data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "How many ordered pairs $(a,b)$ such that $a$ is a positive real number and $b$ is an integer between $2$ and $200$, inclusive, satisfy the equation $(\\log_b a)^{2017}=\\log_b(a^{2017})?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "597",
"style": "rule"
} | {
"index": "1400",
"split": "train"
} |
math | [
{
"content": "Given that $\\cos \\alpha = -\\frac{4}{5}$, and $\\alpha$ is an angle in the third quadrant, find the values of $\\sin \\alpha$ and $\\tan \\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "1401",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive multiple of $17$ that is $3$ more than a multiple of $76$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "459",
"style": "rule"
} | {
"index": "1402",
"split": "train"
} |
math | [
{
"content": "The longest side of a right triangle is 13 meters and one of the other sides is 5 meters. What is the area and the perimeter of the triangle? Also, determine if the triangle has any acute angles. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1403",
"split": "train"
} |
math | [
{
"content": "A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is $3$ , then the minimum possible volume of the cone can be written as $\\f... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "1404",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin ^{10} x+\\cos ^{10} x=\\frac{11}{36}$, find the value of $\\sin ^{14} x+\\cos ^{14} x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{41}{216}",
"style": "rule"
} | {
"index": "1405",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle with sides $51, 52, 53$ . Let $\\Omega$ denote the incircle of $\\bigtriangleup ABC$ . Draw tangents to $\\Omega$ which are parallel to the sides of $ABC$ . Let $r_1, r_2, r_3$ be the inradii of the three corener triangles so formed, Find the largest integer that... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1406",
"split": "train"
} |
math | [
{
"content": "The product of positive integers $a$, $b$, and $c$ equals 3960. What is the minimum possible value of the sum $a + b + c$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "1407",
"split": "train"
} |
math | [
{
"content": "Let $U = \\{2, 4, 3-a^2\\}$ and $P = \\{2, a^2+2-a\\}$. Given that the complement of $P$ in $U$, denoted as $\\complement_U P$, is $\\{-1\\}$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "a = 2",
"style": "rule"
} | {
"index": "1408",
"split": "train"
} |
math | [
{
"content": "A group of adventurers displays their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Additionally, it is known that:\n\n- If an adventurer has diamonds, they either have rubies or sapphires (but not both simult... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "1409",
"split": "train"
} |
math | [
{
"content": "For positive real numbers $a,$ $b,$ and $c,$ compute the maximum value of\n\\[\\frac{abc(a + b + c)}{(a + b)^2 (b + c)^2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "1410",
"split": "train"
} |
math | [
{
"content": "Two congruent squares, $ABCD$ and $EFGH$, each with a side length of $20$ units, overlap to form a $20$ by $35$ rectangle $AEGD$. Calculate the percentage of the area of rectangle $AEGD$ that is shaded. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14\\%",
"style": "rule"
} | {
"index": "1411",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{abcd}$ where $a, b, c, d$ are distinct digits. Find the sum of the elements of $\\mathcal{T}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2520",
"style": "rule"
} | {
"index": "1412",
"split": "train"
} |
math | [
{
"content": "Suppose that $S$ is a series of real numbers between $2$ and $8$ inclusive, and that for any two elements $y > x$ in $S,$ $$ 98y - 102x - xy \\ge 4. $$ What is the maximum possible size for the set $S?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "1413",
"split": "train"
} |
math | [
{
"content": "For any real number \\( x \\), let \\([x]\\) denote the greatest integer less than or equal to \\( x \\). When \\( 0 \\leqslant x \\leqslant 100 \\), how many different integers are in the range of the function \\( f(x) = [2x] + [3x] + [4x] + [5x] \\)? Please output the final answer within \\boxed... | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "1414",
"split": "train"
} |
math | [
{
"content": "The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\\frac{1}{... | {
"ground_truth": "154",
"style": "rule"
} | {
"index": "1415",
"split": "train"
} |
math | [
{
"content": "A man asks: \"How old are you, my son?\"\n\nTo this question, I answered:\n\n\"If my father were seven times as old as I was eight years ago, then one quarter of my father's current age would surely be fourteen years now. Please calculate from this how many years weigh upon my shoulders!\" Please ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "1416",
"split": "train"
} |
math | [
{
"content": "A mischievous child mounted the hour hand on the minute hand's axle and the minute hand on the hour hand's axle of a correctly functioning clock. The question is, how many times within a day does this clock display the correct time? Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "1417",
"split": "train"
} |
math | [
{
"content": "Solve the following system of equations. It has a solution if and only if each term equals zero:\n\n$$\n\\left\\{\\begin{array}{c}\n3 x^{2}+8 x-3=0 \\\\\n3 x^{4}+2 x^{3}-10 x^{2}+30 x-9=0\n\\end{array}\\right.\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "1418",
"split": "train"
} |
math | [
{
"content": "In the game of rock-paper-scissors-lizard-Spock, rock defeats scissors and lizard, paper defeats rock and Spock, scissors defeats paper and lizard, lizard defeats paper and Spock, and Spock defeats rock and scissors. If three people each play a game of rock-paper-scissors-lizard-Spock at the same ... | {
"ground_truth": "\\frac{12}{25}",
"style": "rule"
} | {
"index": "1419",
"split": "train"
} |
math | [
{
"content": "A fair coin is flipped 8 times. What is the probability that exactly 6 of the flips come up heads? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{64}",
"style": "rule"
} | {
"index": "1420",
"split": "train"
} |
math | [
{
"content": "Calculate the product $(\\frac{4}{8})(\\frac{8}{12})(\\frac{12}{16})\\cdots(\\frac{2016}{2020})$. Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{505}",
"style": "rule"
} | {
"index": "1421",
"split": "train"
} |
math | [
{
"content": "In Chinese ancient music, the pentatonic scale is the basic scale, hence the idiom \"五音不全\" (meaning \"lacking in musical talent\"). The five notes of the pentatonic scale in Chinese ancient music are: 宫 (gong), 商 (shang), 角 (jue), 徵 (zhi), 羽 (yu). If these five notes are all used and arranged int... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "1422",
"split": "train"
} |
math | [
{
"content": "Suppose that $a$, $b$, and $c$ are positive real numbers such that $a^{\\log_3 7} = 27$, $b^{\\log_7 11} = 49$, and $c^{\\log_{11}25} = \\sqrt{11}$. Find \\[a^{(\\log_3 7)^2} + b^{(\\log_7 11)^2} + c^{(\\log_{11} 25)^2}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "469",
"style": "rule"
} | {
"index": "1423",
"split": "train"
} |
math | [
{
"content": "How many positive integers $n$ satisfy \\[(n + 8)(n - 3)(n-12)<0\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "1424",
"split": "train"
} |
math | [
{
"content": "Sally has six red cards numbered 1 through 6 and seven blue cards numbered 2 through 8. She attempts to create a stack where the colors alternate, and the number on each red card divides evenly into the number on each neighboring blue card. Can you determine the sum of the numbers on the middle th... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "1425",
"split": "train"
} |
math | [
{
"content": "Compute \\(\\sqrt[4]{506250000}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "1426",
"split": "train"
} |
math | [
{
"content": "Find the sum of all values of $a + b$ , where $(a, b)$ is an ordered pair of positive integers and $a^2+\\sqrt{2017-b^2}$ is a perfect square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "1427",
"split": "train"
} |
math | [
{
"content": "Define $n!!$ to be $n(n-2)(n-4)\\cdots 3\\cdot 1$ for $n$ odd and $n(n-2)(n-4)\\cdots 4\\cdot 2$ for $n$ even. When $\\sum_{i=1}^{2009} \\frac{(2i-1)!!}{(2i)!!}$ is expressed as a fraction in lowest terms, its denominator is $2^ab$ with $b$ odd. Find $\\dfrac{ab}{10}$. Please output the final answ... | {
"ground_truth": "401",
"style": "rule"
} | {
"index": "1428",
"split": "train"
} |
math | [
{
"content": "What are the first three digits to the right of the decimal point in the decimal representation of $\\left(10^{1001}+1\\right)^{3/2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "743",
"style": "rule"
} | {
"index": "1429",
"split": "train"
} |
math | [
{
"content": "Two cross sections of a right octagonal pyramid are obtained by cutting the pyramid with planes parallel to the octagonal base. The areas of the cross sections are $300\\sqrt{2}$ square feet and $675\\sqrt{2}$ square feet. The two planes are $10$ feet apart. How far from the apex of the pyramid is... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "1430",
"split": "train"
} |
math | [
{
"content": "The ratio $\\frac{10^{2000}+10^{2002}}{10^{2001}+10^{2001}}$ is closest to which whole number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "1431",
"split": "train"
} |
math | [
{
"content": "A grid consists of multiple squares, as shown below. Count the different squares possible using the lines in the grid.\n\n[asy]\nunitsize(0.5 cm);\nint i, j;\nfor(i = 0; i <= 5; ++i) {\n draw((0,i)--(5,i));\n draw((i,0)--(i,5));\n}\nfor(i = 1; i <= 4; ++i) {\n for (j = 1; j <= 4; ++j) {\n... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "1432",
"split": "train"
} |
math | [
{
"content": "Solve \\[\\frac{3x^2+2x+1}{x-1}=3x+1\\]for $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}2",
"style": "rule"
} | {
"index": "1433",
"split": "train"
} |
math | [
{
"content": "Rebecca has twenty-four resistors, each with resistance 1 ohm. Every minute, she chooses any two resistors with resistance of $a$ and $b$ ohms respectively, and combine them into one by one of the following methods: - Connect them in series, which produces a resistor with resistance of $a+b$ ohms;... | {
"ground_truth": "1015080877",
"style": "rule"
} | {
"index": "1434",
"split": "train"
} |
math | [
{
"content": "A quadrilateral \\(A B C D\\) is inscribed in a circle with radius 6 and center at point \\(O\\). Its diagonals \\(A C\\) and \\(B D\\) are mutually perpendicular and intersect at point \\(K\\). Points \\(E\\) and \\(F\\) are the midpoints of \\(A C\\) and \\(B D\\), respectively. The segment \\(O... | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "1435",
"split": "train"
} |
math | [
{
"content": "Find the value of \\(\\tan \\left(\\tan^{-1} \\frac{1}{2} + \\tan^{-1} \\frac{1}{2 \\times 2^2} + \\tan^{-1} \\frac{1}{2 \\times 3^2} + \\cdots + \\tan^{-1} \\frac{1}{2 \\times 2009^2}\\right)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2009}{2010}",
"style": "rule"
} | {
"index": "1436",
"split": "train"
} |
math | [
{
"content": "Solve \\[\\frac{x^2+2x+3}{x+4}=x+5\\]for $x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{17}7",
"style": "rule"
} | {
"index": "1437",
"split": "train"
} |
math | [
{
"content": "Given square $ABCD$, points $E$ and $F$ lie on $\\overline{AB}$ so that $\\overline{ED}$ and $\\overline{FD}$ bisect $\\angle ADC$. Calculate the ratio of the area of $\\triangle DEF$ to the area of square $ABCD$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "1438",
"split": "train"
} |
math | [
{
"content": "On an old-fashioned bicycle the front wheel has a radius of $2.5$ feet and the back wheel has a radius of $4$ inches. If there is no slippage, how many revolutions will the back wheel make while the front wheel makes $100$ revolutions? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "750",
"style": "rule"
} | {
"index": "1439",
"split": "train"
} |
math | [
{
"content": "A $5 \\times 8$ rectangle can be rolled to form two different cylinders with different maximum volumes. What is the ratio of the larger volume to the smaller volume? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{5}",
"style": "rule"
} | {
"index": "1440",
"split": "train"
} |
math | [
{
"content": "Suppose $a$, $b$, and $c$ are positive integers such that $a \\geq b \\geq c$ and $a+b+c=2010$. Furthermore, $a!b!c! = m \\cdot 10^n$, where $m$ and $n$ are integers, and $m$ is not divisible by $10$. What is the smallest possible value of $n$?\nA) 499\nB) 500\nC) 502\nD) 504\nE) 506 Please output... | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "1441",
"split": "train"
} |
math | [
{
"content": "For dinner, Priya is eating grilled pineapple spears. Each spear is in the shape of the quadrilateral PINE, with $P I=6 \\mathrm{~cm}, I N=15 \\mathrm{~cm}, N E=6 \\mathrm{~cm}, E P=25 \\mathrm{~cm}$, and \\angle N E P+\\angle E P I=60^{\\circ}$. What is the area of each spear, in \\mathrm{cm}^{2}... | {
"ground_truth": "\\frac{100 \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "1442",
"split": "train"
} |
math | [
{
"content": "Let $V$ be a 10-dimensional real vector space and $U_1,U_2$ two linear subspaces such that $U_1 \\subseteq U_2, \\dim U_1 =3, \\dim U_2=6$ . Let $\\varepsilon$ be the set of all linear maps $T: V\\rightarrow V$ which have $T(U_1)\\subseteq U_1, T(U_2)\\subseteq U_2$ . Calculate the dimen... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "1443",
"split": "train"
} |
math | [
{
"content": "Let $m$ be the largest real solution to the equation\n$\\frac{3}{x-3}+\\frac{5}{x-5}+\\frac{17}{x-17}+\\frac{19}{x-19}=x^2-11x-4$\n There are positive integers $a$, $b$, and $c$ such that $m=a+\\sqrt{b+\\sqrt{c}}$. Find $a+b+c$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "263",
"style": "rule"
} | {
"index": "1444",
"split": "train"
} |
math | [
{
"content": "Trapezoid $EFGH$ has sides $EF=86$, $FG=60$, $GH=26$, and $HE=80$, with $EF$ parallel to $GH$. A circle with center $Q$ on $EF$ is drawn tangent to $FG$ and $HE$. Find $EQ$, expressed as a fraction $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, and compute $p+q$. Please ... | {
"ground_truth": "163",
"style": "rule"
} | {
"index": "1445",
"split": "train"
} |
math | [
{
"content": "Consider a rectangle $ABCD$, and inside it are four squares with non-overlapping interiors. Two squares have the same size and an area of 4 square inches each, located at corners $A$ and $C$ respectively. There is another small square with an area of 1 square inch, and a larger square, twice the s... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "1446",
"split": "train"
} |
math | [
{
"content": "Let $a$, $b$, $c$ be the three sides of a triangle, and let $\\alpha$, $\\beta$, $\\gamma$, be the angles opposite them. If $a^2+b^2=1989c^2$, find\n$\\frac{\\cot \\gamma}{\\cot \\alpha+\\cot \\beta}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "994",
"style": "rule"
} | {
"index": "1447",
"split": "train"
} |
math | [
{
"content": "Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toes and 92 fingers. Ho... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1448",
"split": "train"
} |
math | [
{
"content": "Factor completely: $x^6 - 3x^4 + 3x^2 - 1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(x-1)^3(x+1)^3",
"style": "rule"
} | {
"index": "1449",
"split": "train"
} |
math | [
{
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 7 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{10}",
"style": "rule"
} | {
"index": "1450",
"split": "train"
} |
math | [
{
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$, and when $x > 0$, $f(x) = 2 + f\\left(\\frac{1}{2}\\right)\\log_{2}x$, evaluate $f(-2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "1451",
"split": "train"
} |
math | [
{
"content": "Convert $102012_{(3)}$ to base 10. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "320",
"style": "rule"
} | {
"index": "1452",
"split": "train"
} |
math | [
{
"content": "The area enclosed by the curves $y=e^{x}$, $y=e^{-x}$, and the line $x=1$ is $e^{1}-e^{-1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "e+e^{-1}-2",
"style": "rule"
} | {
"index": "1453",
"split": "train"
} |
math | [
{
"content": "In right triangle $GHI$, we have $\\angle G = 40^\\circ$, $\\angle H = 90^\\circ$, and $HI = 12$. Find the length of $GH$ and $GI$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18.7",
"style": "rule"
} | {
"index": "1454",
"split": "train"
} |
math | [
{
"content": "In triangle $PQR,$ $PQ = 12,$ $QR = 13,$ $PR = 15,$ and point $H$ is the intersection of the medians. Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\\circ$ rotation about $H.$ What is the area of the union of the two regions enclosed by the triangles ... | {
"ground_truth": "20\\sqrt{14}",
"style": "rule"
} | {
"index": "1455",
"split": "train"
} |
math | [
{
"content": "Compute the number of positive integers less than 10! which can be expressed as the sum of at most 4 (not necessarily distinct) factorials. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "648",
"style": "rule"
} | {
"index": "1456",
"split": "train"
} |
math | [
{
"content": "How many ordered pairs $(a, b)$ of positive integers satisfy $a^{2}+b^{2}=50$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "1457",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\sin ( \\frac {7π}{6}-2x)-2\\sin ^{2}x+1(x∈R)$,\n(1) Find the period and the monotonically increasing interval of the function $f(x)$;\n(2) In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. The graph of function $f(x)$ passes ... | {
"ground_truth": "3 \\sqrt {2}",
"style": "rule"
} | {
"index": "1458",
"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{4}{9}",
"style": "rule"
} | {
"index": "1459",
"split": "train"
} |
math | [
{
"content": "Given that each side of a large square is divided into four equal parts, a smaller square is inscribed in such a way that its corners are at the division points one-fourth and three-fourths along each side of the large square, calculate the ratio of the area of this inscribed square to the area of... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "1460",
"split": "train"
} |
math | [
{
"content": "Given that three cultural courses (Chinese, Mathematics, and Foreign Language) and three other arts courses are randomly scheduled in six periods, find the probability that no two adjacent cultural courses are separated by more than one arts course. Please output the final answer within \\boxed{}.... | {
"ground_truth": "\\dfrac{3}{5}",
"style": "rule"
} | {
"index": "1461",
"split": "train"
} |
math | [
{
"content": "A square with side length $x$ is inscribed in a right triangle with sides of length $3$, $4$, and $5$ so that one vertex of the square coincides with the right-angle vertex of the triangle. A square with side length $y$ is inscribed in another right triangle with sides of length $3$, $4$, and $5$ ... | {
"ground_truth": "\\frac{37}{35}",
"style": "rule"
} | {
"index": "1462",
"split": "train"
} |
math | [
{
"content": "A sphere is inscribed in a cone whose axial cross-section is an equilateral triangle. Find the volume of the cone if the volume of the sphere is \\( \\frac{32\\pi}{3} \\ \\text{cm}^3 \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24 \\pi",
"style": "rule"
} | {
"index": "1463",
"split": "train"
} |
math | [
{
"content": "An infinite geometric series has common ratio $1/8$ and sum 60. What is the first term of the series? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{105}{2}",
"style": "rule"
} | {
"index": "1464",
"split": "train"
} |
math | [
{
"content": "Expand the product ${3(x+4)(x+5)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3x^2 + 27x + 60",
"style": "rule"
} | {
"index": "1465",
"split": "train"
} |
math | [
{
"content": "How many rectangles can be formed when the vertices are chosen from points on a 4x4 grid (having 16 points)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "1466",
"split": "train"
} |
math | [
{
"content": "For what real value of $b$ is the expression $\\frac{1}{2}b^2 + 5b - 3$ minimized? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "1467",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\( B, C \\) and \\( D \\) lie on a straight line, with \\(\\angle ACD=100^{\\circ}\\), \\(\\angle ADB=x^{\\circ}\\), \\(\\angle ABD=2x^{\\circ}\\), and \\(\\angle DAC=\\angle BAC=y^{\\circ}\\). The value of \\( x \\) is: Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1468",
"split": "train"
} |
math | [
{
"content": "Find the smallest solution to the equation \\[\\lfloor x^2 \\rfloor - \\lfloor x \\rfloor^2 = 17.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7\\sqrt2",
"style": "rule"
} | {
"index": "1469",
"split": "train"
} |
math | [
{
"content": "Cagney can frost a cupcake every 20 seconds and Lacey can frost a cupcake every 30 seconds. Working together, how many cupcakes can they frost in 5 minutes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1470",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. Given the line $l$: $ρ=- \\frac{6}{3\\cos θ +4\\sin θ}$, and the curve $C$: $\\begin{cases} \\overset{x=3+5\\cos... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "1471",
"split": "train"
} |
math | [
{
"content": "Define $\\displaystyle{f(x) = x + \\sqrt{x + \\sqrt{x + \\sqrt{x + \\sqrt{x + \\ldots}}}}}$ . Find the smallest integer $x$ such that $f(x)\\ge50\\sqrt{x}$ .\n\n(Edit: The official question asked for the \"smallest integer\"; the intended question was the \"smallest positive integer\".) Pleas... | {
"ground_truth": "2500",
"style": "rule"
} | {
"index": "1472",
"split": "train"
} |
math | [
{
"content": "A snowball with a temperature of $0^{\\circ} \\mathrm{C}$ is launched at a speed $v$ towards a wall. Upon impact, $k=0.02\\%$ of the entire snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\\frac{v}{2}$? The specific heat of fusi... | {
"ground_truth": "0.005",
"style": "rule"
} | {
"index": "1473",
"split": "train"
} |
math | [
{
"content": "Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "968",
"style": "rule"
} | {
"index": "1474",
"split": "train"
} |
math | [
{
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\pi} 2^{4} \\sin ^{4}\\left(\\frac{x}{2}\\right) \\cos ^{4}\\left(\\frac{x}{2}\\right) d x\n$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{8}",
"style": "rule"
} | {
"index": "1475",
"split": "train"
} |
math | [
{
"content": "Points were marked on the sides of triangle \\(ABC\\): 12 points on side \\(AB\\), 9 points on side \\(BC\\), and 10 points on side \\(AC\\). None of the vertices of the triangle are marked. How many triangles can be formed with vertices at the marked points? Please output the final answer within ... | {
"ground_truth": "4071",
"style": "rule"
} | {
"index": "1476",
"split": "train"
} |
math | [
{
"content": "Anthony made $5$ of his first $12$ free throw attempts. If he makes $2/3$ of his next $24$ attempts, by how many percentage points will he increase his overall success rate percentage? Express your answer to the nearest whole number. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "17\\%",
"style": "rule"
} | {
"index": "1477",
"split": "train"
} |
math | [
{
"content": "Suppose $A, B, C$, and $D$ are four circles of radius $r>0$ centered about the points $(0, r),(r, 0)$, $(0,-r)$, and $(-r, 0)$ in the plane. Let $O$ be a circle centered at $(0,0)$ with radius $2 r$. In terms of $r$, what is the area of the union of circles $A, B, C$, and $D$ subtracted by the are... | {
"ground_truth": "8 r^{2}",
"style": "rule"
} | {
"index": "1478",
"split": "train"
} |
math | [
{
"content": "Define an operation between sets A and B: $A*B = \\{x | x = x_1 + x_2, \\text{ where } x_1 \\in A, x_2 \\in B\\}$. If $A = \\{1, 2, 3\\}$ and $B = \\{1, 2\\}$, then the sum of all elements in $A*B$ is ____. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "1479",
"split": "train"
} |
math | [
{
"content": "Let $p(x)=x^{2}-x+1$. Let $\\alpha$ be a root of $p(p(p(p(x))))$. Find the value of $(p(\\alpha)-1) p(\\alpha) p(p(\\alpha)) p(p(p(\\alpha)))$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "1480",
"split": "train"
} |
math | [
{
"content": "In the $x-y$ plane, draw a circle of radius 2 centered at $(0,0)$. Color the circle red above the line $y=1$, color the circle blue below the line $y=-1$, and color the rest of the circle white. Now consider an arbitrary straight line at distance 1 from the circle. We color each point $P$ of the l... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "1481",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = ax^7 + bx - 2$, if $f(2008) = 10$, then the value of $f(-2008)$ is. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-12",
"style": "rule"
} | {
"index": "1482",
"split": "train"
} |
math | [
{
"content": "The numbers 1, 3, 6, 10, $\\ldots$, are called triangular numbers, as shown geometrically here. What is the $20^{\\text{th}}$ triangular number?\n\n[asy]\n\ndot((0,0));\nlabel(\"1\",(0,-1.5));\n\ndot((3,0));\ndot((4,0));\ndot((3,1));\nlabel(\"3\",(3.5,-1.5));\n\ndot((7,0));\ndot((8,0));\ndot((9,0... | {
"ground_truth": "210",
"style": "rule"
} | {
"index": "1483",
"split": "train"
} |
math | [
{
"content": "Given that $F$ is the right focus of the hyperbola $C$: ${{x}^{2}}-\\dfrac{{{y}^{2}}}{8}=1$, and $P$ is a point on the left branch of $C$, $A(0,4)$. When the perimeter of $\\Delta APF$ is minimized, the area of this triangle is \\_\\_\\_. Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "\\dfrac{36}{7}",
"style": "rule"
} | {
"index": "1484",
"split": "train"
} |
math | [
{
"content": "Find the area of rhombus $ABCD$ given that the circumradii of triangles $ABD$ and $ACD$ are $12.5$ and $25$, respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "400",
"style": "rule"
} | {
"index": "1485",
"split": "train"
} |
math | [
{
"content": "Let the roots of the cubic equation \\(27x^3 - 81x^2 + 63x - 14 = 0\\) be in geometric progression. Find the difference between the square of the largest root and the square of the smallest root. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "1486",
"split": "train"
} |
math | [
{
"content": "Find the minimum value of the function\n\n$$\nf(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\\ldots+(x-104)^{2}\n$$\n\nIf the result is a non-integer, round it to the nearest integer. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49608",
"style": "rule"
} | {
"index": "1487",
"split": "train"
} |
math | [
{
"content": "Find the sum of the roots of the equation \\[(2x^3 + x^2 - 8x + 20)(5x^3 - 25x^2 + 19) = 0.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\tfrac{9}{2}",
"style": "rule"
} | {
"index": "1488",
"split": "train"
} |
math | [
{
"content": "Let $ ABP, BCQ, CAR$ be three non-overlapping triangles erected outside of acute triangle $ ABC$. Let $ M$ be the midpoint of segment $ AP$. Given that $ \\angle PAB \\equal{} \\angle CQB \\equal{} 45^\\circ$, $ \\angle ABP \\equal{} \\angle QBC \\equal{} 75^\\circ$, $ \\angle RAC \\equal{} 105^\\... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "1489",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of all integers greater than 4 and less than 21. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "1490",
"split": "train"
} |
math | [
{
"content": "In parallelogram \\(ABCD\\), \\(AB = 1\\), \\(BC = 4\\), and \\(\\angle ABC = 60^\\circ\\). Suppose that \\(AC\\) is extended from \\(A\\) to a point \\(E\\) beyond \\(C\\) so that triangle \\(ADE\\) has the same area as the parallelogram. Find the length of \\(DE\\). Please output the final answe... | {
"ground_truth": "2\\sqrt{3}",
"style": "rule"
} | {
"index": "1491",
"split": "train"
} |
math | [
{
"content": "Seven points are evenly spaced out on a circle and connected as shown below to form a 7-pointed star. What is the sum of the angle measurements of the seven tips of the star, in degrees? One such angle is marked as $\\alpha$ below.\n\n[asy]\ndotfactor=4;\ndraw(Circle((0,0),1));\nreal x = 2*pi/7;... | {
"ground_truth": "540",
"style": "rule"
} | {
"index": "1492",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, $c$ are the opposite sides of angles $A$, $B$, $C$ in triangle $ABC$, and $\\frac{a-c}{b-\\sqrt{2}c}=\\frac{sin(A+C)}{sinA+sinC}$. \n$(Ⅰ)$ Find the measure of angle $A$; \n$(Ⅱ)$ If $a=\\sqrt{2}$, $O$ is the circumcenter of triangle $ABC$, find the minimum value of $|3\\overr... | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "1493",
"split": "train"
} |
math | [
{
"content": "Consider the infinite series defined by the following progression:\n\\[2 + \\frac{1}{3} + \\frac{1}{9} + \\frac{1}{3^2} + \\frac{1}{9^2} + \\frac{1}{3^3} + \\frac{1}{9^3} + \\cdots\\]\nDetermine the limit of this series as it extends to infinity.\nA) $\\frac{1}{3}$\nB) $3$\nC) $\\frac{21}{8}$\nD) ... | {
"ground_truth": "\\frac{21}{8}",
"style": "rule"
} | {
"index": "1494",
"split": "train"
} |
math | [
{
"content": "In a photograph, Aristotle, David, Flora, Munirah, and Pedro are seated in a random order in a row of 5 chairs. If David is seated in the middle of the row, what is the probability that Pedro is seated beside him? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "1495",
"split": "train"
} |
math | [
{
"content": "Point $Q$ is located inside triangle $DEF$ such that angles $QDE, QEF,$ and $QFD$ are all congruent. The sides of the triangle have lengths $DE = 15, EF = 16,$ and $FD = 17.$ Find $\\tan \\angle QDE.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{272}{385}",
"style": "rule"
} | {
"index": "1496",
"split": "train"
} |
math | [
{
"content": "The diameter, in inches, of a sphere with twice the volume of a sphere of radius 9 inches can be expressed in the form $a\\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "1497",
"split": "train"
} |
math | [
{
"content": "The sum of two numbers is 25 and their product is 126. What is the absolute value of the difference of the two numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "1498",
"split": "train"
} |
math | [
{
"content": "In the book \"The Nine Chapters on the Mathematical Art,\" a right square pyramid, whose base is a rectangle and has a side edge perpendicular to the base, is called a \"yangma.\" Given a right square pyramid $M-ABCD$, where side edge $MA$ is perpendicular to the base $ABCD$, and $MA = BC = AB = 2... | {
"ground_truth": "36\\pi - 16\\sqrt{2}\\pi",
"style": "rule"
} | {
"index": "1499",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.