data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given \\( \\theta_{1}, \\theta_{2}, \\theta_{3}, \\theta_{4} \\in \\mathbf{R}^{+} \\) and \\( \\theta_{1} + \\theta_{2} + \\theta_{3} + \\theta_{4} = \\pi \\), find the minimum value of \\( \\left(2 \\sin^{2} \\theta_{1} + \\frac{1}{\\sin^{2} \\theta_{1}}\\right)\\left(2 \\sin^{2} \\theta_{2} + \\... | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "7600",
"split": "train"
} |
math | [
{
"content": "Let $ABCDE$ be a convex pentagon, and let $H_A,$ $H_B,$ $H_C,$ $H_D$ denote the centroids of triangles $BCD,$ $ACE,$ $ABD,$ and $ABC,$ respectively. Determine the ratio $\\frac{[H_A H_B H_C H_D]}{[ABCDE]}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{9}",
"style": "rule"
} | {
"index": "7601",
"split": "train"
} |
math | [
{
"content": "Calculate $500,000,000,000 - 3 \\times 111,111,111,111$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "166,666,666,667",
"style": "rule"
} | {
"index": "7602",
"split": "train"
} |
math | [
{
"content": "On a table, there are 2 candles, each 20 cm long, but of different diameters. The candles burn evenly, with the thin candle burning completely in 4 hours and the thick candle in 5 hours. After how much time will the thin candle become twice as short as the thick candle if they are lit simultaneous... | {
"ground_truth": "20/3",
"style": "rule"
} | {
"index": "7603",
"split": "train"
} |
math | [
{
"content": "A triangle has one side of length 5 cm, another side of length 12 cm, and includes a right angle. What is the shortest possible length of the third side of the triangle? Express your answer in centimeters as a decimal to the nearest hundredth. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "10.91",
"style": "rule"
} | {
"index": "7604",
"split": "train"
} |
math | [
{
"content": "The points $(-1,4)$ and $(2,-3)$ are adjacent vertices of a square. What is the area of the square? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "7605",
"split": "train"
} |
math | [
{
"content": "What is the value of $$\\frac{1}{2}\\times4\\times\\frac{1}{8}\\times16\\times\\frac{1}{32}\\times64\\times\\frac{1}{128}\\times256\\times\\frac{1}{512}\\times1024?$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "7606",
"split": "train"
} |
math | [
{
"content": "Given that 28×15=420, directly write out the results of the following multiplications:\n2.8×1.5=\\_\\_\\_\\_\\_\\_、0.28×1.5=\\_\\_\\_\\_\\_\\_、0.028×0.15=\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.0042",
"style": "rule"
} | {
"index": "7607",
"split": "train"
} |
math | [
{
"content": "Rhombus $ABCD$ has side length $3$ and $\\angle B = 110$°. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$?\n**A)** $0.81$\n**B)** $1.62$\n**C)** $2.43$\n**D)** $2.16$\n**E)** $3.24$ Please output the ... | {
"ground_truth": "2.16",
"style": "rule"
} | {
"index": "7608",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $(\\frac{{2x}^{2}+2x}{{x}^{2}-1}-\\frac{{x}^{2}-x}{{x}^{2}-2x+1})÷\\frac{x}{x+1}$, where $x=|\\sqrt{3}-2|+(\\frac{1}{2})^{-1}-(π-3.14)^0-\\sqrt[3]{27}+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2\\sqrt{3}}{3} + 1",
"style": "rule"
} | {
"index": "7609",
"split": "train"
} |
math | [
{
"content": "Formulas for shortened multiplication (other). \nCommon fractions Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "198719871987",
"style": "rule"
} | {
"index": "7610",
"split": "train"
} |
math | [
{
"content": "All positive odd numbers are arranged in the following table (the number of numbers in the next row is twice the number of numbers in the previous row) \nFirst row 1 \nSecond row 3 5 \nThird row 7 9 11 13 \n… \nThen, the third number in the sixth row is ． Please output the fi... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "7611",
"split": "train"
} |
math | [
{
"content": "A rectangle is divided into 40 identical squares. The rectangle contains more than one row of squares. Andrew coloured all the squares in the middle row. How many squares did he not colour? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "7612",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a=3 \\sqrt {3}$, $c=2$, $B=150^{\\circ}$, find the length of side $b$ and the area of $\\triangle ABC$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3 \\sqrt {3}}{2}",
"style": "rule"
} | {
"index": "7613",
"split": "train"
} |
math | [
{
"content": "Find $y$ so that the vectors $\\begin{pmatrix} 1 \\\\ -3 \\\\ -4 \\end{pmatrix}$ and $\\begin{pmatrix} -2 \\\\ y \\\\ -1 \\end{pmatrix}$ are orthogonal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "7614",
"split": "train"
} |
math | [
{
"content": "Suppose $x, y$, and $z$ are real numbers greater than 1 such that $$\\begin{aligned} x^{\\log _{y} z} & =2, \\\\ y^{\\log _{z} x} & =4, \\text { and } \\\\ z^{\\log _{x} y} & =8 \\end{aligned}$$ Compute $\\log _{x} y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "7615",
"split": "train"
} |
math | [
{
"content": "The letter T is formed by placing a $2\\:\\text{inch}\\!\\times\\!6\\:\\text{inch}$ rectangle vertically and a $3\\:\\text{inch}\\!\\times\\!2\\:\\text{inch}$ rectangle horizontally on top of the vertical rectangle at its middle, as shown. What is the perimeter of this T, in inches?\n```\n[No grap... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "7616",
"split": "train"
} |
math | [
{
"content": "A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. Please output the final answer within \\boxed... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7617",
"split": "train"
} |
math | [
{
"content": "Given the quadratic function $f(x)=ax^{2}+(2b+1)x-a-2 (a,b \\in R, a \\neq 0)$ has at least one root in the interval $[3,4]$, calculate the minimum value of $a^{2}+b^{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{100}",
"style": "rule"
} | {
"index": "7618",
"split": "train"
} |
math | [
{
"content": "An arithmetic sequence consists of positive terms, with the sum of the first $n$ terms denoted by $S_n$, satisfying $2S_2 = a_2(a_2 + 1)$, and given that $a_1 = 1$, find the minimum value of $\\frac{2S_n + 13}{n}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{33}{4}",
"style": "rule"
} | {
"index": "7619",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$, the sum of the first $n$ terms is $S_n$. If $\\frac{S_6}{S_3} = 3$, then calculate $\\frac{S_{12}}{S_{9}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "7620",
"split": "train"
} |
math | [
{
"content": "A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) Please output the final answer w... | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7621",
"split": "train"
} |
math | [
{
"content": "The difference between the larger root and the smaller root of $x^2 - px + \\frac{p^2 - 1}{4} = 0$ is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7622",
"split": "train"
} |
math | [
{
"content": "A car travels due east at $\\frac 23$ mile per minute on a long, straight road. At the same time, a circular storm, whose radius is $51$ miles, moves southeast at $\\frac 12\\sqrt{2}$ mile per minute. At time $t=0$, the center of the storm is $110$ miles due north of the car. At time $t=t_1$ minut... | {
"ground_truth": "198",
"style": "rule"
} | {
"index": "7623",
"split": "train"
} |
math | [
{
"content": "Suppose $a, b$, and $c$ are real numbers such that $$\\begin{aligned} a^{2}-b c & =14 \\\\ b^{2}-c a & =14, \\text { and } \\\\ c^{2}-a b & =-3 \\end{aligned}$$ Compute $|a+b+c|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17}{5}",
"style": "rule"
} | {
"index": "7624",
"split": "train"
} |
math | [
{
"content": "A basketball player made 5 baskets during a game. Each basket was worth either 2 or 3 points. How many different numbers could represent the total points scored by the player? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7625",
"split": "train"
} |
math | [
{
"content": "The forecast predicts an 80 percent chance of rain for each day of a three-day festival. If it doesn't rain, there is a 50% chance it will be sunny and a 50% chance it will be cloudy. Mina and John want exactly one sunny day during the festival for their outdoor activities. What is the probability... | {
"ground_truth": "0.243",
"style": "rule"
} | {
"index": "7626",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\sum_{k = 1}^\\infty \\frac{6^k}{(3^k - 2^k)(3^{k + 1} - 2^{k + 1})}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7627",
"split": "train"
} |
math | [
{
"content": "There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. Th... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7628",
"split": "train"
} |
math | [
{
"content": "Given that \\(\\frac{810 \\times 811 \\times 812 \\times \\cdots \\times 2010}{810^{n}}\\) is an integer, find the maximum value of \\(n\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "7629",
"split": "train"
} |
math | [
{
"content": "How many 12 step paths are there from point $A$ to point $C$ which pass through point $B$ on a grid, where $A$ is at the top left corner, $B$ is 5 steps to the right and 2 steps down from $A$, and $C$ is 7 steps to the right and 4 steps down from $A$? Please output the final answer within \\boxed{... | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "7630",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=x^{3}+ax^{2}+bx+a^{2}$ where $a,b \\in \\mathbb{R}$. If the function $f(x)$ has an extremum of $10$ at $x=1$, then the value of $b$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-11",
"style": "rule"
} | {
"index": "7631",
"split": "train"
} |
math | [
{
"content": "Find the number of integers $x$ such that the following three conditions all hold: - $x$ is a multiple of 5 - $121<x<1331$ - When $x$ is written as an integer in base 11 with no leading 0 s (i.e. no 0 s at the very left), its rightmost digit is strictly greater than its leftmost digit. Please outp... | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "7632",
"split": "train"
} |
math | [
{
"content": "Given that $\\theta=\\arctan \\frac{5}{12}$, find the principal value of the argument of the complex number $z=\\frac{\\cos 2 \\theta+i \\sin 2 \\theta}{239+i}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "7633",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $a$, $b$, $c$ form an arithmetic sequence in that order. \n$(1)$ If the vectors $\\overrightarrow{m}=(3,\\sin B)$ and $\\overrightarrow{n}=(2,\\sin C)$ are collinear, find the value o... | {
"ground_truth": "2 \\sqrt {3}",
"style": "rule"
} | {
"index": "7634",
"split": "train"
} |
math | [
{
"content": "Find $h(x)$, with terms in order of decreasing degree, if \\[3x^4+2x-1+h(x)=5x^2-6x-1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3x^4+5x^2-8x",
"style": "rule"
} | {
"index": "7635",
"split": "train"
} |
math | [
{
"content": "Find all values of $r$ such that $\\lfloor r \\rfloor + r = 16.5$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8.5",
"style": "rule"
} | {
"index": "7636",
"split": "train"
} |
math | [
{
"content": "The number of rounds of golf played by each golfer of an amateur golf association is shown in the chart below. What is the average number of rounds played by each golfer? Express your answer to the nearest whole number. [asy]\nsize(150);\ndraw((0,7)--(0,0)--(10,0));\nfor(int i = 1; i <= 5; ++i){\n... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "7637",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer that is neither prime nor a cube and that has an even number of prime factors, all greater than 60? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3721",
"style": "rule"
} | {
"index": "7638",
"split": "train"
} |
math | [
{
"content": "Simplify $\\dfrac{3+4i}{1+2i}$. Your answer should be of the form $a+bi$, where $a$ and $b$ are both real numbers and written as improper fractions (if necessary). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{11}{5} - \\dfrac{2}{5}i",
"style": "rule"
} | {
"index": "7639",
"split": "train"
} |
math | [
{
"content": "The real function $f$ has the property that, whenever $a,$ $b,$ $n$ are positive integers such that $a + b = 2^n,$ the equation\n\\[f(a) + f(b) = n^2\\]holds. What is $f(2002)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "7640",
"split": "train"
} |
math | [
{
"content": "Find $AB$ in the triangle below.\n\n[asy]\nunitsize(1inch);\npair A,B,C;\nA = (0,0);\nB = (1,0);\nC = (0.5,sqrt(3)/2);\ndraw (A--B--C--A,linewidth(0.9));\ndraw(rightanglemark(B,A,C,3));\nlabel(\"$A$\",A,S);\nlabel(\"$B$\",B,S);\nlabel(\"$C$\",C,N);\nlabel(\"$18$\", (A+C)/2,W);\nlabel(\"$30^\\circ$... | {
"ground_truth": "18\\sqrt{3}",
"style": "rule"
} | {
"index": "7641",
"split": "train"
} |
math | [
{
"content": "In the circle above, $M$ is the midpoint of arc $CAB$ and segment $MP$ is perpendicular to chord $AB$ at $P$. If the measure of chord $AC$ is $x$ and that of segment $AP$ is $(x+1)$, then segment $PB$ has measure equal to Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2x+1",
"style": "rule"
} | {
"index": "7642",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be an equilateral triangle with side length 1. Points $D, E, F$ lie inside triangle $A B C$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $D E F$ is equilateral. Suppose that there exists a unique equilateral triangle $X Y Z$ with $X$... | {
"ground_truth": "\\frac{1}{1+\\sqrt[3]{2}}",
"style": "rule"
} | {
"index": "7643",
"split": "train"
} |
math | [
{
"content": "On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \\cdot 7 + 12 = 26$. What number will be on the board after an hour? Please outpu... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "7644",
"split": "train"
} |
math | [
{
"content": "In 2006, the revenues of an insurance company increased by 25% and the expenses increased by 15% compared to the previous year. The company's profit (revenue - expenses) increased by 40%. What percentage of the revenues were the expenses in 2006? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "55.2",
"style": "rule"
} | {
"index": "7645",
"split": "train"
} |
math | [
{
"content": "In triangle $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\\sin\\frac{A+C}{2}=b\\sin C$. \n$(1)$ Find angle $B$; \n$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\\sqrt{3}$. Find the perimeter of $\\triangle ABC$.... | {
"ground_truth": "3 + \\sqrt{3}",
"style": "rule"
} | {
"index": "7646",
"split": "train"
} |
math | [
{
"content": "Given that 2 students exercised for 0 days, 4 students exercised for 1 day, 2 students exercised for 2 days, 5 students exercised for 3 days, 4 students exercised for 4 days, 7 students exercised for 5 days, 3 students exercised for 6 days, and 2 students exercised for 7 days, find the mean number... | {
"ground_truth": "3.66",
"style": "rule"
} | {
"index": "7647",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=2\\sqrt{3}\\sin x\\cos x+2\\cos^{2}x-1$.\n\n(I) Find the axis of symmetry and the center of symmetry of $f(x)$;\n\n(II) Find the maximum and minimum values of $f(x)$ on the interval $\\left[-\\frac{\\pi }{6}, \\frac{\\pi }{4}\\right]$. Please output the final answer within... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7648",
"split": "train"
} |
math | [
{
"content": "Points \\( A, B, C, \\) and \\( D \\) lie on a straight line in that order. For a point \\( E \\) outside the line,\n\n\\[ \\angle AEB = \\angle BEC = \\angle CED = 45^\\circ. \\]\n\nLet \\( F \\) be the midpoint of segment \\( AC \\), and \\( G \\) be the midpoint of segment \\( BD \\). What is t... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "7649",
"split": "train"
} |
math | [
{
"content": "Find the smallest $n$ such that $n$! ends in 290 zeroes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1170",
"style": "rule"
} | {
"index": "7650",
"split": "train"
} |
math | [
{
"content": "What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "7651",
"split": "train"
} |
math | [
{
"content": "An electric car is charged 3 times per week for 52 weeks. The cost to charge the car each time is $0.78. What is the total cost to charge the car over these 52 weeks? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\$121.68",
"style": "rule"
} | {
"index": "7652",
"split": "train"
} |
math | [
{
"content": "Let $a, b$, and $c$ be the 3 roots of $x^{3}-x+1=0$. Find $\\frac{1}{a+1}+\\frac{1}{b+1}+\\frac{1}{c+1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "7653",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive real number $x$ such that\n\\[\\lfloor x^2 \\rfloor - x \\lfloor x \\rfloor = 8.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{89}{9}",
"style": "rule"
} | {
"index": "7654",
"split": "train"
} |
math | [
{
"content": "The sum of the maximum and minimum values of the function $y=2\\sin \\left( \\frac{\\pi x}{6}- \\frac{\\pi}{3}\\right)$ where $(0\\leqslant x\\leqslant 9)$ is to be determined. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2-\\sqrt{3}",
"style": "rule"
} | {
"index": "7655",
"split": "train"
} |
math | [
{
"content": "Two congruent cylinders each have radius 8 inches and height 3 inches. The radius of one cylinder and the height of the other are both increased by the same nonzero number of inches. The resulting volumes are equal. How many inches is the increase? Express your answer as a common fraction. Please ... | {
"ground_truth": "\\frac{16}{3}",
"style": "rule"
} | {
"index": "7656",
"split": "train"
} |
math | [
{
"content": "A sector with acute central angle $\\theta$ is cut from a circle of radius 6. The radius of the circle circumscribed about the sector is\n$\\textbf{(A)}\\ 3\\cos\\theta \\qquad \\textbf{(B)}\\ 3\\sec\\theta \\qquad \\textbf{(C)}\\ 3 \\cos \\frac12 \\theta \\qquad \\textbf{(D)}\\ 3 \\sec \\frac12 \... | {
"ground_truth": "3 \\sec \\frac{1}{2} \\theta",
"style": "rule"
} | {
"index": "7657",
"split": "train"
} |
math | [
{
"content": "How many ordered pairs of integers $(x, y)$ satisfy the equation $x^{2020} + y^2 = 2y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "7658",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be an isosceles triangle with apex $A$. Let $I$ be the incenter. If $A I=3$ and the distance from $I$ to $B C$ is 2 , then what is the length of $B C$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{5}",
"style": "rule"
} | {
"index": "7659",
"split": "train"
} |
math | [
{
"content": "Given $\\overrightarrow{a}=(1+\\cos \\omega x,-1)$, $\\overrightarrow{b}=( \\sqrt {3},\\sin \\omega x)$ ($\\omega > 0$), and the function $f(x)= \\overrightarrow{a}\\cdot \\overrightarrow{b}$, with the smallest positive period of $f(x)$ being $2\\pi$. \n(1) Find the expression of the function $f(... | {
"ground_truth": "\\frac {3 \\sqrt {3}+4}{10}",
"style": "rule"
} | {
"index": "7660",
"split": "train"
} |
math | [
{
"content": "The line $l_1$: $x+my+6=0$ is parallel to the line $l_2$: $(m-2)x+3y+2m=0$. Find the value of $m$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "7661",
"split": "train"
} |
math | [
{
"content": "In trapezoid \\(ABCD\\) with bases \\(AD\\) and \\(BC\\), the diagonals intersect at point \\(E\\). The areas of \\(\\triangle ADE\\) and \\(\\triangle BCE\\) are given as 12 and 3, respectively. Find the area of the trapezoid. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "7662",
"split": "train"
} |
math | [
{
"content": "The length of the segment between the points $(2a, a-4)$ and $(4, -1)$ is $2\\sqrt{10}$ units. What is the product of all possible values for $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "7663",
"split": "train"
} |
math | [
{
"content": "Compute: $5^2-3(4)+3^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "7664",
"split": "train"
} |
math | [
{
"content": "Given that the sequence $\\{a_{n}\\}$ is an arithmetic sequence with a non-zero common difference, and $a_{1}+a_{10}=a_{9}$, find $\\frac{{a}_{1}+{a}_{2}+…+{a}_{9}}{{a}_{10}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{27}{8}",
"style": "rule"
} | {
"index": "7665",
"split": "train"
} |
math | [
{
"content": "Let $R$ be the set of all possible remainders when a number of the form $2^n$, $n$ a nonnegative integer, is divided by 1000. Let $S$ be the sum of the elements in $R$. Find the remainder when $S$ is divided by 1000. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "7666",
"split": "train"
} |
math | [
{
"content": "If the point $\\left(m,n\\right)$ in the first quadrant is symmetric with respect to the line $x+y-2=0$ and lies on the line $2x+y+3=0$, calculate the minimum value of $\\frac{1}{m}+\\frac{8}{n}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{25}{9}",
"style": "rule"
} | {
"index": "7667",
"split": "train"
} |
math | [
{
"content": "In the polar coordinate system, circle $C$ is centered at point $C\\left(2, -\\frac{\\pi}{6}\\right)$ with a radius of $2$.\n\n$(1)$ Find the polar equation of circle $C$;\n\n$(2)$ Find the length of the chord cut from circle $C$ by the line $l$: $\\theta = -\\frac{5\\pi}{12} (\\rho \\in \\mathbb{... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "7668",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be a convex trapezoid such that \\(\\angle BAD = \\angle ADC = 90^{\\circ}\\), \\(AB = 20\\), \\(AD = 21\\), and \\(CD = 28\\). Point \\(P \\neq A\\) is chosen on segment \\(AC\\) such that \\(\\angle BPD = 90^{\\circ}\\). Compute \\(AP\\). Please output the final answer within \\bo... | {
"ground_truth": "143/5",
"style": "rule"
} | {
"index": "7669",
"split": "train"
} |
math | [
{
"content": "How many integers between $2020$ and $2400$ have four distinct digits arranged in increasing order? (For example, $2347$ is one integer.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7670",
"split": "train"
} |
math | [
{
"content": "Define $E(n)$ as the sum of the even digits of $n$ and $O(n)$ as the sum of the odd digits of $n$. Find the value of $E(1) + O(1) + E(2) + O(2) + \\dots + E(150) + O(150)$.\nA) 1200\nB) 1300\nC) 1350\nD) 1400\nE) 1450 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1350",
"style": "rule"
} | {
"index": "7671",
"split": "train"
} |
math | [
{
"content": "The number $839$ can be written as $19q+r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q-r$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "7672",
"split": "train"
} |
math | [
{
"content": "Determine all positive integers relatively prime to all the terms of the infinite sequence \\[ a_n=2^n+3^n+6^n -1,\\ n\\geq 1. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7673",
"split": "train"
} |
math | [
{
"content": "Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile inte... | {
"ground_truth": "279",
"style": "rule"
} | {
"index": "7674",
"split": "train"
} |
math | [
{
"content": "Find the length of side $XY$ in the triangle below.\n\n[asy]\nunitsize(1inch);\npair X,Y,Z;\nX = (0,0);\nY= (2,0);\nZ = (0,sqrt(3));\ndraw (X--Y--Z--X,linewidth(0.9));\ndraw(rightanglemark(Y,X,Z,3));\nlabel(\"$X$\",X,S);\nlabel(\"$Y$\",Y,S);\nlabel(\"$Z$\",Z,N);\nlabel(\"$12$\",Z/2,W);\nlabel(\"$6... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "7675",
"split": "train"
} |
math | [
{
"content": "A function $f$ satisfies $f(4x) = 4f(x)$ for all positive real values of $x$, and $f(x) = 2 - |x - 3|$ for $2 \\leq x \\leq 4$. Find the smallest \\( x \\) for which \\( f(x) = f(2022) \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2022",
"style": "rule"
} | {
"index": "7676",
"split": "train"
} |
math | [
{
"content": "Nine points are evenly spaced at intervals of one unit around a $3 \\times 3$ square grid, such that each side of the square has three equally spaced points. Two of the 9 points are chosen at random. What is the probability that the two points are one unit apart?\nA) $\\frac{1}{3}$\nB) $\\frac{1}{... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "7677",
"split": "train"
} |
math | [
{
"content": "Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72000",
"style": "rule"
} | {
"index": "7678",
"split": "train"
} |
math | [
{
"content": "Find the product of the values of $x$ that satisfy the equation $|5x| + 7 = 47$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-64",
"style": "rule"
} | {
"index": "7679",
"split": "train"
} |
math | [
{
"content": "In rectangle \\(ABCD\\), \\(E\\) and \\(F\\) are chosen on \\(\\overline{AB}\\) and \\(\\overline{CD}\\), respectively, so that \\(AEFD\\) is a square. If \\(\\frac{AB}{BE} = \\frac{BE}{BC}\\), determine the value of \\(\\frac{AB}{BC}\\). Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\frac{3 + \\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "7680",
"split": "train"
} |
math | [
{
"content": "The sequence $3, 8, 13, a, b, 33$ is arithmetic. What is the sum of values $a$ and $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "7681",
"split": "train"
} |
math | [
{
"content": "Calculate the expression $16 \\times 0.5 - (4.5 - 0.125 \\times 8)$.\nA) $4.5$\nB) $4$\nC) $4 \\frac{1}{2}$\nD) $6$\nE) $7$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 \\frac{1}{2}",
"style": "rule"
} | {
"index": "7682",
"split": "train"
} |
math | [
{
"content": "Given that $\\dfrac{\\pi}{2} < \\alpha < \\beta < \\dfrac{3\\pi}{4}, \\cos(\\alpha - \\beta) = \\dfrac{12}{13}, \\sin(\\alpha + \\beta) = -\\dfrac{3}{5}$, find the value of $\\sin 2\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\dfrac{56}{65}",
"style": "rule"
} | {
"index": "7683",
"split": "train"
} |
math | [
{
"content": "In a party with $1982$ people, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\[ 1979 \\]",
"style": "rule"
} | {
"index": "7684",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be acute angles such that\n\\begin{align*}\n3 \\sin^2 a + 2 \\sin^2 b &= 1, \\\\\n3 \\sin 2a - 2 \\sin 2b &= 0.\n\\end{align*}Find $a + 2b,$ as measured in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "7685",
"split": "train"
} |
math | [
{
"content": "Mady now has boxes each capable of holding up to 5 balls instead of 4. Under the same process as described, Mady adds balls and resets boxes. Determine the total number of balls in the boxes after her $2010$th step. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7686",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\\frac{c}{b\\sin B}$$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "7687",
"split": "train"
} |
math | [
{
"content": "The rodent control task force went into the woods one day and caught $200$ rabbits and $18$ squirrels. The next day they went into the woods and caught $3$ fewer rabbits and two more squirrels than the day before. Each day they went into the woods and caught $3$ fewer rabbits and two more ... | {
"ground_truth": "5491",
"style": "rule"
} | {
"index": "7688",
"split": "train"
} |
math | [
{
"content": "In a town every two residents who are not friends have a friend in common, and no one is a friend of everyone else. Let us number the residents from 1 to $n$ and let $a_{i}$ be the number of friends of the $i$-th resident. Suppose that $\\sum_{i=1}^{n} a_{i}^{2}=n^{2}-n$. Let $k$ be the smallest n... | {
"ground_truth": "\\[\nk = 5\n\\]",
"style": "rule"
} | {
"index": "7689",
"split": "train"
} |
math | [
{
"content": "For positive integers $n,$ let $\\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\\tau(1)+ \\tau(2) + \\cdots + \\tau(n).$ Let $c$ denote the number of positive integers $n \\leq 1000$ with $S(n)$ odd, and let $d$ denote the number of po... | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "7690",
"split": "train"
} |
math | [
{
"content": "In this subtraction problem, \\( P, Q, R, S, T \\) represent single digits. What is the value of \\( P + Q + R + S + T \\)?\n\n\\[\n\\begin{array}{rrrrr}\n7 & Q & 2 & S & T \\\\\n-P & 3 & R & 9 & 6 \\\\\n\\hline\n2 & 2 & 2 & 2 & 2 \n\\end{array}\n\\] Please output the final answer within \\boxed{}... | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "7691",
"split": "train"
} |
math | [
{
"content": "If the average of a set of sample data 4, 5, 7, 9, $a$ is 6, then the variance $s^2$ of this set of data is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{16}{5}",
"style": "rule"
} | {
"index": "7692",
"split": "train"
} |
math | [
{
"content": "Inside an isosceles triangle $\\mathrm{ABC}$ with equal sides $\\mathrm{AB} = \\mathrm{BC}$ and an angle of 80 degrees at vertex $\\mathrm{B}$, a point $\\mathrm{M}$ is taken such that the angle $\\mathrm{MAC}$ is 10 degrees and the angle $\\mathrm{MCA}$ is 30 degrees. Find the measure of the angl... | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "7693",
"split": "train"
} |
math | [
{
"content": "During the Spring Festival, a supermarket holds a promotional lottery event. When the amount spent by a customer reaches a certain threshold, they can participate in a lottery. The rules of the event are: from a box containing 3 black balls, 2 red balls, and 1 white ball (identical except for colo... | {
"ground_truth": "\\frac{8}{9}",
"style": "rule"
} | {
"index": "7694",
"split": "train"
} |
math | [
{
"content": "Find an ordered pair $(u,v)$ that solves the system: \\begin{align*} 5u &= -7 - 2v,\\\\ 3u &= 4v - 25 \\end{align*} Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-3,4)",
"style": "rule"
} | {
"index": "7695",
"split": "train"
} |
math | [
{
"content": "At the end of a professional bowling tournament, the top 6 bowlers have a playoff. First #6 bowls #5. The loser receives $6^{th}$ prize and the winner bowls #4 in another game. The loser of this game receives $5^{th}$ prize and the winner bowls #3. The loser of this game receives $4^{th}$ prize an... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "7696",
"split": "train"
} |
math | [
{
"content": "A $1 \\times 3$ rectangle is inscribed in a semicircle with the longer side on the diameter. What is the area of the semicircle?\nA) $\\frac{9\\pi}{8}$\nB) $\\frac{12\\pi}{8}$\nC) $\\frac{13\\pi}{8}$\nD) $\\frac{15\\pi}{8}$\nE) $\\frac{16\\pi}{8}$ Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{13\\pi}{8}",
"style": "rule"
} | {
"index": "7697",
"split": "train"
} |
math | [
{
"content": "Given \\(\\triangle DEF\\), where \\(DE=28\\), \\(EF=30\\), and \\(FD=16\\), calculate the area of \\(\\triangle DEF\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "221.25",
"style": "rule"
} | {
"index": "7698",
"split": "train"
} |
math | [
{
"content": "The integers \\( r \\) and \\( k \\) are randomly selected, where \\(-5 < r < 10\\) and \\(0 < k < 10\\). What is the probability that the division \\( r \\div k \\) results in \\( r \\) being a square number? Express your answer as a common fraction. Please output the final answer within \\boxed{... | {
"ground_truth": "\\frac{8}{63}",
"style": "rule"
} | {
"index": "7699",
"split": "train"
} |
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