data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Find the value of the following expression:\n\n$$\\left| \\, |{ -|{-1 + 1}| - 1 }| + 1\\right|.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "32100",
"split": "train"
} |
math | [
{
"content": "Estimate \n99×71≈ \n25×39≈ \n124÷3≈ \n398÷5≈ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "32101",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of\n\\[\\frac{x^2}{y - 1} + \\frac{y^2}{x - 1}\\]for real numbers $x > 1$ and $y > 1.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "32102",
"split": "train"
} |
math | [
{
"content": "How many of the first $1000$ positive integers can be expressed in the form\n\\[\\lfloor 2x \\rfloor + \\lfloor 4x \\rfloor + \\lfloor 6x \\rfloor + \\lfloor 8x \\rfloor\\]where $x$ is a real number, and $\\lfloor z \\rfloor$ denotes the greatest integer less than or equal to $z$? Please output th... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "32103",
"split": "train"
} |
math | [
{
"content": "An equiangular hexagon has side lengths $1,1, a, 1,1, a$ in that order. Given that there exists a circle that intersects the hexagon at 12 distinct points, we have $M<a<N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\\frac{N}{M}$. Please output the final a... | {
"ground_truth": "\\frac{3 \\sqrt{3}+3}{2}",
"style": "rule"
} | {
"index": "32104",
"split": "train"
} |
math | [
{
"content": "A coordinate system and parametric equations problem (4-4):\n\nIn the rectangular coordinate system $xoy$, a curve $C_1$ is defined by the parametric equations $\\begin{cases} x = 1 + \\cos \\alpha \\\\ y = \\sin^2 \\alpha - \\frac{9}{4} \\end{cases}$, where $\\alpha$ is the parameter and $\\alpha... | {
"ground_truth": "\\sqrt{2} - 1",
"style": "rule"
} | {
"index": "32105",
"split": "train"
} |
math | [
{
"content": "We call a pair of natural numbers $(a, p)$ good if the number $a^{3} + p^{3}$ is divisible by $a^{2} - p^{2}$, and $a > p$.\n\n(a) (1 point) Give any possible value of $a$ for which the pair $(a, 11)$ is good.\n\n(b) (3 points) Find the number of good pairs, where $p$ is a prime number less than 1... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "32106",
"split": "train"
} |
math | [
{
"content": "The distance between Luga and Volkhov is 194 km, between Volkhov and Lodeynoe Pole is 116 km, between Lodeynoe Pole and Pskov is 451 km, and between Pskov and Luga is 141 km. What is the distance between Pskov and Volkhov? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "335",
"style": "rule"
} | {
"index": "32107",
"split": "train"
} |
math | [
{
"content": "A positive number $x$ has the property that $x\\%$ of $x$ is $4$. What is $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "32108",
"split": "train"
} |
math | [
{
"content": "Find the integer $n$, $0 \\le n \\le 7$, such that \\[n \\equiv -3737 \\pmod{8}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "32109",
"split": "train"
} |
math | [
{
"content": "Suppose that $PQ$ and $RS$ are two chords of a circle intersecting at a point $O$ . It is given that $PO=3 \\text{cm}$ and $SO=4 \\text{cm}$ . Moreover, the area of the triangle $POR$ is $7 \\text{cm}^2$ . Find the area of the triangle $QOS$ . Please output the final answer within \\bo... | {
"ground_truth": "112/9",
"style": "rule"
} | {
"index": "32110",
"split": "train"
} |
math | [
{
"content": "In a magic square, the numbers in each row, the numbers in each column, and the numbers on each diagonal have the same sum. In the magic square shown, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.2",
"style": "rule"
} | {
"index": "32111",
"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": "32112",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$ with $\\overline{AB}=\\overline{AC}=3.6$ , a point $D$ is taken on $AB$ at a distance $1.2$ from $A$ . Point $D$ is joined to $E$ in the prolongation of $AC$ so that triangle $AED$ is equal in area to $ABC$ . Then $\\overline{AE}$ is: Please output the fina... | {
"ground_truth": "10.8",
"style": "rule"
} | {
"index": "32113",
"split": "train"
} |
math | [
{
"content": "For each positive integer n, let $f(n) = \\sum_{k = 1}^{100} \\lfloor \\log_{10} (kn) \\rfloor$. Find the largest value of $n$ for which $f(n) \\le 300$.\nNote: $\\lfloor x \\rfloor$ is the greatest integer less than or equal to $x$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "109",
"style": "rule"
} | {
"index": "32114",
"split": "train"
} |
math | [
{
"content": "The graph of $y = ax^2 + bx + c$ is described, where $a$, $b$, and $c$ are integers. Given that the vertex of this parabola is at $(-2, 3)$ and one point on the graph is $(1, 6)$, determine the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "32115",
"split": "train"
} |
math | [
{
"content": "The sum of the lengths of all the edges of a cube is 60 cm. Find the number of cubic centimeters in the volume of the cube. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "32116",
"split": "train"
} |
math | [
{
"content": "How many ways are there to distribute 6 distinguishable balls into 2 indistinguishable boxes if no box can hold more than 4 balls? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "32117",
"split": "train"
} |
math | [
{
"content": "Nonzero real numbers $x$, $y$, $a$, and $b$ satisfy $x < a$ and $y < b$. How many of the following inequalities must be true?\n$\\textbf{(I)}\\ x+y < a+b$\n$\\textbf{(II)}\\ x-y < a-b$\n$\\textbf{(III)}\\ xy < ab$\n$\\textbf{(IV)}\\ \\frac{x}{y} < \\frac{a}{b}$ Please output the final answer withi... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "32118",
"split": "train"
} |
math | [
{
"content": "What value of $x$ will give the maximum value for $-x^2- 6x + 12$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "32119",
"split": "train"
} |
math | [
{
"content": "If $\\{a_1,a_2,a_3,\\ldots,a_n\\}$ is a set of real numbers, indexed so that $a_1 < a_2 < a_3 < \\cdots < a_n,$ its complex power sum is defined to be $a_1i + a_2i^2+ a_3i^3 + \\cdots + a_ni^n,$ where $i^2 = - 1.$ Let $S_n$ be the sum of the complex power sums of all nonempty subsets of $\\{1,2,\\... | {
"ground_truth": "368",
"style": "rule"
} | {
"index": "32120",
"split": "train"
} |
math | [
{
"content": "The numbers \\(a, b, c, d\\) belong to the interval \\([-7.5, 7.5]\\). Find the maximum value of the expression \\(a + 2b + c + 2d - ab - bc - cd - da\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "32121",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the maximum value of $x^3 y^2 z.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{432}",
"style": "rule"
} | {
"index": "32122",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$, if $\\frac{a_{11}}{a_{10}} < -1$, and the sum of its first $n$ terms $S_n$ has a maximum value, find the maximum value of $n$ for which $S_n > 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "32123",
"split": "train"
} |
math | [
{
"content": "The Eagles beat the Falcons 3 times and the Falcons won 4 times in their initial meetings. They then played $N$ more times, and the Eagles ended up winning 90% of all the games played, including the additional games, to find the minimum possible value for $N$. Please output the final answer within... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "32124",
"split": "train"
} |
math | [
{
"content": "For any integer $a$ , let $f(a) = |a^4 - 36a^2 + 96a - 64|$ . What is the sum of all values of $f(a)$ that are prime?\n\n*Proposed by Alexander Wang* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "32125",
"split": "train"
} |
math | [
{
"content": "If $\\sin \\theta= \\frac {3}{5}$ and $\\frac {5\\pi}{2} < \\theta < 3\\pi$, then $\\sin \\frac {\\theta}{2}=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac {3 \\sqrt {10}}{10}",
"style": "rule"
} | {
"index": "32126",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = x^3 - 3x^2 - 9x + 2$,\n(I) Determine the intervals of monotonicity for the function $f(x)$;\n(II) Find the minimum value of function $f(x)$ on the interval [-2, 2]. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-20",
"style": "rule"
} | {
"index": "32127",
"split": "train"
} |
math | [
{
"content": "If you walk for $45$ minutes at a rate of $4 \\text{ mph}$ and then run for $30$ minutes at a rate of $10\\text{ mph}$, how many miles will you have gone at the end of one hour and $15$ minutes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "32128",
"split": "train"
} |
math | [
{
"content": "If $x<-2$, then $|1-|1+x||$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2-x",
"style": "rule"
} | {
"index": "32129",
"split": "train"
} |
math | [
{
"content": "Express $0.3\\overline{206}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5057}{9990}",
"style": "rule"
} | {
"index": "32130",
"split": "train"
} |
math | [
{
"content": "Consider a large square where each side is divided into four equal parts. At each division, a point is placed. An inscribed square is constructed such that its corners are at these division points nearest to the center of each side of the large square. Calculate the ratio of the area of the inscri... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32131",
"split": "train"
} |
math | [
{
"content": "Understanding and trying: When calculating $\\left(-4\\right)^{2}-\\left(-3\\right)\\times \\left(-5\\right)$, there are two methods; Method 1, please calculate directly: $\\left(-4\\right)^{2}-\\left(-3\\right)\\times \\left(-5\\right)$, Method 2, use letters to represent numbers, transform or ca... | {
"ground_truth": "-1.35",
"style": "rule"
} | {
"index": "32132",
"split": "train"
} |
math | [
{
"content": "A food factory processes mushrooms, with a cost of 20 yuan per kilogram of mushrooms, and a processing fee of t yuan per kilogram (t is a constant, and $2 \\leq t \\leq 5$). Assume the factory price per kilogram of mushrooms is x yuan ($25 \\leq x \\leq 40$). According to market research, the sale... | {
"ground_truth": "100e^4",
"style": "rule"
} | {
"index": "32133",
"split": "train"
} |
math | [
{
"content": "Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \\subseteq\\{1,2,3, \\ldots, 8\\}$. (b) $|A \\cap B|=|B \\cap C|=|C \\cap A|=2$. (c) $|A|=|B|=|C|=4$. Here, $|S|$ denotes the number of elements in the set $S$. Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "45360",
"style": "rule"
} | {
"index": "32134",
"split": "train"
} |
math | [
{
"content": "Given that $\\overrightarrow{OA}$, $\\overrightarrow{OB}$, and $\\overrightarrow{OC}$ are all unit vectors, and satisfy $\\frac{1}{2}\\overrightarrow{OA}+\\overrightarrow{OB}+\\overrightarrow{OC}=\\overrightarrow{0}$, calculate the value of $\\overrightarrow{AB} \\cdot \\overrightarrow{AC}$. Pleas... | {
"ground_truth": "\\frac{5}{8}",
"style": "rule"
} | {
"index": "32135",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n\\[\n\\int_{0}^{\\frac{\\pi}{4}} \\frac{2 \\cos x + 3 \\sin x}{(2 \\sin x - 3 \\cos x)^{3}} \\, dx\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{17}{18}",
"style": "rule"
} | {
"index": "32136",
"split": "train"
} |
math | [
{
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{3}{4} \\right)^x\\]\non the interval \\( (0, 50\\pi) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "32137",
"split": "train"
} |
math | [
{
"content": "Let $N$ denote the sum of the decimal digits of $\\binom{1000}{100}$. Estimate the value of $N$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "621",
"style": "rule"
} | {
"index": "32138",
"split": "train"
} |
math | [
{
"content": "Regular hexagon $GHJKLMN$ is the base of a right pyramid $QGHJKLMN$. If $QGM$ is an equilateral triangle with side length 10, then what is the volume of the pyramid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "187.5",
"style": "rule"
} | {
"index": "32139",
"split": "train"
} |
math | [
{
"content": "What are the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{2003}+1\\right)^{11/8}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "375",
"style": "rule"
} | {
"index": "32140",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be a quadrilateral with an inscribed circle $\\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$ . Let $R_1$ , $R_2$ , $R_3$ , $R_4$ be the circumradii of triangles $APB$ , $BPC$ , $CPD$ , $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "32141",
"split": "train"
} |
math | [
{
"content": "Let $\\theta$ be the angle between the line\n\\[\\frac{x - 2}{4} = \\frac{y + 1}{5} = \\frac{z - 4}{7}\\]\nand the plane $3x + 4y - 7z = 5.$ Find $\\sin \\theta.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{\\sqrt{6660}}",
"style": "rule"
} | {
"index": "32142",
"split": "train"
} |
math | [
{
"content": "In the sequence of positive integers \\(1, 2, 3, 4, \\cdots\\), remove multiples of 3 and 4, but keep all multiples of 5 (for instance, 15 and 120 should not be removed). The remaining numbers form a new sequence: \\(a_{1} = 1, a_{2} = 2, a_{3} = 5, a_{4} = 7, \\cdots\\). Find \\(a_{1999}\\). Plea... | {
"ground_truth": "3331",
"style": "rule"
} | {
"index": "32143",
"split": "train"
} |
math | [
{
"content": "Two cyclists, $k$ miles apart, and starting at the same time, would be together in $r$ hours if they traveled in the same direction, but would pass each other in $t$ hours if they traveled in opposite directions. The ratio of the speed of the faster cyclist to that of the slower is: Please output ... | {
"ground_truth": "\\frac {r + t}{r - t}",
"style": "rule"
} | {
"index": "32144",
"split": "train"
} |
math | [
{
"content": "Find the number of solutions to the equation\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,150 \\pi)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "32145",
"split": "train"
} |
math | [
{
"content": "A black and white chocolate bar consists of individual pieces arranged in $n$ horizontal rows and $m$ vertical columns, painted in a checkerboard pattern. Ian ate all the black pieces, and Max ate all the white pieces. What is the sum of $m + n$ if it is known that Ian ate $8 \\frac{1}{3} \\%$ mor... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32146",
"split": "train"
} |
math | [
{
"content": "If \\( x, \\) \\( y, \\) and \\( k \\) are positive real numbers such that \n\\[\n4 = k^2\\left(\\frac{x^2}{y^2}+\\frac{y^2}{x^2}\\right)+k\\left(\\frac{x}{y}+\\frac{y}{x}\\right),\n\\]\nfind the maximum possible value of \\( k \\). Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{-1+\\sqrt{17}}{4}",
"style": "rule"
} | {
"index": "32147",
"split": "train"
} |
math | [
{
"content": "Let $A,$ $B,$ $C$ be the angles of a triangle. Evaluate\n\\[\\begin{vmatrix} \\sin^2 A & \\cot A & 1 \\\\ \\sin^2 B & \\cot B & 1 \\\\ \\sin^2 C & \\cot C & 1 \\end{vmatrix}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "32148",
"split": "train"
} |
math | [
{
"content": "The circumference of a circle has 50 numbers written on it, each of which is either +1 or -1. We want to find the product of these numbers. What is the minimum number of questions needed to determine this product if we can ask about the product of three consecutive numbers at a time? Please output... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "32149",
"split": "train"
} |
math | [
{
"content": "Calculate the result of the expression \\( 2015 \\frac{1999}{2015} \\times \\frac{1}{4} - \\frac{2011}{2015} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "503",
"style": "rule"
} | {
"index": "32150",
"split": "train"
} |
math | [
{
"content": "From points A and B, a motorcyclist and a cyclist respectively set off towards each other simultaneously and met at a distance of 4 km from B. At the moment the motorcyclist arrived in B, the cyclist was at a distance of 15 km from A. Find the distance AB. Please output the final answer within \\b... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "32151",
"split": "train"
} |
math | [
{
"content": "Initially, a prime number is displayed on the computer screen. Every second, the number on the screen is replaced with a number obtained by adding the last digit of the previous number, increased by 1. After how much time, at the latest, will a composite number appear on the screen? Please output ... | {
"ground_truth": "996",
"style": "rule"
} | {
"index": "32152",
"split": "train"
} |
math | [
{
"content": "Two circles \\( C_{1} \\) and \\( C_{2} \\) touch each other externally and the line \\( l \\) is a common tangent. The line \\( m \\) is parallel to \\( l \\) and touches the two circles \\( C_{1} \\) and \\( C_{3} \\). The three circles are mutually tangent. If the radius of \\( C_{2} \\) is 9 a... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "32153",
"split": "train"
} |
math | [
{
"content": "A circle $C$ is defined by the equation $x^2 + y^2 = 1$. After the transformation $\\begin{cases} x' = 2x \\\\ y' = \\sqrt{2}y \\end{cases}$, we obtain the curve $C_1$. Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axi... | {
"ground_truth": "\\frac{12\\sqrt{2}}{5}",
"style": "rule"
} | {
"index": "32154",
"split": "train"
} |
math | [
{
"content": "There are 7 volunteers to be arranged to participate in the promotional activities for the Shanghai World Expo on Saturday and Sunday, with 6 people participating each day and 3 people arranged for each day. There are __________ different arrangements possible (answer with a number). Please output... | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "32155",
"split": "train"
} |
math | [
{
"content": "Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought? Please out... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "32156",
"split": "train"
} |
math | [
{
"content": "Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "32157",
"split": "train"
} |
math | [
{
"content": "Given that sin(α + $\\frac {π}{4}$) = $\\frac { \\sqrt {3}}{3}$, and α ∈ (0, π), find the value of cosα. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {-2 \\sqrt {3}+ \\sqrt {6}}{6}",
"style": "rule"
} | {
"index": "32158",
"split": "train"
} |
math | [
{
"content": "Complex numbers $a, b, c$ form an equilateral triangle with side length 18 in the complex plane. If $|a+b+c|=36$, find $|b c+c a+a b|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "32159",
"split": "train"
} |
math | [
{
"content": "A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week... | {
"ground_truth": "729",
"style": "rule"
} | {
"index": "32160",
"split": "train"
} |
math | [
{
"content": "Marcus and four of his relatives are at a party. Each pair of the five people are either friends or enemies. For any two enemies, there is no person that they are both friends with. In how many ways is this possible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "32161",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin(\\omega x+\\varphi)$, which is monotonically increasing on the interval ($\\frac{\\pi}{6}$,$\\frac{{2\\pi}}{3}$), and the lines $x=\\frac{\\pi}{6}$ and $x=\\frac{{2\\pi}}{3}$ are the two symmetric axes of the graph of the function $y=f(x)$, find $f(-\\frac{{5\\pi}}{{... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "32162",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\lfloor{\\pi}\\rfloor$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "32163",
"split": "train"
} |
math | [
{
"content": "The positive integer $i$ is chosen at random such that the probability of a positive integer $k$ being chosen is $\\frac{3}{2}$ times the probability of $k+1$ being chosen. What is the probability that the $i^{\\text {th }}$ digit after the decimal point of the decimal expansion of $\\frac{1}{7}$ ... | {
"ground_truth": "\\frac{108}{665}",
"style": "rule"
} | {
"index": "32164",
"split": "train"
} |
math | [
{
"content": "Given that the terminal side of angle $\\alpha$ passes through point P $\\left(\\frac{4}{5}, -\\frac{3}{5}\\right)$.\n\n(1) Find the value of $\\sin\\alpha$.\n\n(2) Calculate the value of $\\frac{\\sin\\left(\\frac{\\pi}{2}-\\alpha\\right)}{\\sin(\\alpha +\\pi)} \\cdot \\frac{\\tan(\\alpha-\\pi)}{... | {
"ground_truth": "\\frac{5}{4}",
"style": "rule"
} | {
"index": "32165",
"split": "train"
} |
math | [
{
"content": "Determine the largest constant $K\\geq 0$ such that $$ \\frac{a^a(b^2+c^2)}{(a^a-1)^2}+\\frac{b^b(c^2+a^2)}{(b^b-1)^2}+\\frac{c^c(a^2+b^2)}{(c^c-1)^2}\\geq K\\left (\\frac{a+b+c}{abc-1}\\right)^2 $$ holds for all positive real numbers $a,b,c$ such that $ab+bc+ca=abc$ .\n\n*Proposed by Ori... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "32166",
"split": "train"
} |
math | [
{
"content": "To protect the ecological environment, a mountainous area in our city has started to implement the policy of returning farmland to forest since 2005. It is known that at the end of 2004, the forest coverage area of this mountainous area was $a$ acres. \n(1) Assuming the annual natural growth rate... | {
"ground_truth": "7.2\\%",
"style": "rule"
} | {
"index": "32167",
"split": "train"
} |
math | [
{
"content": "A square with sides of length $1$ is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points on three of the sides, as shown. Find $x$, the length of the longer parallel side of each trapezoid. Please output the final answer wit... | {
"ground_truth": "\\frac{5}{6}",
"style": "rule"
} | {
"index": "32168",
"split": "train"
} |
math | [
{
"content": "Find the number of positive integers $n$ that satisfy\n\\[(n - 2)(n - 4)(n - 6) \\dotsm (n - 98) < 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32169",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\begin{cases} 2^{x}, & x < 2 \\\\ f(x-1), & x\\geqslant 2 \\end{cases}$, then $f(\\log_{2}7)=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {7}{2}",
"style": "rule"
} | {
"index": "32170",
"split": "train"
} |
math | [
{
"content": "Given that $2x + 5y = 20$ and $5x + 2y = 26$, find $20x^2 + 60xy + 50y^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{59600}{49}",
"style": "rule"
} | {
"index": "32171",
"split": "train"
} |
math | [
{
"content": "Neo has an infinite supply of red pills and blue pills. When he takes a red pill, his weight will double, and when he takes a blue pill, he will lose one pound. If Neo originally weighs one pound, what is the minimum number of pills he must take to make his weight 2015 pounds? Please output the fi... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "32172",
"split": "train"
} |
math | [
{
"content": "Suppose functions $g$ and $f$ have the properties that $g(x)=3f^{-1}(x)$ and $f(x)=\\frac{24}{x+3}$. For what value of $x$ does $g(x)=15$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "32173",
"split": "train"
} |
math | [
{
"content": "A rectangular piece of paper $ABCD$ has sides of lengths $AB = 1$ , $BC = 2$ . The rectangle is folded in half such that $AD$ coincides with $BC$ and $EF$ is the folding line. Then fold the paper along a line $BM$ such that the corner $A$ falls on line $EF$ . How large, in degrees, ... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32174",
"split": "train"
} |
math | [
{
"content": "Given that $F_{1}$ and $F_{2}$ are two foci of the hyperbola $C: x^{2}-\\frac{{y}^{2}}{3}=1$, $P$ and $Q$ are two points on $C$ symmetric with respect to the origin, and $\\angle PF_{2}Q=120^{\\circ}$, find the area of quadrilateral $PF_{1}QF_{2}$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "6\\sqrt{3}",
"style": "rule"
} | {
"index": "32175",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$ where $a_1=1$ and $a_na_{n-1}=a_{n-1}+(-1)^n$ for $n\\geqslant 2, n\\in\\mathbb{N}^*$, find the value of $\\frac{a_3}{a_5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "32176",
"split": "train"
} |
math | [
{
"content": "Simplify $\\sin ^{2}\\left( \\alpha-\\frac{\\pi}{6} \\right)+\\sin ^{2}\\left( \\alpha+\\frac{\\pi}{6} \\right)-\\sin ^{2}\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32177",
"split": "train"
} |
math | [
{
"content": "The Badgers play the Cougars in a series of seven basketball games. Each team has an equal chance of winning each game. What is the probability that the Badgers will win at least four games? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32178",
"split": "train"
} |
math | [
{
"content": "The image of the point with coordinates $(2, -4)$ under the reflection across the line $y = mx + b$ is the point with coordinates $(-4, 8)$. The line of reflection passes through the point $(0, 3)$. Find $m+b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.5",
"style": "rule"
} | {
"index": "32179",
"split": "train"
} |
math | [
{
"content": "Determine the largest natural number $r$ with the property that among any five subsets with $500$ elements of the set $\\{1,2,\\ldots,1000\\}$ there exist two of them which share at least $r$ elements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "32180",
"split": "train"
} |
math | [
{
"content": "Evaluate\n\\[\n\\left(c^c - c (c - 1)^{c-1}\\right)^c\n\\]\nwhen \\( c = 4 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "148^4",
"style": "rule"
} | {
"index": "32181",
"split": "train"
} |
math | [
{
"content": "Given \\( m = \\frac{\\sin x}{\\sin (y-z)}, n = \\frac{\\sin y}{\\sin (z-x)}, p = \\frac{\\sin z}{\\sin (x-y)} \\), find the value of \\( m n + n p + p m \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32182",
"split": "train"
} |
math | [
{
"content": "Evaluate the integral \\(\\int_{0}^{1} \\ln x \\ln (1-x) \\, dx\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - \\frac{\\pi^2}{6}",
"style": "rule"
} | {
"index": "32183",
"split": "train"
} |
math | [
{
"content": "The number $m$ is a three-digit positive integer and is the product of the three distinct prime factors $x$, $y$ and $10x+y$, where $x$ and $y$ are each less than 10. What is the largest possible value of $m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "795",
"style": "rule"
} | {
"index": "32184",
"split": "train"
} |
math | [
{
"content": "How many different divisors does the number 86,400,000 have (including 1 and the number 86,400,000 itself)? Find the sum of all these divisors. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "319823280",
"style": "rule"
} | {
"index": "32185",
"split": "train"
} |
math | [
{
"content": "Define the sequence $a_1, a_2, a_3, \\ldots$ by $a_n = \\sum\\limits_{k=1}^n \\sin{k}$, where $k$ represents radian measure. Find the index of the 100th term for which $a_n < 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "628",
"style": "rule"
} | {
"index": "32186",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $(xOy)$, the slope angle of line $l$ passing through point $M(2,1)$ is $\\frac{\\pi}{4}$. Establish a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, using the same unit length for both coordinate systems. The po... | {
"ground_truth": "\\frac{\\sqrt{30}}{7}",
"style": "rule"
} | {
"index": "32187",
"split": "train"
} |
math | [
{
"content": "Barney Schwinn notices that the odometer on his bicycle reads $1441$, a palindrome, because it reads the same forward and backward. After riding $4$ more hours that day and $6$ the next, he notices that the odometer shows another palindrome, $1661$. What was his average speed in miles per hour? Pl... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "32188",
"split": "train"
} |
math | [
{
"content": "Simplify $\frac{1}{2+\frac{2}{3}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "32189",
"split": "train"
} |
math | [
{
"content": "Liam needs to add 125 to 96 and then round the result to the nearest hundred. He plans to round the numbers before adding them. He rounds 125 to 100 and 96 to 100, resulting in an answer of $100 + 100 = 200$. Is this answer correct? What is the correct answer? Please output the final answer within... | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "32190",
"split": "train"
} |
math | [
{
"content": "There are two buildings facing each other, each 5 stories high. How many ways can Kevin string ziplines between the buildings so that: (a) each zipline starts and ends in the middle of a floor. (b) ziplines can go up, stay flat, or go down, but can't touch each other (this includes touching at the... | {
"ground_truth": "252",
"style": "rule"
} | {
"index": "32191",
"split": "train"
} |
math | [
{
"content": "Splicing two identical small cubes into a cuboid, the surface area is reduced by approximately \\_\\_\\_\\_\\_\\_%. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16.7",
"style": "rule"
} | {
"index": "32192",
"split": "train"
} |
math | [
{
"content": "At a state contest, 21 Mathletes stay in the same hotel. Each Mathlete gets his/her own room and the room numbers are 1 through 21. When all the Mathletes have arrived, except the ones staying in rooms 12 and 13, what is the median room number of the other 19 Mathletes? Please output the final ans... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32193",
"split": "train"
} |
math | [
{
"content": "In a drawer Sandy has $5$ pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the $10$ socks in the drawer. On Tuesday Sandy selects $2$ of the remaining $8$ socks at random and on Wednesday two of the remaining $6$ socks at random. The probabil... | {
"ground_truth": "341",
"style": "rule"
} | {
"index": "32194",
"split": "train"
} |
math | [
{
"content": "For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\\text{4}}) = 10 = 12_{\\text{8}}$, and $g(2020) = \\text{the digit sum of... | {
"ground_truth": "151",
"style": "rule"
} | {
"index": "32195",
"split": "train"
} |
math | [
{
"content": "In the plane rectangular coordinate system $xOy$, the parametric equations of curve $C$ are $\\left\\{\\begin{array}{l}{x=\\frac{4}{1+{t}^{2}}},\\\\{y=\\frac{4t}{1+{t}^{2}}}\\end{array}\\right.$ $(t\\in R)$.<br/>$(Ⅰ)$ Find the rectangular coordinate equation of curve $C$;<br/>$(Ⅱ)$ Given that the ... | {
"ground_truth": "\\frac{\\sqrt{15}}{3}",
"style": "rule"
} | {
"index": "32196",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=4, B C=8$, and $C A=5$. Let $M$ be the midpoint of $B C$, and let $D$ be the point on the circumcircle of $A B C$ so that segment $A D$ intersects the interior of $A B C$, and $\\angle B A D=\\angle C A M$. Let $A D$ intersect side $B C$ at $X$. Compute the rati... | {
"ground_truth": "$\\frac{9}{41}$",
"style": "rule"
} | {
"index": "32197",
"split": "train"
} |
math | [
{
"content": "A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2601",
"style": "rule"
} | {
"index": "32198",
"split": "train"
} |
math | [
{
"content": "Given two circles $O\\_1$: $(x-2)^2 + y^2 = 16$ and $O\\_2$: $x^2 + y^2 = r^2 (0 < r < 2)$, a moving circle $M$ is tangent to both circles $O\\_1$ and $O\\_2$. The locus of the center of the moving circle $M$ forms two ellipses, with eccentricities $e\\_1$ and $e\\_2 (e\\_1 > e\\_2)$. Find the min... | {
"ground_truth": "\\frac{3+2\\sqrt{2}}{4}",
"style": "rule"
} | {
"index": "32199",
"split": "train"
} |
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