [
{
"content": "Given $\\overrightarrow{a}=(\\tan (\\theta+ \\frac {\\pi}{12}),1)$ and $\\overrightarrow{b}=(1,-2)$, where $\\overrightarrow{a} \\perp \\overrightarrow{b}$, find the value of $\\tan (2\\theta+ \\frac {5\\pi}{12})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac{1}{7}",
"style": "rule"
} | {
"index": "38022",
"split": "train"
} |
math | [
{
"content": "Given positive numbers $a$, $b$, $c$ satisfying: $a^2+ab+ac+bc=6+2\\sqrt{5}$, find the minimum value of $3a+b+2c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{10}+2\\sqrt{2}",
"style": "rule"
} | {
"index": "38023",
"split": "train"
} |
math | [
{
"content": "At a painting club meeting, 7 friends are present. They need to create two separate teams: one team of 4 members, and another team of 2 members for different competitions. How many distinct ways can they form these teams? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "38024",
"split": "train"
} |
math | [
{
"content": "Using the same Rotokas alphabet, how many license plates of five letters are possible that begin with G, K, or P, end with T, cannot contain R, and have no letters that repeat? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "630",
"style": "rule"
} | {
"index": "38025",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ such that $\\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4096",
"style": "rule"
} | {
"index": "38026",
"split": "train"
} |
math | [
{
"content": "Calculate the probability that the line $y=kx+k$ intersects with the circle ${{\\left( x-1 \\right)}^{2}}+{{y}^{2}}=1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{3}",
"style": "rule"
} | {
"index": "38027",
"split": "train"
} |
math | [
{
"content": "Scenario: In a math activity class, the teacher presented a set of questions and asked the students to explore the pattern by reading the following solution process:\n\n$\\sqrt{1+\\frac{5}{4}}=\\sqrt{\\frac{9}{4}}=\\sqrt{{(\\frac{3}{2})}^{2}}=\\frac{3}{2}$;\n$\\sqrt{1+\\frac{7}{9}}=\\sqrt{\\frac{1... | {
"ground_truth": "\\frac{1012}{1011}",
"style": "rule"
} | {
"index": "38028",
"split": "train"
} |
math | [
{
"content": "When three standard dice are tossed, the numbers $a, b, c$ are obtained. Find the probability that the product $abc = 72$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{24}",
"style": "rule"
} | {
"index": "38029",
"split": "train"
} |
math | [
{
"content": "Express the quotient $1021_3 \\div 11_3$ in base $3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22_3",
"style": "rule"
} | {
"index": "38030",
"split": "train"
} |
math | [
{
"content": "The entire exterior of a solid $6 \\times 6 \\times 3$ rectangular prism is painted. Then, the prism is cut into $1 \\times 1 \\times 1$ cubes. How many of these cubes have no painted faces? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "38031",
"split": "train"
} |
math | [
{
"content": "Find the point in the $xz$-plane that is equidistant from the points $(1,-1,0),$ $(2,1,2),$ and $(3,2,-1).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( \\frac{31}{10}, 0, \\frac{1}{5} \\right)",
"style": "rule"
} | {
"index": "38032",
"split": "train"
} |
math | [
{
"content": "Two fair dice, each with at least $6$ faces are rolled. On each face of each die is printed a distinct integer from $1$ to the number of faces on that die, inclusive. The probability of rolling a sum of $7$ is $\\frac34$ of the probability of rolling a sum of $10,$ and the probability of rolling a... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "38033",
"split": "train"
} |
math | [
{
"content": "How many four-digit numbers are there in which at least one digit occurs more than once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4464",
"style": "rule"
} | {
"index": "38034",
"split": "train"
} |
math | [
{
"content": "The number of points common to the graphs of\n$(x-y+2)(3x+y-4)=0$ and $(x+y-2)(2x-5y+7)=0$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "38035",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sin(2x-\\frac{\\pi}{3})-1$, find the probability that a real number $a$ randomly selected from the interval $\\left[0,\\frac{\\pi}{2}\\right]$ satisfies $f(a) > 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38036",
"split": "train"
} |
math | [
{
"content": "Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "h",
"style": "rule"
} | {
"index": "38037",
"split": "train"
} |
math | [
{
"content": "Given the curve \n\\[\n(x - \\arcsin \\alpha)(x - \\arccos \\alpha) + (y - \\arcsin \\alpha)(y + \\arccos \\alpha) = 0 \n\\]\nis intersected by the line \\( x = \\frac{\\pi}{4} \\), determine the minimum value of the length of the chord intercepted as \\( \\alpha \\) varies. Please output the fina... | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "38038",
"split": "train"
} |
math | [
{
"content": "A delicious circular pie with diameter $12\\text{ cm}$ is cut into three equal-sized sector-shaped pieces. Let $l$ be the number of centimeters in the length of the longest line segment that may be drawn in one of these pieces. What is $l^2$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "38039",
"split": "train"
} |
math | [
{
"content": "Let $\\zeta=e^{2 \\pi i / 99}$ and $\\omega=e^{2 \\pi i / 101}$. The polynomial $$x^{9999}+a_{9998} x^{9998}+\\cdots+a_{1} x+a_{0}$$ has roots $\\zeta^{m}+\\omega^{n}$ for all pairs of integers $(m, n)$ with $0 \\leq m<99$ and $0 \\leq n<101$. Compute $a_{9799}+a_{9800}+\\cdots+a_{9998}$. Please o... | {
"ground_truth": "14849-\\frac{9999}{200}\\binom{200}{99}",
"style": "rule"
} | {
"index": "38040",
"split": "train"
} |
math | [
{
"content": "A hydra consists of several heads and several necks, where each neck joins two heads. When a hydra's head $A$ is hit by a sword, all the necks from head $A$ disappear, but new necks grow up to connect head $A$ to all the heads which weren't connected to $A$ . Heracle defeats a hydra by cu... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38041",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be nonzero real numbers. Find the minimum value of\n\\[a^2 + b^2 + \\frac{1}{a^2} + \\frac{b}{a}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "38042",
"split": "train"
} |
math | [
{
"content": "Ewan writes out a sequence where he counts by 11s starting at 3. The resulting sequence is $3, 14, 25, 36, \\ldots$. What is a number that will appear in Ewan's sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "113",
"style": "rule"
} | {
"index": "38043",
"split": "train"
} |
math | [
{
"content": "Consider a $4 \\times 4$ grid of squares. Aziraphale and Crowley play a game on this grid, alternating turns, with Aziraphale going first. On Aziraphales turn, he may color any uncolored square red, and on Crowleys turn, he may color any uncolored square blue. The game ends when all the squares ar... | {
"ground_truth": "\\[ 6 \\]",
"style": "rule"
} | {
"index": "38044",
"split": "train"
} |
math | [
{
"content": "Peter Ivanovich, along with 49 other men and 50 women, are seated in a random order around a round table. We call a man satisfied if a woman is sitting next to him. Find:\n\na) The probability that Peter Ivanovich is satisfied.\n\nb) The expected number of satisfied men. Please output the final an... | {
"ground_truth": "\\frac{1250}{33}",
"style": "rule"
} | {
"index": "38045",
"split": "train"
} |
math | [
{
"content": "Consider all ordered pairs of integers $(a, b)$ such that $1 \\leq a \\leq b \\leq 100$ and $$\\frac{(a+b)(a+b+1)}{a b}$$ is an integer. Among these pairs, find the one with largest value of $b$. If multiple pairs have this maximal value of $b$, choose the one with largest $a$. For example choose ... | {
"ground_truth": "(35,90)",
"style": "rule"
} | {
"index": "38046",
"split": "train"
} |
math | [
{
"content": "The young man paid with $100 for a gift that cost $18 and received $79 in change from Mr. Wang. Mr. Wang then returned the counterfeit $100 bill to the neighbor. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "97",
"style": "rule"
} | {
"index": "38047",
"split": "train"
} |
math | [
{
"content": "Given a fixed circle \\\\(F_{1}:(x+2)^{2}+y^{2}=24\\\\) and a moving circle \\\\(N\\\\) passing through point \\\\(F_{2}(2,0)\\\\) and tangent to circle \\\\(F_{1}\\\\), denote the locus of the center of circle \\\\(N\\\\) as \\\\(E\\\\).\n\n\\\\((I)\\\\) Find the equation of the locus \\\\(E\\\\)... | {
"ground_truth": "-\\frac{5}{9}",
"style": "rule"
} | {
"index": "38048",
"split": "train"
} |
math | [
{
"content": "Take a unit sphere \\(S\\), i.e., a sphere with radius 1. Circumscribe a cube \\(C\\) about \\(S\\), and inscribe a cube \\(D\\) in \\(S\\) such that every edge of cube \\(C\\) is parallel to some edge of cube \\(D\\). What is the shortest possible distance from a point on a face of \\(C\\) to a p... | {
"ground_truth": "1 - \\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "38049",
"split": "train"
} |
math | [
{
"content": "A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\\log 101$ and $\\log 2018$ . The last side of his triangle has side length $\\log n$ , where $n$ is an integer. How many possible values are there for $n$ ?\n\n*2018 CCA Math Bonanza Ind... | {
"ground_truth": "203797",
"style": "rule"
} | {
"index": "38050",
"split": "train"
} |
math | [
{
"content": "A natural number $N$ greater than 20 is a palindromic number in both base 14 and base 20 (A palindromic number is a number that reads the same forwards and backwards, such as 12321 and 3443, but not 12331). What is the smallest value of $N$ (expressed in decimal)? Please output the final answer wi... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "38051",
"split": "train"
} |
math | [
{
"content": "A bivalent metal element is used in a chemical reaction. When 3.5g of the metal is added into 50g of a dilute hydrochloric acid solution with a mass percentage of 18.25%, there is some metal leftover after the reaction finishes. When 2.5g of the metal is added into the same mass and mass percentag... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38052",
"split": "train"
} |
math | [
{
"content": "Find all pairs $(a,\\, b)$ of positive integers such that $2a-1$ and $2b+1$ are coprime and $a+b$ divides $4ab+1.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(a, a+1)",
"style": "rule"
} | {
"index": "38053",
"split": "train"
} |
math | [
{
"content": "Find the number of even digits in the base-7 representation of $403_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "38054",
"split": "train"
} |
math | [
{
"content": "In a lathe workshop, parts are turned from steel blanks, one part from one blank. The shavings left after processing three blanks can be remelted to get exactly one blank. How many parts can be made from nine blanks? What about from fourteen blanks? How many blanks are needed to get 40 parts? Plea... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "38055",
"split": "train"
} |
math | [
{
"content": "Given that Adam has a triangular field ABC with AB = 5, BC = 8, and CA = 11, and he intends to separate the field into two parts by building a straight fence from A to a point D on side BC such that AD bisects ∠BAC, find the area of the part of the field ABD. Please output the final answer within ... | {
"ground_truth": "\\frac{5 \\sqrt{21}}{4}",
"style": "rule"
} | {
"index": "38056",
"split": "train"
} |
math | [
{
"content": "The diagram shows three rectangles and three straight lines. What is the value of \\( p + q + r \\)?\nA) 135 \nB) 180 \nC) 210 \nD) 225 \nE) 270 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "38057",
"split": "train"
} |
math | [
{
"content": "Jacob flipped a fair coin five times. In the first three flips, the coin came up heads exactly twice. In the last three flips, the coin also came up heads exactly twice. What is the probability that the third flip was heads? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "38058",
"split": "train"
} |
math | [
{
"content": "The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "38059",
"split": "train"
} |
math | [
{
"content": "The matrices\n\\[\\begin{pmatrix} a & 1 & b \\\\ 2 & 2 & 3 \\\\ c & 5 & d \\end{pmatrix} \\quad \\text{and} \\quad \\begin{pmatrix} -5 & e & -11 \\\\ f & -13 & g \\\\ 2 & h & 4 \\end{pmatrix}\\]are inverses. Find $a + b + c + d + e + f + g + h.$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "38060",
"split": "train"
} |
math | [
{
"content": "A hockey team has 6 more red helmets than blue helmets. The ratio of red helmets to blue helmets is $5:3$. What is the total number of red helmets and blue helmets? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "38061",
"split": "train"
} |
math | [
{
"content": "Compute the definite integral:\n\n$$\n\\int_{-\\pi}^{0} 2^{8} \\sin ^{6} x \\cos ^{2} x \\, dx\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\pi",
"style": "rule"
} | {
"index": "38062",
"split": "train"
} |
math | [
{
"content": "At a variety show, there are seven acts: dance, comic dialogue, sketch, singing, magic, acrobatics, and opera. When arranging the program order, the conditions are that dance, comic dialogue, and sketch cannot be adjacent to each other. How many different arrangements of the program are possible? ... | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": "38063",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = 3x^2 + 2x + 1$, if $\\int_{-1}^{1} f(x)\\,dx = 2f(a)$, then $a = \\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38064",
"split": "train"
} |
math | [
{
"content": "Given the three interior angles \\( A, B, C \\) of \\(\\triangle ABC\\) satisfy \\( A = 3B = 9C \\), find the value of \\( \\cos A \\cos B + \\cos B \\cos C + \\cos C \\cos A \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1/4",
"style": "rule"
} | {
"index": "38065",
"split": "train"
} |
math | [
{
"content": "How many ways are there to put 4 indistinguishable balls into 2 indistinguishable boxes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "38066",
"split": "train"
} |
math | [
{
"content": "Two divisions of an oil extraction company compete with each other for the amount of oil extracted. Depending on the availability of free tanks, each of them can extract the amount of tons defined by the following payoff matrix:\n\n$$\n\\text { Extraction }=\\left(\\begin{array}{cc}\n12 & 22 \\\\\... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "38067",
"split": "train"
} |
math | [
{
"content": "Find the value of \\(10 \\cdot \\operatorname{ctg}(\\operatorname{arcctg} 3 + \\operatorname{arcctg} 7 + \\operatorname{arcctg} 13 + \\operatorname{arcctg} 21)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "38068",
"split": "train"
} |
math | [
{
"content": "Let $f(n)$ be the number of times you have to hit the $\\sqrt{ }$ key on a calculator to get a number less than 2 starting from $n$. For instance, $f(2)=1, f(5)=2$. For how many $1<m<2008$ is $f(m)$ odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "242",
"style": "rule"
} | {
"index": "38069",
"split": "train"
} |
math | [
{
"content": "Three-digit powers of $2$ and $5$ are used in this \"cross-number\" puzzle. What is the only possible digit for the outlined square?\n\\[\\begin{array}{lcl} \\textbf{ACROSS} & & \\textbf{DOWN} \\\\ \\textbf{2}.~ 2^m & & \\textbf{1}.~ 5^n \\end{array}\\] Please output the final answer within \\boxe... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "38070",
"split": "train"
} |
math | [
{
"content": "Given that the volume of the tetrahedron \\(ABCD\\) is 40, \\(AB = 7\\), \\(BC = 8\\), \\(AC = 9\\), and the orthogonal projection of vertex \\(D\\) onto the plane \\(ABC\\) is precisely the incenter \\(H\\) of triangle \\(ABC\\), what is the surface area of the tetrahedron? Please output the fina... | {
"ground_truth": "60 + 12 \\sqrt{5}",
"style": "rule"
} | {
"index": "38071",
"split": "train"
} |
math | [
{
"content": "Triangle $PQR$ has positive integer side lengths with $PQ = PR$. Let $J$ be the intersection of the bisectors of $\\angle Q$ and $\\angle R$. Suppose $QJ = 10$. Find the smallest possible perimeter of $\\triangle PQR$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1818",
"style": "rule"
} | {
"index": "38072",
"split": "train"
} |
math | [
{
"content": "A computer software retailer has 1200 copies of a new software package to sell. From past experience, she knows that:\n\n- Half of them will sell right away at the original price she sets,\n- Two-thirds of the remainder will sell later when their price is reduced by 40%, and\n- The remaining copie... | {
"ground_truth": "$80.90",
"style": "rule"
} | {
"index": "38073",
"split": "train"
} |
math | [
{
"content": "Given the ellipse $E$: $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$, its minor axis length is $2$, and the eccentricity is $\\frac{\\sqrt{6}}{3}$. The line $l$ passes through the point $(-1,0)$ and intersects the ellipse $E$ at points $A$ and $B$. $O$ is the coordinate origin.\n\n(1) ... | {
"ground_truth": "\\frac{\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "38074",
"split": "train"
} |
math | [
{
"content": "A circle with center $O$ has radius 25. Chord $\\overline{AB}$ of length 30 and chord $\\overline{CD}$ of length 14 intersect at point $P$. The distance between the midpoints of the two chords is 12. The quantity $OP^2$ can be represented as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime p... | {
"ground_truth": "57",
"style": "rule"
} | {
"index": "38075",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$ where $a_n > 0$, $a_1=1$, $a_{n+2}= \\frac {1}{a_n+1}$, and $a_{100}=a_{96}$, find the value of $a_{2014}+a_3$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "38076",
"split": "train"
} |
math | [
{
"content": "If both $a$ and $b$ in the fraction $\\frac{a+b}{a^2+b^2}$ are enlarged by a factor of $2$, then calculate the new value of the fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "38077",
"split": "train"
} |
math | [
{
"content": "In the plane Cartesian coordinate system \\(xOy\\), point \\(P\\) is a moving point on the line \\(y = -x - 2\\). Two tangents to the parabola \\(y = \\frac{x^2}{2}\\) are drawn through point \\(P\\), and the points of tangency are \\(A\\) and \\(B\\). Find the minimum area of the triangle \\(PAB\... | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "38078",
"split": "train"
} |
math | [
{
"content": "Consider the set $E$ of all positive integers $n$ such that when divided by $9,10,11$ respectively, the remainders(in that order) are all $>1$ and form a non constant geometric progression. If $N$ is the largest element of $E$ , find the sum of digits of $E$ Please output the final ans... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "38079",
"split": "train"
} |
math | [
{
"content": "What is the probability, expressed as a fraction, of drawing a marble that is either green or white from a bag containing 4 green, 3 white, 5 red, and 6 blue marbles? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{18}",
"style": "rule"
} | {
"index": "38080",
"split": "train"
} |
math | [
{
"content": "Given that $F$ is the right focus of the hyperbola $C$: $x^{2}- \\frac{y^{2}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,6 \\sqrt {6})$ is a point on the $y$-axis. The minimum area of $\\triangle APF$ is $\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "6+9 \\sqrt {6}",
"style": "rule"
} | {
"index": "38081",
"split": "train"
} |
math | [
{
"content": "In trapezoid $ABCD$, $CD$ is three times the length of $AB$. If the area of trapezoid $ABCD$ is $18$ square units, what is the area of $\\triangle ABC?$\n\n[asy]\ndraw((0,0)--(1,6)--(10,6)--(15,0)--cycle);\ndraw((10,6)--(0,0));\nlabel(\"$C$\",(0,0),W);\nlabel(\"$A$\",(1,6),NW);\nlabel(\"$B$\",(10,... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "38082",
"split": "train"
} |
math | [
{
"content": "Given $a\\in \\mathbb{R}$, $i$ is the imaginary unit. If $z=1+ai$ and $z\\cdot \\overline{z}=4$, solve for the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{3}",
"style": "rule"
} | {
"index": "38083",
"split": "train"
} |
math | [
{
"content": "Rhombus $ABCD$ has side length $2$ and $\\angle B = 120^\\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "38084",
"split": "train"
} |
math | [
{
"content": "Find the number of ways in which the nine numbers $$1,12,123,1234, \\ldots, 123456789$$ can be arranged in a row so that adjacent numbers are relatively prime. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "38085",
"split": "train"
} |
math | [
{
"content": "What expression describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "a>\\frac{1}{2}",
"style": "rule"
} | {
"index": "38086",
"split": "train"
} |
math | [
{
"content": "In right triangle \\\\(ABC\\\\), where \\\\(\\angle C = 90^{\\circ}\\\\) and \\\\(\\angle A = 30^{\\circ}\\\\), the eccentricity of the ellipse, which has \\\\(A\\\\) and \\\\(B\\\\) as its foci and passes through point \\\\(C\\\\), is \\_\\_\\_\\_\\_. Please output the final answer within \\boxed... | {
"ground_truth": "\\sqrt{3} - 1",
"style": "rule"
} | {
"index": "38087",
"split": "train"
} |
math | [
{
"content": "In $\\Delta ABC$, $\\angle A$ satisfies the condition $\\sqrt{3}\\sin A+\\cos A=1$, and $AB=2$, $BC=2\\sqrt{3}$. Determine the area $S=$_______ of $\\Delta ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "38088",
"split": "train"
} |
math | [
{
"content": "A line initially 1 inch long grows according to a new law, where the first term is the initial length and an additional linear term is introduced.\n\\[1+\\frac{1}{3}\\sqrt{3}+2\\cdot\\frac{1}{9}+3\\cdot\\frac{1}{27}\\sqrt{3}+4\\cdot\\frac{1}{81}+\\cdots\\]\nDetermine the limit of the length of the... | {
"ground_truth": "\\frac{1}{3}(4+\\sqrt{3})",
"style": "rule"
} | {
"index": "38089",
"split": "train"
} |
math | [
{
"content": "Convex quadrilateral $A B C D$ has right angles $\\angle A$ and $\\angle C$ and is such that $A B=B C$ and $A D=C D$. The diagonals $A C$ and $B D$ intersect at point $M$. Points $P$ and $Q$ lie on the circumcircle of triangle $A M B$ and segment $C D$, respectively, such that points $P, M$, and $... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "38090",
"split": "train"
} |
math | [
{
"content": "A dice is thrown twice. Let $a$ be the number of dots that appear on the first throw and $b$ be the number of dots that appear on the second throw. Given the system of equations $\\begin{cases} ax+by=2 \\\\\\\\ 2x+y=3\\\\end{cases}$.\n\n(I) Find the probability that the system of equations has onl... | {
"ground_truth": "\\frac{7}{12}",
"style": "rule"
} | {
"index": "38091",
"split": "train"
} |
math | [
{
"content": "A geometric sequence of positive integers starts with the first term as 5, and the fifth term of the sequence is 320. Determine the second term of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38092",
"split": "train"
} |
math | [
{
"content": "Given the system of equations:\n$$\n\\begin{cases}\nx - 2y = z - 2u \\\\\n2yz = ux\n\\end{cases}\n$$\nfor each set of positive real number solutions \\{x, y, z, u\\}, where $z \\geq y$, there exists a positive real number $M$ such that $M \\leq \\frac{z}{y}$. Find the maximum value of $M$. Please ... | {
"ground_truth": "6 + 4\\sqrt{2}",
"style": "rule"
} | {
"index": "38093",
"split": "train"
} |
math | [
{
"content": "A circle is tangent to the lines $4x - 3y = 30$ and $4x - 3y = -10.$ The center of the circle lies on the line $2x + y = 0.$ Find the center of the circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1,-2)",
"style": "rule"
} | {
"index": "38094",
"split": "train"
} |
math | [
{
"content": "How many positive integers $k$ are there such that $$\\frac{k}{2013}(a+b)=\\operatorname{lcm}(a, b)$$ has a solution in positive integers $(a, b)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1006",
"style": "rule"
} | {
"index": "38095",
"split": "train"
} |
math | [
{
"content": "For a real number $x$ , let $f(x)=\\int_0^{\\frac{\\pi}{2}} |\\cos t-x\\sin 2t|\\ dt$ .\n\n(1) Find the minimum value of $f(x)$ .\n\n(2) Evaluate $\\int_0^1 f(x)\\ dx$ .\n\n*2011 Tokyo Institute of Technology entrance exam, Problem 2* Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\frac{1}{4} + \\frac{1}{2} \\ln 2",
"style": "rule"
} | {
"index": "38096",
"split": "train"
} |
math | [
{
"content": "The integers from 1 to \\( n \\), inclusive, are equally spaced in order around a circle. The diameter through the position of the integer 7 also goes through the position of 23. What is the value of \\( n \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "38097",
"split": "train"
} |
math | [
{
"content": "A certain interest group conducted a survey on the reading of classic literature by people of different age groups in a certain region. The relevant data is shown in the table below:\n\n| Age Interval | $[0,10)$ | $[10,15)$ | $[15,20)$ | $[20,25)$ | $[25,30)$ |\n|--------------|----------|--------... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "38098",
"split": "train"
} |
math | [
{
"content": "Four distinct points, $A$, $B$, $C$, and $D$, are to be selected from $1996$ points evenly spaced around a circle. All quadruples are equally likely to be chosen. What is the probability that the chord $\\overline{AB}$ intersects the chord $\\overline{CD}$? Please output the final answer within \\... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "38099",
"split": "train"
} |
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