data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Triangle $\\triangle P Q R$, with $P Q=P R=5$ and $Q R=6$, is inscribed in circle $\\omega$. Compute the radius of the circle with center on $\\overline{Q R}$ which is tangent to both $\\omega$ and $\\overline{P Q}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{20}{9}",
"style": "rule"
} | {
"index": "300",
"split": "train"
} |
math | [
{
"content": "Consider numbers of the form \\(10n + 1\\), where \\(n\\) is a positive integer. We shall call such a number 'grime' if it cannot be expressed as the product of two smaller numbers, possibly equal, both of which are of the form \\(10k + 1\\), where \\(k\\) is a positive integer. How many 'grime nu... | {
"ground_truth": "87",
"style": "rule"
} | {
"index": "301",
"split": "train"
} |
math | [
{
"content": "In the plane Cartesian coordinate system \\(xOy\\), the set of points\n$$\n\\begin{aligned}\nK= & \\{(x, y) \\mid(|x|+|3 y|-6) \\cdot \\\\\n& (|3 x|+|y|-6) \\leqslant 0\\}\n\\end{aligned}\n$$\ncorresponds to an area in the plane with the measurement of ______. Please output the final answer within... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "302",
"split": "train"
} |
math | [
{
"content": "The distances from point \\( P \\), which lies inside an equilateral triangle, to its vertices are 3, 4, and 5. Find the area of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9 + \\frac{25\\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "303",
"split": "train"
} |
math | [
{
"content": "Two teachers and 4 students need to be divided into 2 groups, each consisting of 1 teacher and 2 students. Calculate the number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "304",
"split": "train"
} |
math | [
{
"content": "I have a drawer with 4 shirts, 5 pairs of shorts, and 6 pairs of socks in it. If I reach in and randomly remove three articles of clothing, what is the probability that I get one shirt, one pair of shorts, and one pair of socks? (Treat pairs of socks as one article of clothing.) Please output the ... | {
"ground_truth": "\\frac{24}{91}",
"style": "rule"
} | {
"index": "305",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(2x +3y)^{20}$ , find the number of coefficients divisible by $144$ .\n\n*Proposed by Hannah Shen* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "306",
"split": "train"
} |
math | [
{
"content": "A student correctly added the two two-digit numbers on the left of the board and got the answer 137. What answer will she obtain if she adds the two four-digit numbers on the right of the board? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13837",
"style": "rule"
} | {
"index": "307",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be a convex quadrilateral such that \\(AB + BC = 2021\\) and \\(AD = CD\\). We are also given that \\(\\angle ABC = \\angle CDA = 90^\\circ\\). Determine the length of the diagonal \\(BD\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2021}{\\sqrt{2}}",
"style": "rule"
} | {
"index": "308",
"split": "train"
} |
math | [
{
"content": "Hou Yi shot three arrows at each of three targets. On the first target, he scored 29 points, and on the second target, he scored 43 points. How many points did he score on the third target? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "309",
"split": "train"
} |
math | [
{
"content": "Zhang Hua, Li Liang, and Wang Min each sent out x, y, z New Year's cards, respectively. If it is known that the least common multiple of x, y, z is 60, the greatest common divisor of x and y is 4, and the greatest common divisor of y and z is 3, then how many New Year's cards did Zhang Hua send ou... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "310",
"split": "train"
} |
math | [
{
"content": "What is the average number of pairs of consecutive integers in a randomly selected subset of $5$ distinct integers chosen from the set $\\{ 1, 2, 3, \\dots, 30\\}$? (For example the set $\\{1, 17, 18, 19, 30\\}$ has $2$ pairs of consecutive integers.) Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "311",
"split": "train"
} |
math | [
{
"content": "Four positive integers $A$, $B$, $C$ and $D$ have a sum of 36. If $A+2 = B-2 = C \\times 2 = D \\div 2$, what is the value of the product $A \\times B \\times C \\times D$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3840",
"style": "rule"
} | {
"index": "312",
"split": "train"
} |
math | [
{
"content": "If $f(x)=\\dfrac{5x+1}{x-1}$, find the value of $f(7)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "313",
"split": "train"
} |
math | [
{
"content": "For an arithmetic sequence $a_1,$ $a_2,$ $a_3,$ $\\dots,$ let\n\\[S_n = a_1 + a_2 + a_3 + \\dots + a_n,\\]and let\n\\[T_n = S_1 + S_2 + S_3 + \\dots + S_n.\\]If you are told the value of $S_{2019},$ then you can uniquely determine the value of $T_n$ for some integer $n.$ What is this integer $n$?... | {
"ground_truth": "3028",
"style": "rule"
} | {
"index": "314",
"split": "train"
} |
math | [
{
"content": "Let $S_n$ be the sum of the first $n$ terms of a geometric sequence $\\{a_n\\}$. Given that $a_3 = 8a_6$, find the value of $\\frac{S_4}{S_2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "315",
"split": "train"
} |
math | [
{
"content": "If triangle $ABC$ has sides of length $AB = 6,$ $AC = 5,$ and $BC = 4,$ then calculate\n\\[\\frac{\\cos \\frac{A - B}{2}}{\\sin \\frac{C}{2}} - \\frac{\\sin \\frac{A - B}{2}}{\\cos \\frac{C}{2}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "316",
"split": "train"
} |
math | [
{
"content": "For each integer $n$ greater than 1, let $G(n)$ be the number of solutions of the equation $\\sin x = \\sin (n^2 x)$ on the interval $[0, 2\\pi]$. What is $\\sum_{n=2}^{100} G(n)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "676797",
"style": "rule"
} | {
"index": "317",
"split": "train"
} |
math | [
{
"content": "Let $N$ be the number of ways in which the letters in \"HMMTHMMTHMMTHMMTHMMTHMMT\" (\"HMMT\" repeated six times) can be rearranged so that each letter is adjacent to another copy of the same letter. For example, \"MMMMMMTTTTTTHHHHHHHHHHHH\" satisfies this property, but \"HMMMMMTTTTTTHHHHHHHHHHHM\"... | {
"ground_truth": "78556",
"style": "rule"
} | {
"index": "318",
"split": "train"
} |
math | [
{
"content": "Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \\geq v \\geq f(x) - x$ , then $f(z) = v + z$ , for some number $z$ between $x$ and $y$ . $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, the... | {
"ground_truth": "1988",
"style": "rule"
} | {
"index": "319",
"split": "train"
} |
math | [
{
"content": "Compute the limit of the function:\n\n$$\\lim _{x \\rightarrow \\frac{\\pi}{2}} \\frac{e^{\\sin 2 x}-e^{\\tan 2 x}}{\\ln \\left(\\frac{2 x}{\\pi}\\right)}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2\\pi",
"style": "rule"
} | {
"index": "320",
"split": "train"
} |
math | [
{
"content": "Person A, Person B, Person C, and Person D share 2013 candies. Person A gets 10 more candies than twice the amount Person B has, 18 more candies than three times the amount Person C has, and 55 less candies than five times the amount Person D has. How many candies does Person A get? Please output ... | {
"ground_truth": "990",
"style": "rule"
} | {
"index": "321",
"split": "train"
} |
math | [
{
"content": "Alice and Carol each have a rectangular sheet of paper. Alice has a sheet of paper measuring 10 inches by 12 inches and rolls it into a tube by taping the two 10-inch sides together. Carol rolls her sheet, which measures 8 inches by 15 inches, by taping the two 15-inch sides together. Calculate $\... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "322",
"split": "train"
} |
math | [
{
"content": "Given $(x^2+1)(x-2)^8 = a + a_1(x-1) + a_2(x-1)^2 + \\ldots + a_{10}(x-1)^{10}$, find the value of $a_1 + a_2 + \\ldots + a_{10}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "323",
"split": "train"
} |
math | [
{
"content": "Find all the solutions to\n\\[\\sqrt{(2 + \\sqrt{3})^x} + \\sqrt{(2 - \\sqrt{3})^x} = 4.\\]Enter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2,-2",
"style": "rule"
} | {
"index": "324",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $AB = 3$, $BC = 4$, $AC = 5$, and $BD$ is the angle bisector from vertex $B$. If $BD = k \\sqrt{2}$, then find $k$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{12}{7}",
"style": "rule"
} | {
"index": "325",
"split": "train"
} |
math | [
{
"content": "Find the matrix $\\mathbf{M},$ with real entries, such that\n\\[\\mathbf{M}^3 - 4 \\mathbf{M}^2 + 5 \\mathbf{M} = \\begin{pmatrix} 10 & 20 \\\\ 5 & 10 \\end{pmatrix}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} 2 & 4 \\\\ 1 & 2 \\end{pmatrix}",
"style": "rule"
} | {
"index": "326",
"split": "train"
} |
math | [
{
"content": "In the equation $\\frac{1}{m} + \\frac{1}{n} = \\frac{1}{4}$, where $m$ and $n$ are positive integers, determine the sum of all possible values for $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "327",
"split": "train"
} |
math | [
{
"content": "Find the value of \\( \\cos (\\angle OBC + \\angle OCB) \\) in triangle \\( \\triangle ABC \\), where angle \\( \\angle A \\) is an obtuse angle, \\( O \\) is the orthocenter, and \\( AO = BC \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "328",
"split": "train"
} |
math | [
{
"content": "If $x + 2y = 30$, what is the value of $\\frac{x}{5} + \\frac{2y}{3} + \\frac{2y}{5} + \\frac{x}{3}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "329",
"split": "train"
} |
math | [
{
"content": "The concept of negative numbers first appeared in the ancient Chinese mathematical book \"Nine Chapters on the Mathematical Art.\" If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as $-5$ yuan. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "330",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "331",
"split": "train"
} |
math | [
{
"content": "Find the least upper bound for the set of values \\((x_1 x_2 + 2x_2 x_3 + x_3 x_4) / (x_1^2 + x_2^2 + x_3^2 + x_4^2)\\), where \\(x_i\\) are real numbers, not all zero. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}+1}{2}",
"style": "rule"
} | {
"index": "332",
"split": "train"
} |
math | [
{
"content": "For any real number a and positive integer k, define\n$\\binom{a}{k} = \\frac{a(a-1)(a-2)\\cdots(a-(k-1))}{k(k-1)(k-2)\\cdots(2)(1)}$\nWhat is\n$\\binom{-\\frac{1}{2}}{100} \\div \\binom{\\frac{1}{2}}{100}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-199",
"style": "rule"
} | {
"index": "333",
"split": "train"
} |
math | [
{
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2880",
"style": "rule"
} | {
"index": "334",
"split": "train"
} |
math | [
{
"content": "On a table, there are 2004 boxes, each containing one ball. It is known that some of the balls are white, and their number is even. You are allowed to point to any two boxes and ask if there is at least one white ball in them. What is the minimum number of questions needed to guarantee the identif... | {
"ground_truth": "2003",
"style": "rule"
} | {
"index": "335",
"split": "train"
} |
math | [
{
"content": "The yearly changes in the population census of a town for four consecutive years are, respectively, 25% increase, 25% increase, 25% decrease, 25% decrease. The net change over the four years, to the nearest percent, is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "336",
"split": "train"
} |
math | [
{
"content": "Given an ellipse E: $$\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$$ ($$a > b > 0$$) passing through point Q ($$\\frac{\\sqrt{2}}{2}, \\frac{\\sqrt{3}}{2}$$), the product of the slopes of the lines connecting the moving point P on the ellipse to the two endpoints of the minor axis is $$-\\frac{1}{2}$$.... | {
"ground_truth": "\\frac{4}{3}",
"style": "rule"
} | {
"index": "337",
"split": "train"
} |
math | [
{
"content": "Thirty girls - 13 in red dresses and 17 in blue dresses - were dancing around a Christmas tree. Afterwards, each was asked if the girl to her right was in a blue dress. It turned out that only those who stood between two girls in dresses of the same color answered correctly. How many girls could h... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "338",
"split": "train"
} |
math | [
{
"content": "Find all odd positive integers $n>1$ such that there is a permutation $a_{1}, a_{2}, \\ldots, a_{n}$ of the numbers $1,2, \\ldots, n$, where $n$ divides one of the numbers $a_{k}^{2}-a_{k+1}-1$ and $a_{k}^{2}-a_{k+1}+1$ for each $k, 1 \\leq k \\leq n$ (we assume $a_{n+1}=a_{1}$ ). Please output th... | {
"ground_truth": "n=3",
"style": "rule"
} | {
"index": "339",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, $sin2C=\\sqrt{3}sinC$. \n$(1)$ Find the value of $\\angle C$; \n$(2)$ If $b=6$ and the perimeter of $\\triangle ABC$ is $6\\sqrt{3}+6$, find the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6\\sqrt{3}",
"style": "rule"
} | {
"index": "340",
"split": "train"
} |
math | [
{
"content": "Find the number of six-digit palindromes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "900",
"style": "rule"
} | {
"index": "341",
"split": "train"
} |
math | [
{
"content": "What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "342",
"split": "train"
} |
math | [
{
"content": "Given real numbers $x$ and $y$ that satisfy the system of inequalities $\\begin{cases} x - 2y - 2 \\leqslant 0 \\\\ x + y - 2 \\leqslant 0 \\\\ 2x - y + 2 \\geqslant 0 \\end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determine the minimum value... | {
"ground_truth": "\\frac{10 + 4\\sqrt{6}}{3}",
"style": "rule"
} | {
"index": "343",
"split": "train"
} |
math | [
{
"content": "Suppose a regular tetrahedron \\( P-ABCD \\) has all edges equal in length. Using \\(ABCD\\) as one face, construct a cube \\(ABCD-EFGH\\) on the other side of the regular tetrahedron. Determine the cosine of the angle between the skew lines \\( PA \\) and \\( CF \\). Please output the final answe... | {
"ground_truth": "\\frac{2 + \\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "344",
"split": "train"
} |
math | [
{
"content": "The decimal expansion of $8/11$ is a repeating decimal. What is the least number of digits in a repeating block of 8/11? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "345",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $201$ are multiples of either $6$ or $8$, but not both at once? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "346",
"split": "train"
} |
math | [
{
"content": "The number $24!$ has many positive integer divisors. What is the probability that a divisor randomly chosen from these is odd? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{23}",
"style": "rule"
} | {
"index": "347",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(1-x)^{2}(2-x)^{8}$, find the coefficient of $x^{8}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "348",
"split": "train"
} |
math | [
{
"content": "Express $0.6\\overline{03}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{104}{165}",
"style": "rule"
} | {
"index": "349",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin (2x+\\varphi)$, if the graph is shifted to the left by $\\dfrac {\\pi}{6}$ units and the resulting graph is symmetric about the $y$-axis, determine the possible value of $\\varphi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac {\\pi}{6}",
"style": "rule"
} | {
"index": "350",
"split": "train"
} |
math | [
{
"content": "How many positive integers less than $1000$ are either a perfect cube or a perfect square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "351",
"split": "train"
} |
math | [
{
"content": "A line passing through any two vertices of a cube has a total of 28 lines. Calculate the number of pairs of skew lines among them. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "174",
"style": "rule"
} | {
"index": "352",
"split": "train"
} |
math | [
{
"content": "When the value of $y$ is doubled and then this increased value is divided by 5, the result is 10. What is the value of $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "353",
"split": "train"
} |
math | [
{
"content": "Given that $O$ is any point in space, and $A$, $B$, $C$, $D$ are four points such that no three of them are collinear, but they are coplanar, and $\\overrightarrow{OA}=2x\\cdot \\overrightarrow{BO}+3y\\cdot \\overrightarrow{CO}+4z\\cdot \\overrightarrow{DO}$, find the value of $2x+3y+4z$. Please o... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "354",
"split": "train"
} |
math | [
{
"content": "Determine the largest prime number less than 5000 of the form \\( a^n - 1 \\), where \\( a \\) and \\( n \\) are positive integers, and \\( n \\) is greater than 1. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "127",
"style": "rule"
} | {
"index": "355",
"split": "train"
} |
math | [
{
"content": "For any integer $n>1$, the number of prime numbers greater than $n!+1$ and less than $n!+n$ is:\n$\\text{(A) } 0\\quad\\qquad \\text{(B) } 1\\quad\\\\ \\text{(C) } \\frac{n}{2} \\text{ for n even, } \\frac{n+1}{2} \\text{ for n odd}\\quad\\\\ \\text{(D) } n-1\\quad \\text{(E) } n$ Please output th... | {
"ground_truth": "",
"style": "rule"
} | {
"index": "356",
"split": "train"
} |
math | [
{
"content": "A factory assigns five newly recruited employees, including A and B, to three different workshops. Each workshop must be assigned at least one employee, and A and B must be assigned to the same workshop. The number of different ways to assign the employees is \\_\\_\\_\\_\\_\\_. Please output the ... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "357",
"split": "train"
} |
math | [
{
"content": "Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? Please output the final answer within \\boxed{}.... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "358",
"split": "train"
} |
math | [
{
"content": "To open the safe, you need to enter a code — a number consisting of seven digits: twos and threes. The safe will open if there are more twos than threes, and the code is divisible by both 3 and 4. Create a code that opens the safe. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "2222232",
"style": "rule"
} | {
"index": "359",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \\colon \\mathbb{R}^+ \\to \\mathbb{R}^+$ such that, for all $x,y \\in \\mathbb{R}^+$,\n$$f(xy+f(x))=xf(y)+2.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x + 1",
"style": "rule"
} | {
"index": "360",
"split": "train"
} |
math | [
{
"content": "In the diagram, three lines meet at the points \\( A, B \\), and \\( C \\). If \\( \\angle ABC = 50^\\circ \\) and \\( \\angle ACB = 30^\\circ \\), the value of \\( x \\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "361",
"split": "train"
} |
math | [
{
"content": "Susie pays for $4$ muffins and $3$ bananas. Calvin spends twice as much paying for $2$ muffins and $16$ bananas. A muffin is how many times as expensive as a banana? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "362",
"split": "train"
} |
math | [
{
"content": "What is the median of the numbers in the list $19^{20}, \\frac{20}{19}, 20^{19}, 2019, 20 \\times 19$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2019",
"style": "rule"
} | {
"index": "363",
"split": "train"
} |
math | [
{
"content": "A sequence \\( b_1, b_2, b_3, \\ldots \\) is defined recursively by \\( b_1 = 2 \\), \\( b_2 = 3 \\), and for \\( k \\geq 3 \\),\n\\[ b_k = \\frac{1}{2} b_{k-1} + \\frac{1}{3} b_{k-2}. \\]\nEvaluate \\( b_1 + b_2 + b_3 + \\dotsb. \\) Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "364",
"split": "train"
} |
math | [
{
"content": "Calculate the sum:\n\\[\n\\sum_{n=1}^\\infty \\frac{n^3 + n^2 + n - 1}{(n+3)!}\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "365",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=1$, $(\\overrightarrow{a}+\\overrightarrow{b}) \\perp \\overrightarrow{a}$, and $(2\\overrightarrow{a}+\\overrightarrow{b}) \\perp \\overrightarrow{b}$, calculate the angle between vectors $\\overrightarro... | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "366",
"split": "train"
} |
math | [
{
"content": "George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the second time? Please outp... | {
"ground_truth": "4\\frac{1}{2}",
"style": "rule"
} | {
"index": "367",
"split": "train"
} |
math | [
{
"content": "Through the end of a chord that divides the circle in the ratio 3:5, a tangent is drawn. Find the acute angle between the chord and the tangent. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67.5",
"style": "rule"
} | {
"index": "368",
"split": "train"
} |
math | [
{
"content": "If the sines of the internal angles of $\\triangle ABC$ form an arithmetic sequence, what is the minimum value of $\\cos C$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "369",
"split": "train"
} |
math | [
{
"content": "Suppose that $b$ is a positive integer greater than or equal to $2.$ When $197$ is converted to base $b$, the resulting representation has $4$ digits. What is the number of possible values for $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "370",
"split": "train"
} |
math | [
{
"content": "Given that $-4\\leq x\\leq-2$ and $2\\leq y\\leq4$, what is the largest possible value of $\\frac{x+y}{x}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "371",
"split": "train"
} |
math | [
{
"content": "For how many integer values of $n$ between 1 and 180 inclusive does the decimal representation of $\\frac{n}{180}$ terminate? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "372",
"split": "train"
} |
math | [
{
"content": "The FISS World Cup is a very popular football event among high school students worldwide. China successfully obtained the hosting rights for the International Middle School Sports Federation (FISS) World Cup in 2024, 2026, and 2028. After actively bidding by Dalian City and official recommendation... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "373",
"split": "train"
} |
math | [
{
"content": "In the triangle $ABC$ it is known that $\\angle A = 75^o, \\angle C = 45^o$ . On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$ . Let $M$ be the midpoint of the segment $AT$ . Find the measure of the $\\angle BMC$ .\n\n(Anton Trygub) Please output the final... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "374",
"split": "train"
} |
math | [
{
"content": "Given three numbers $1$, $3$, $4$, find the value of x such that the set $\\{1, 3, 4, x\\}$ forms a proportion. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "375",
"split": "train"
} |
math | [
{
"content": "Bob has a seven-digit phone number and a five-digit postal code. The sum of the digits in his phone number and the sum of the digits in his postal code are the same. Bob's phone number is 346-2789. What is the largest possible value for Bob's postal code, given that no two digits in the postal cod... | {
"ground_truth": "98765",
"style": "rule"
} | {
"index": "376",
"split": "train"
} |
math | [
{
"content": "On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the fro... | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "377",
"split": "train"
} |
math | [
{
"content": "Given $a$, $b$, $c > 0$ and $$a(a+b+c)+bc=4-2 \\sqrt {3}$$, calculate the minimum value of $2a+b+c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{3}-2",
"style": "rule"
} | {
"index": "378",
"split": "train"
} |
math | [
{
"content": "We write the equation on the board:\n\n$$\n(x-1)(x-2) \\ldots (x-2016) = (x-1)(x-2) \\ldots (x-2016) .\n$$\n\nWe want to erase some of the 4032 factors in such a way that the equation on the board has no real solutions. What is the minimal number of factors that need to be erased to achieve this? ... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "379",
"split": "train"
} |
math | [
{
"content": "What is the sum of the seven smallest distinct positive integer multiples of 9? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "252",
"style": "rule"
} | {
"index": "380",
"split": "train"
} |
math | [
{
"content": "The center of one sphere is on the surface of another sphere with an equal radius. How does the volume of the intersection of the two spheres compare to the volume of one of the spheres? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{16}",
"style": "rule"
} | {
"index": "381",
"split": "train"
} |
math | [
{
"content": "Among five numbers, if we take the average of any four numbers and add the remaining number, the sums will be 74, 80, 98, 116, and 128, respectively. By how much is the smallest number less than the largest number among these five numbers? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "382",
"split": "train"
} |
math | [
{
"content": "Let three non-identical complex numbers \\( z_1, z_2, z_3 \\) satisfy the equation \\( 4z_1^2 + 5z_2^2 + 5z_3^2 = 4z_1z_2 + 6z_2z_3 + 4z_3z_1 \\). Denote the lengths of the sides of the triangle in the complex plane, with vertices at \\( z_1, z_2, z_3 \\), from smallest to largest as \\( a, b, c \... | {
"ground_truth": "2:\\sqrt{5}:\\sqrt{5}",
"style": "rule"
} | {
"index": "383",
"split": "train"
} |
math | [
{
"content": "The product \\( \\left(1-\\frac{1}{3}\\right)\\left(1-\\frac{1}{4}\\right)\\left(1-\\frac{1}{5}\\right) \\) is equal to what? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "384",
"split": "train"
} |
math | [
{
"content": "The sum of two positive integers $a$ and $b$ is 1001. What is the largest possible value of $\\gcd(a,b)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "143",
"style": "rule"
} | {
"index": "385",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $AB= 400$, $BC=480$, and $AC=560$. An interior point $P$ is identified, and segments are drawn through $P$ parallel to the sides of the triangle. These three segments are of equal length $d$. Determine $d$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "218\\frac{2}{9}",
"style": "rule"
} | {
"index": "386",
"split": "train"
} |
math | [
{
"content": "Complex numbers \\( p, q, r \\) form an equilateral triangle with a side length of 24 in the complex plane. If \\( |p + q + r| = 48 \\), find \\( |pq + pr + qr| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "768",
"style": "rule"
} | {
"index": "387",
"split": "train"
} |
math | [
{
"content": "Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "388",
"split": "train"
} |
math | [
{
"content": "Two different natural numbers are selected from the set $\\{1, 2, 3, \\ldots, 8\\}$. What is the probability that the greatest common factor of these two numbers is one? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{7}",
"style": "rule"
} | {
"index": "389",
"split": "train"
} |
math | [
{
"content": "Twelve tiles numbered $1$ through $12$ are turned face down. One tile is turned up at random, and a die with 8 faces numbered 1 to 8 is rolled. Determine the probability that the product of the numbers on the tile and the die will be a square. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{7}{48}",
"style": "rule"
} | {
"index": "390",
"split": "train"
} |
math | [
{
"content": "Given that $α \\in (0, \\frac{π}{3})$ satisfies the equation $\\sqrt{6} \\sin α + \\sqrt{2} \\cos α = \\sqrt{3}$, find the values of:\n1. $\\cos (α + \\frac{π}{6})$\n2. $\\cos (2α + \\frac{π}{12})$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{30} + \\sqrt{2}}{8}",
"style": "rule"
} | {
"index": "391",
"split": "train"
} |
math | [
{
"content": "How many whole numbers are between $\\sqrt[3]{50}$ and $\\sqrt{200}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "392",
"split": "train"
} |
math | [
{
"content": "Let \\(a, b, c\\) be the side lengths of a right triangle, with \\(a \\leqslant b < c\\). Determine the maximum constant \\(k\\) such that the inequality \\(a^{2}(b+c) + b^{2}(c+a) + c^{2}(a+b) \\geqslant k a b c\\) holds for all right triangles, and specify when equality occurs. Please output the... | {
"ground_truth": "2 + 3\\sqrt{2}",
"style": "rule"
} | {
"index": "393",
"split": "train"
} |
math | [
{
"content": "Given that $α \\in \\left( \\frac{π}{2}, π \\right)$ and $3\\cos 2α = \\sin \\left( \\frac{π}{4} - α \\right)$, find the value of $\\sin 2α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{17}{18}",
"style": "rule"
} | {
"index": "394",
"split": "train"
} |
math | [
{
"content": "How many sequences of 0s and 1s are there of length 10 such that there are no three 0s or 1s consecutively anywhere in the sequence? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "178",
"style": "rule"
} | {
"index": "395",
"split": "train"
} |
math | [
{
"content": "The graphs of a function $f(x)=3x+b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-3,a)$. Given that $b$ and $a$ are both integers, what is the value of $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "396",
"split": "train"
} |
math | [
{
"content": "A car travels the 120 miles from $A$ to $B$ at 60 miles per hour, and then returns to $A$ on the same road. If the average rate of the round trip is 45 miles per hour, what is the rate, in miles per hour, of the car traveling back from $B$ to $A$? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "397",
"split": "train"
} |
math | [
{
"content": "Determine the length of side $PQ$ in the right-angled triangle $PQR$, where $PR = 15$ units and $\\angle PQR = 45^\\circ$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "398",
"split": "train"
} |
math | [
{
"content": "Below is the graph of $y = a \\sin (bx + c) + d$ for some positive constants $a,$ $b,$ $c,$ and $d.$ Find $a.$\n\n[asy]import TrigMacros;\n\nsize(400);\n\nreal f(real x)\n{\n\treturn 2*sin(3*x + pi) + 1;\n}\n\ndraw(graph(f,-3*pi,3*pi,n=700,join=operator ..),red);\ntrig_axes(-3*pi,3*pi,-4,4,pi/2,1... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "399",
"split": "train"
} |
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