data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Define: \\(\\triangle a = a + (a + 1) + (a + 2) + \\cdots + (2a - 2) + (2a - 1)\\). For example: \\(\\triangle 5 = 5 + 6 + 7 + 8 + 9\\). What is the result of \\(\\triangle 1 + \\triangle 2 + \\triangle 3 + \\cdots + \\triangle 19 + \\triangle 20\\)? Please output the final answer within \\boxed{}... | {
"ground_truth": "4200",
"style": "rule"
} | {
"index": "32600",
"split": "train"
} |
math | [
{
"content": "Simplify $\\sqrt[3]{1+27} \\cdot \\sqrt[3]{1+\\sqrt[3]{27}} \\cdot \\sqrt{4}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\cdot \\sqrt[3]{112}",
"style": "rule"
} | {
"index": "32601",
"split": "train"
} |
math | [
{
"content": "Sherlock Holmes and Dr. Watson recover a suitcase with a three-digit combination lock from a mathematician turned criminal. Embedded in the suitcase above the lock is the cryptic message \"AT SEA BASE. SEAS EBB SEA: BASS. \"\n\nDr. Watson comments, \"This probably isn't about ocean fish. Perhaps i... | {
"ground_truth": "871",
"style": "rule"
} | {
"index": "32602",
"split": "train"
} |
math | [
{
"content": "Given that the sequence {a<sub>n</sub>} is a decreasing geometric sequence and satisfies the conditions $$a_{2}a_{7}= \\frac {1}{2}$$ and $$a_{3}+a_{6}= \\frac {9}{4}$$, find the maximum value of a<sub>1</sub>a<sub>2</sub>a<sub>3</sub>…a<sub>2n</sub>. Please output the final answer within \\boxed{... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "32603",
"split": "train"
} |
math | [
{
"content": "In triangle \\( DEF \\) where \\( DE = 5, EF = 12, DF = 13 \\), and point \\( H \\) is the centroid. After rotating triangle \\( DEF \\) by \\( 180^\\circ \\) around \\( H \\), vertices \\( D', E', F' \\) are formed. Calculate the area of the union of triangles \\( DEF \\) and \\( D'E'F' \\). Plea... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "32604",
"split": "train"
} |
math | [
{
"content": "Convert the binary number $1110011_2$ to its decimal equivalent. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "115",
"style": "rule"
} | {
"index": "32605",
"split": "train"
} |
math | [
{
"content": "Given that \\( p \\) and \\( q \\) are positive integers such that \\( p + q > 2017 \\), \\( 0 < p < q \\leq 2017 \\), and \\((p, q) = 1\\), find the sum of all fractions of the form \\(\\frac{1}{pq}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "32606",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4087",
"style": "rule"
} | {
"index": "32607",
"split": "train"
} |
math | [
{
"content": "Point $(x,y)$ is chosen randomly from the rectangular region with vertices at $(0,0)$, $(3036,0)$, $(3036,3037)$, and $(0,3037)$. What is the probability that $x > 3y$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{506}{3037}",
"style": "rule"
} | {
"index": "32608",
"split": "train"
} |
math | [
{
"content": "In how many ways is it possible to arrange the digits of 1150 to get a four-digit multiple of 5? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32609",
"split": "train"
} |
math | [
{
"content": "There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "32610",
"split": "train"
} |
math | [
{
"content": "For how many integers $n$ between 1 and 100 is the greatest common divisor of 15 and $n$ equal to 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "32611",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = \\sqrt{3}\\cos x\\sin x - \\frac{1}{2}\\cos 2x$.\n\n(1) Find the smallest positive period of $f(x)$.\n\n(2) Find the maximum and minimum values of $f(x)$ on the interval $\\left[0, \\frac{\\pi}{2}\\right]$ and the corresponding values of $x$. Please output the final answ... | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "32612",
"split": "train"
} |
math | [
{
"content": "Let $A_{n}=\\{a_{1}, a_{2}, a_{3}, \\ldots, a_{n}, b\\}$, for $n \\geq 3$, and let $C_{n}$ be the 2-configuration consisting of \\( \\{a_{i}, a_{i+1}\\} \\) for all \\( 1 \\leq i \\leq n-1, \\{a_{1}, a_{n}\\} \\), and \\( \\{a_{i}, b\\} \\) for \\( 1 \\leq i \\leq n \\). Let $S_{e}(n)$ be the numb... | {
"ground_truth": "\\[\nS_{1}(101) = 101, \\quad S_{2}(101) = 101, \\quad S_{3}(101) = 0\n\\]",
"style": "rule"
} | {
"index": "32613",
"split": "train"
} |
math | [
{
"content": "What is the total number of digits used when the first 1500 positive odd integers are written? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5445",
"style": "rule"
} | {
"index": "32614",
"split": "train"
} |
math | [
{
"content": "Let \\( A \\) and \\( B \\) be two moving points on the ellipse \\( x^{2}+3 y^{2}=1 \\), and \\( OA \\) is perpendicular to \\( OB \\) (where \\( O \\) is the origin). Then, the product of the maximum and minimum values of \\( |AB| \\) is ______. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{2 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "32615",
"split": "train"
} |
math | [
{
"content": "During the process of choosing trial points using the 0.618 method, if the trial interval is $[3, 6]$ and the first trial point is better than the second, then the third trial point should be at _____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5.292",
"style": "rule"
} | {
"index": "32616",
"split": "train"
} |
math | [
{
"content": "Find all integers $n$ and $m$, $n > m > 2$, and such that a regular $n$-sided polygon can be inscribed in a regular $m$-sided polygon so that all the vertices of the $n$-gon lie on the sides of the $m$-gon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(m, n) = (m, 2m), (3, 4)",
"style": "rule"
} | {
"index": "32617",
"split": "train"
} |
math | [
{
"content": "Find the minimum sample size for which the precision of the estimate of the population mean $a$ based on the sample mean with a confidence level of 0.975 is $\\delta=0.3$, given that the standard deviation $\\sigma=1.2$ of the normally distributed population is known. Please output the final answe... | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "32618",
"split": "train"
} |
math | [
{
"content": "Kate has four red socks and four blue socks. If she randomly divides these eight socks into four pairs, what is the probability that none of the pairs will be mismatched? That is, what is the probability that each pair will consist either of two red socks or of two blue socks? Please output the fi... | {
"ground_truth": "3 / 35",
"style": "rule"
} | {
"index": "32619",
"split": "train"
} |
math | [
{
"content": "As a result of five measurements of the rod's length with one device (without systematic errors), the following results (in mm) were obtained: $92, 94, 103, 105, 106$. Find: a) the sample mean length of the rod; b) the sample variance and the unbiased corrected variance of the measurement errors. ... | {
"ground_truth": "42.5",
"style": "rule"
} | {
"index": "32620",
"split": "train"
} |
math | [
{
"content": "Given \\\\(a < 0\\\\), \\\\((3x^{2}+a)(2x+b) \\geqslant 0\\\\) holds true over the interval \\\\((a,b)\\\\), then the maximum value of \\\\(b-a\\\\) is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{3}",
"style": "rule"
} | {
"index": "32621",
"split": "train"
} |
math | [
{
"content": "In the diagram below, $WXYZ$ is a trapezoid where $\\overline{WX}\\parallel \\overline{ZY}$ and $\\overline{WY}\\perp\\overline{ZY}$. If $YZ = 20$, $\\tan Z = 2$, and $\\tan X = 2.5$, then what is the length of $XY$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{116}",
"style": "rule"
} | {
"index": "32622",
"split": "train"
} |
math | [
{
"content": "The operation $ \\diamond $ is defined for positive integers $a$ and $b$ such that $a \\diamond b = a^2 - b$. Determine how many positive integers $x$ exist such that $20 \\diamond x$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "32623",
"split": "train"
} |
math | [
{
"content": "An empty $2020 \\times 2020 \\times 2020$ cube is given, and a $2020 \\times 2020$ grid of square unit cells is drawn on each of its six faces. A beam is a $1 \\times 1 \\times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions:\nThe two faces of ... | {
"ground_truth": "\\[\n3030\n\\]",
"style": "rule"
} | {
"index": "32624",
"split": "train"
} |
math | [
{
"content": "Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is its area? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25\\sqrt{3}",
"style": "rule"
} | {
"index": "32625",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system xOy, the equation of line l is given as x+1=0, and curve C is a parabola with the coordinate origin O as the vertex and line l as the axis. Establish a polar coordinate system with the coordinate origin O as the pole and the non-negative semi-axis of the x-axis a... | {
"ground_truth": "\\frac { \\sqrt {2}}{2}",
"style": "rule"
} | {
"index": "32626",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sqrt{3}\\cos^2\\left(\\frac{\\pi}{2}+x\\right)-2\\sin(\\pi+x)\\cos x-\\sqrt{3}$. \n$(1)$ Find the extreme values of $f(x)$ on the interval $\\left[\\frac{\\pi}{4}, \\frac{\\pi}{2}\\right]$. \n$(2)$ If $f(x_0-\\frac{\\pi}{6})=\\frac{10}{13}$, where $x_0\\in\\left[\\fra... | {
"ground_truth": "-\\frac{5+12\\sqrt{3}}{26}",
"style": "rule"
} | {
"index": "32627",
"split": "train"
} |
math | [
{
"content": "The numbers $1,2,\\dots,9$ are randomly placed into the $9$ squares of a $3 \\times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{1}{14}",
"style": "rule"
} | {
"index": "32628",
"split": "train"
} |
math | [
{
"content": "In a batch of 100 products, there are 98 qualified products and 2 defective ones. During product inspection, 3 products are randomly selected from the 100 products. \n(1) How many different ways are there to select the 3 products? \n(2) How many ways are there to select exactly 1 defective produ... | {
"ground_truth": "9604",
"style": "rule"
} | {
"index": "32629",
"split": "train"
} |
math | [
{
"content": "A triangle has one side of length $13$, and the angle opposite this side is $60^{\\circ}$. The ratio of the other two sides is $4:3$. Calculate the area of this triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "39 \\sqrt{3}",
"style": "rule"
} | {
"index": "32630",
"split": "train"
} |
math | [
{
"content": "For a sequence $x_1,x_2,\\ldots,x_n$ of real numbers, we define its $\\textit{price}$ as \\[\\max_{1\\le i\\le n}|x_1+\\cdots +x_i|.\\] Given $n$ real numbers, Dave and George want to arrange them into a sequence with a low price. Diligent Dave checks all possible ways and finds the minimum possib... | {
"ground_truth": "c=2",
"style": "rule"
} | {
"index": "32631",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a rectangle with $A B=20$ and $A D=23$. Let $M$ be the midpoint of $C D$, and let $X$ be the reflection of $M$ across point $A$. Compute the area of triangle $X B D$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "575",
"style": "rule"
} | {
"index": "32632",
"split": "train"
} |
math | [
{
"content": "If $10^{2y} = 25$, then $10^{ - y}$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "32633",
"split": "train"
} |
math | [
{
"content": "If a computer executes the following program:\n1. Initial values: \\( x = 3 \\), \\( S = 0 \\).\n2. \\( x = x + 2 \\).\n3. \\( S = S + x \\).\n4. If \\( S \\geq 10000 \\), go to step 5; otherwise, go back to step 2.\n5. Print \\( x \\).\n6. Stop.\n\nThen the value printed at step 5 is: Please outp... | {
"ground_truth": "201",
"style": "rule"
} | {
"index": "32634",
"split": "train"
} |
math | [
{
"content": "In a new diagram, triangle $A'B'C'$ has an area of 36 square units. The points $A', B', C', D'$ are aligned such that $A'C' = 12$ units and $C'D' = 30$ units. What is the area of triangle $B'C'D'$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "32635",
"split": "train"
} |
math | [
{
"content": "A rectangular box has interior dimensions 6-inches by 5-inches by 10-inches. The box is filled with as many solid 3-inch cubes as possible, with all of the cubes entirely inside the rectangular box. What percent of the volume of the box is taken up by the cubes? Please output the final answer with... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "32636",
"split": "train"
} |
math | [
{
"content": "Exactly three of the interior angles of a convex polygon are obtuse. What is the maximum number of sides of such a polygon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32637",
"split": "train"
} |
math | [
{
"content": "A point in three-space has distances $2,6,7,8,9$ from five of the vertices of a regular octahedron. What is its distance from the sixth vertex? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{21}",
"style": "rule"
} | {
"index": "32638",
"split": "train"
} |
math | [
{
"content": "A sample is divided into 5 groups, with a total of 160 data points in the first, second, and third groups, and a total of 260 data points in the third, fourth, and fifth groups, and the frequency of the third group is 0.20. Calculate the frequency of the third group. Please output the final answer... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "32639",
"split": "train"
} |
math | [
{
"content": "Arrange 3 male students and 4 female students in a row. Under the following different requirements, calculate the number of different arrangement methods: \n(1) Person A and Person B must stand at the two ends; \n(2) All male students must be grouped together; \n(3) Male students must not stand... | {
"ground_truth": "1200",
"style": "rule"
} | {
"index": "32640",
"split": "train"
} |
math | [
{
"content": "Assume that $a$, $b$, $c$, and $d$ are positive integers such that $a^5 = b^4$, $c^3 = d^2$, and $c - a = 19$. Determine $d - b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "757",
"style": "rule"
} | {
"index": "32641",
"split": "train"
} |
math | [
{
"content": "The perimeter of a triangle is 30, and all sides are different integers. There are a total of triangles. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "32642",
"split": "train"
} |
math | [
{
"content": "Triangle $A B C$ has $A B=4, B C=5$, and $C A=6$. Points $A^{\\prime}, B^{\\prime}, C^{\\prime}$ are such that $B^{\\prime} C^{\\prime}$ is tangent to the circumcircle of $\\triangle A B C$ at $A, C^{\\prime} A^{\\prime}$ is tangent to the circumcircle at $B$, and $A^{\\prime} B^{\\prime}$ is tang... | {
"ground_truth": "\\frac{80}{3}",
"style": "rule"
} | {
"index": "32643",
"split": "train"
} |
math | [
{
"content": "Ben rolls six fair 12-sided dice, and each of the dice has faces numbered from 1 to 12. What is the probability that exactly three of the dice show a prime number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{857500}{2985984}",
"style": "rule"
} | {
"index": "32644",
"split": "train"
} |
math | [
{
"content": "During the night shift, four duty personnel ate a whole barrel of pickles. If Assistant Mur ate half as much, one-tenth of the barrel would remain. If Lab Technician Trott ate half as much, one-eighth of the barrel would remain. If Intern Glupp ate half as much, one-quarter of the barrel would rem... | {
"ground_truth": "\\frac{1}{40}",
"style": "rule"
} | {
"index": "32645",
"split": "train"
} |
math | [
{
"content": "Oleg drew an empty $50 \\times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 being irrational. Then, in each cell of the table, he wrote the sum of the number... | {
"ground_truth": "1250",
"style": "rule"
} | {
"index": "32646",
"split": "train"
} |
math | [
{
"content": "The measure of angle $ACB$ is 60 degrees. If ray $CA$ is rotated 300 degrees about point $C$ in a clockwise direction, what will be the positive measure of the new acute angle $ACB$, in degrees? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "32647",
"split": "train"
} |
math | [
{
"content": "Given $\\delta(x) = 3x + 8$ and $\\phi(x) = 8x + 7$, what is $x$ if $\\delta(\\phi(x)) = 7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{11}{12}",
"style": "rule"
} | {
"index": "32648",
"split": "train"
} |
math | [
{
"content": "8 people are sitting around a circular table for a meeting, including one leader, one deputy leader, and one recorder. If the recorder is sitting between the leader and the deputy leader, how many different seating arrangements are possible (seating arrangements that can be made identical through ... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "32649",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_n\\}$, where $a_{n+1} + (-1)^n a_n = 2n - 1$, calculate the sum of the first 12 terms of $\\{a_n\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "32650",
"split": "train"
} |
math | [
{
"content": "Given that the price of a gallon of gasoline initially increased by $30\\%$ in January, then decreased by $10\\%$ in February, increased by $15\\%$ in March, and returned to its original value at the end of April, find the value of $x\\%$ that represents the percentage decrease in April to the nea... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "32651",
"split": "train"
} |
math | [
{
"content": "A supermarket has 6 checkout lanes, each with two checkout points numbered 1 and 2. Based on daily traffic, the supermarket plans to select 3 non-adjacent lanes on Monday, with at least one checkout point open in each lane. How many different arrangements are possible for the checkout lanes on Mon... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "32652",
"split": "train"
} |
math | [
{
"content": "The circumcenter of a regular tetrahedron \\( ABCD \\) is \\( O \\). If \\( E \\) is the midpoint of \\( BC \\), what is the measure of the dihedral angle between \\( A-BO-E \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}\\pi",
"style": "rule"
} | {
"index": "32653",
"split": "train"
} |
math | [
{
"content": "Real numbers between 0 and 1, inclusive, are chosen based on the outcome of flipping two fair coins. If two heads are flipped, then the chosen number is 0; if a head and a tail are flipped (in any order), the number is 0.5; if two tails are flipped, the number is 1. Another number is chosen indepe... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "32654",
"split": "train"
} |
math | [
{
"content": "Suppose $m$ and $n$ are positive integers for which the sum of the first $m$ multiples of $n$ is 120, and the sum of the first $m^{3}$ multiples of $n^{3}$ is 4032000. Determine the sum of the first $m^{2}$ multiples of $n^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "20800",
"style": "rule"
} | {
"index": "32655",
"split": "train"
} |
math | [
{
"content": "By permuting the digits of 20130518, how many different eight-digit positive odd numbers can be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3600",
"style": "rule"
} | {
"index": "32656",
"split": "train"
} |
math | [
{
"content": "Multiply \\(333\\) by \\(111\\) and express the result. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36963",
"style": "rule"
} | {
"index": "32657",
"split": "train"
} |
math | [
{
"content": "Pete liked the puzzle; he decided to glue it together and hang it on the wall. In one minute, he glued together two pieces (initial or previously glued). As a result, the entire puzzle was assembled into one complete picture in 2 hours. How much time would it take to assemble the picture if Pete g... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "32658",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\cos \\frac {\\pi}{7}\\cos \\frac {2\\pi}{7}\\cos \\frac {4\\pi}{7}=$ ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac {1}{8}",
"style": "rule"
} | {
"index": "32659",
"split": "train"
} |
math | [
{
"content": "If the six-digit number $\\overline{201 a b 7}$ is divisible by 11 and 13, then the two-digit number $\\overline{a b}$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "32660",
"split": "train"
} |
math | [
{
"content": "It is known that when 2008 is divided by certain natural numbers, the remainder is always 10. How many such natural numbers are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "32661",
"split": "train"
} |
math | [
{
"content": "An equilateral triangle and a square both have perimeters of 48 inches. What is the ratio of the length of the side of the triangle to the length of the side of the square? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac43",
"style": "rule"
} | {
"index": "32662",
"split": "train"
} |
math | [
{
"content": "A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F,... | {
"ground_truth": "512",
"style": "rule"
} | {
"index": "32663",
"split": "train"
} |
math | [
{
"content": "At a meeting of cactus enthusiasts, 80 cactophiles presented their collections, each consisting of cacti of different species. It turned out that no single species of cactus is found in all collections simultaneously, but any 15 people have cacti of the same species. What is the minimum total numb... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "32664",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d \\neq 0$, and $a_1=1$, $a_3$, $a_{13}$ form a geometric sequence. Find the minimum value of $\\frac{2S_n+8}{a_n+3}$ for all positive integers $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{2}",
"style": "rule"
} | {
"index": "32665",
"split": "train"
} |
math | [
{
"content": "Given a parabola $C: y^2 = 2px (p > 0)$ that passes through the point $(1, -2)$, a line $l$ through focus $F$ intersects the parabola $C$ at points $A$ and $B$. If $Q$ is the point $(-\\frac{7}{2}, 0)$ and $BQ \\perp BF$, find the value of $|BF| - |AF|$. Please output the final answer within \\box... | {
"ground_truth": "-\\frac{3}{2}",
"style": "rule"
} | {
"index": "32666",
"split": "train"
} |
math | [
{
"content": "Given the functions $f(x)=2(x+1)$ and $g(x)=x+ \\ln x$, points $A$ and $B$ are located on the graphs of $f(x)$ and $g(x)$ respectively, and their y-coordinates are always equal. Calculate the minimum distance between points $A$ and $B$. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "32667",
"split": "train"
} |
math | [
{
"content": "Given $\\tan \\alpha =2$, find the values of the following expressions: \n$(1) \\frac{\\sin \\alpha - 3\\cos \\alpha}{\\sin \\alpha + \\cos \\alpha}$ \n$(2) 2\\sin ^{2} \\alpha - \\sin \\alpha \\cos \\alpha + \\cos ^{2} \\alpha$ Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{7}{5}",
"style": "rule"
} | {
"index": "32668",
"split": "train"
} |
math | [
{
"content": "Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and nickel both come up heads? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{4}",
"style": "rule"
} | {
"index": "32669",
"split": "train"
} |
math | [
{
"content": "Given points $A(-3, -4)$ and $B(6, 3)$ in the xy-plane; point $C(1, m)$ is taken so that $AC + CB$ is a minimum. Find the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{8}{9}",
"style": "rule"
} | {
"index": "32670",
"split": "train"
} |
math | [
{
"content": "How many real triples $(a, b, c)$ are there such that the polynomial $p(x)=x^{4}+a x^{3}+b x^{2}+a x+c$ has exactly three distinct roots, which are equal to $\\tan y, \\tan 2 y$, and $\\tan 3 y$ for some real $y$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "32671",
"split": "train"
} |
math | [
{
"content": "The coordinates of the 3 vertices of triangle are \\( P(-8, 5) \\), \\( Q(-15, -19) \\), and \\( R(1, -7) \\). The equation of the angle bisector of \\(\\angle P\\) can be written as \\(a x + b y + c = 0\\), where \\(a, b, c \\in \\mathbf{Z}^{+}\\) and \\((a, b, c)=1\\). What is the value of \\(a ... | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "32672",
"split": "train"
} |
math | [
{
"content": "The Pythagoras High School band has $100$ female and $80$ male members. The Pythagoras High School orchestra has $80$ female and $100$ male members. There are $60$ females who are members in both band and orchestra. Altogether, there are $230$ students who are in either band or orchestra or both. ... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32673",
"split": "train"
} |
math | [
{
"content": "How many of the numbers\n\\[ \na_1\\cdot 5^1+a_2\\cdot 5^2+a_3\\cdot 5^3+a_4\\cdot 5^4+a_5\\cdot 5^5+a_6\\cdot 5^6\n\\]\nare negative if $a_1,a_2,a_3,a_4,a_5,a_6 \\in \\{-1,0,1 \\}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "364",
"style": "rule"
} | {
"index": "32674",
"split": "train"
} |
math | [
{
"content": "In a triangle, the area is numerically equal to the perimeter. What is the radius of the inscribed circle?\n$\\text{(A) } 2\\quad \\text{(B) } 3\\quad \\text{(C) } 4\\quad \\text{(D) } 5\\quad \\text{(E) } 6$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "32675",
"split": "train"
} |
math | [
{
"content": "A rectangular piece of paper with side lengths 5 by 8 is folded along the dashed lines so that the folded flaps just touch at the corners as indicated by the dotted lines. Find the area of the resulting trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55/2",
"style": "rule"
} | {
"index": "32676",
"split": "train"
} |
math | [
{
"content": "The two spinners shown are spun once and each lands on one of the numbered sectors. What is the probability that the sum of the numbers in the two sectors is prime? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{9}",
"style": "rule"
} | {
"index": "32677",
"split": "train"
} |
math | [
{
"content": "In a sequence $a_1, a_2, . . . , a_{1000}$ consisting of $1000$ distinct numbers a pair $(a_i, a_j )$ with $i < j$ is called *ascending* if $a_i < a_j$ and *descending* if $a_i > a_j$ . Determine the largest positive integer $k$ with the property that every sequence of $1000$ disti... | {
"ground_truth": "333",
"style": "rule"
} | {
"index": "32678",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = \\sin^6\\left(\\frac{x}{4}\\right) + \\cos^6\\left(\\frac{x}{4}\\right) \\) for all real numbers \\( x \\). Determine \\( f^{(2008)}(0) \\) (i.e., \\( f \\) differentiated 2008 times and then evaluated at \\( x = 0 \\)). Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "32679",
"split": "train"
} |
math | [
{
"content": "Given two real numbers $1<p<q$ so that $\\frac{1}{p} + \\frac{1}{q} = 1$ and $pq = \\frac{9}{2}$, what is $q$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "q = 3",
"style": "rule"
} | {
"index": "32680",
"split": "train"
} |
math | [
{
"content": "Find the total number of occurrences of the digits $0,1 \\ldots, 9$ in the entire guts round. If your answer is $X$ and the actual value is $Y$, your score will be $\\max \\left(0,20-\\frac{|X-Y|}{2}\\right)$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "559",
"style": "rule"
} | {
"index": "32681",
"split": "train"
} |
math | [
{
"content": "Find the positive value of $y$ which satisfies\n\\[\\log_7 (y - 3) + \\log_{\\sqrt{7}} (y^2 - 3) + \\log_{\\frac{1}{7}} (y - 3) = 3.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{\\sqrt{343} + 3}",
"style": "rule"
} | {
"index": "32682",
"split": "train"
} |
math | [
{
"content": "Find all the integers $n>1$ with the following property: the numbers $1,2, \\ldots, n$ can be arranged in a line so that, of any two adjacent numbers, one is divisible by the other. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2, 3, 4, 6",
"style": "rule"
} | {
"index": "32683",
"split": "train"
} |
math | [
{
"content": "Let's call two positive integers almost neighbors if each of them is divisible (without remainder) by their difference. In a math lesson, Vova was asked to write down in his notebook all the numbers that are almost neighbors with \\(2^{10}\\). How many numbers will he have to write down? Please ou... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "32684",
"split": "train"
} |
math | [
{
"content": "Find the number of real solutions of the equation\n\\[\\frac{x}{50} = \\cos x.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "32685",
"split": "train"
} |
math | [
{
"content": "If a die is rolled, event \\( A = \\{1, 2, 3\\} \\) consists of rolling one of the faces 1, 2, or 3. Similarly, event \\( B = \\{1, 2, 4\\} \\) consists of rolling one of the faces 1, 2, or 4.\n\nThe die is rolled 10 times. It is known that event \\( A \\) occurred exactly 6 times.\n\na) Find the ... | {
"ground_truth": "\\frac{16}{3}",
"style": "rule"
} | {
"index": "32686",
"split": "train"
} |
math | [
{
"content": "A middle school cafeteria regularly purchases rice from a grain store at a price of 1500 yuan per ton. Each time rice is purchased, a transportation fee of 100 yuan is required. The cafeteria needs 1 ton of rice per day, and the storage cost for rice is 2 yuan per ton per day (less than one day is... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32687",
"split": "train"
} |
math | [
{
"content": "There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \\( N \\) be the number of two-humped camels. How many possible values c... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "32688",
"split": "train"
} |
math | [
{
"content": "The Athenas are playing a 44 game season. They have 20 wins and 15 losses so far. What is the smallest number of their remaining games that they must win to make the playoffs, given they must win at least 60% of all of their games? Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "32689",
"split": "train"
} |
math | [
{
"content": "In the 17th FIFA World Cup, 35 teams participated, each with 23 players. How many players participated in total? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "805",
"style": "rule"
} | {
"index": "32690",
"split": "train"
} |
math | [
{
"content": "Write $0.\\overline{43}$ as a simplified fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{43}{99}",
"style": "rule"
} | {
"index": "32691",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ has sides of length 2. Set $S$ is the set of all line segments that have length 2 and whose endpoints are on adjacent sides of the square. The midpoints of the line segments in set $S$ enclose a region whose area to the nearest hundredth is $k$. Find $100k$. Please output the final a... | {
"ground_truth": "86",
"style": "rule"
} | {
"index": "32692",
"split": "train"
} |
math | [
{
"content": "Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "\\frac{5}{9}",
"style": "rule"
} | {
"index": "32693",
"split": "train"
} |
math | [
{
"content": "Biejia and Vasha are playing a game. Biejia selects 100 non-negative numbers \\(x_1, x_2, \\cdots, x_{100}\\) (they can be the same), whose sum equals 1. Vasha then pairs these numbers into 50 pairs in any way he chooses, computes the product of the two numbers in each pair, and writes the largest... | {
"ground_truth": "1/396",
"style": "rule"
} | {
"index": "32694",
"split": "train"
} |
math | [
{
"content": "Given any set $A = \\{a_1, a_2, a_3, a_4\\}$ of four distinct positive integers, we denote the sum $a_1 +a_2 +a_3 +a_4$ by $s_A$. Let $n_A$ denote the number of pairs $(i, j)$ with $1 \\leq i < j \\leq 4$ for which $a_i +a_j$ divides $s_A$. Find all sets $A$ of four distinct positive integers whi... | {
"ground_truth": "\\{k, 5k, 7k, 11k\\} \\text{ and } \\{k, 11k, 19k, 29k\\}",
"style": "rule"
} | {
"index": "32695",
"split": "train"
} |
math | [
{
"content": "How many pairs of positive integers $(a, b)$ with $a \\leq b$ satisfy $\\frac{1}{a} + \\frac{1}{b} = \\frac{1}{6}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32696",
"split": "train"
} |
math | [
{
"content": "Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$ ,\n$(*)$ $\\frac{S_{m+n}}{m+n}=\\frac{S_m}{m}\\frac{S_n}{n}$ \nfor $(m,n)=(2,3),(3,2),(2,5)$ , or $(5,2)$ . Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $x+y+z=... | {
"ground_truth": "\\((m, n) = (5, 2), (2, 5), (3, 2), (2, 3)\\)",
"style": "rule"
} | {
"index": "32697",
"split": "train"
} |
math | [
{
"content": "In a single-elimination tournament consisting of $2^{9}=512$ teams, there is a strict ordering on the skill levels of the teams, but Joy does not know that ordering. The teams are randomly put into a bracket and they play out the tournament, with the better team always beating the worse team. Joy ... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "32698",
"split": "train"
} |
math | [
{
"content": "A scientist walking through a forest recorded as integers the heights of $5$ trees standing in a row. She observed that each tree was either twice as tall or half as tall as the one to its right. Unfortunately some of her data was lost when rain fell on her notebook. Her notes are shown below, wit... | {
"ground_truth": "24.2",
"style": "rule"
} | {
"index": "32699",
"split": "train"
} |
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