data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given $\\overrightarrow{a}=( \\sqrt {2},m)(m > 0)$, $\\overrightarrow{b}=(\\sin x,\\cos x)$, and the maximum value of the function $f(x)= \\overrightarrow{a} \\cdot \\overrightarrow{b}$ is $2$.\n1. Find $m$ and the smallest positive period of the function $f(x)$;\n2. In $\\triangle ABC$, $f(A- \\f... | {
"ground_truth": "\\frac { \\sqrt {3}}{4}",
"style": "rule"
} | {
"index": "32700",
"split": "train"
} |
math | [
{
"content": "$A, B, C$ and $D$ are distinct positive integers such that the product $AB = 60$, the product $CD = 60$ and $A - B = C + D$ . What is the value of $A$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "32701",
"split": "train"
} |
math | [
{
"content": "A cube is dissected into 6 pyramids by connecting a given point in the interior of the cube with each vertex of the cube, so that each face of the cube forms the base of a pyramid. The volumes of five of these pyramids are 200, 500, 1000, 1100, and 1400. What is the volume of the sixth pyramid? Pl... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "32702",
"split": "train"
} |
math | [
{
"content": "Given \\(\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)\\), \\(\\sin \\beta = 2 \\cos (\\alpha + \\beta) \\cdot \\sin \\alpha \\left(\\alpha + \\beta \\neq \\frac{\\pi}{2}\\right)\\), find the maximum value of \\(\\tan \\beta\\). Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "32703",
"split": "train"
} |
math | [
{
"content": "When 100 is divided by a positive integer $x$, the remainder is 10. When 1000 is divided by $x$, what is the remainder? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32704",
"split": "train"
} |
math | [
{
"content": "A typical triangle is defined as one in which each angle is at least \\(30^{\\circ}\\).\n\nA triangle is identified using the following procedure. The perimeter of a circle is divided into 180 equal arcs, and 3 distinct points are randomly chosen from these division points. (Any three points are c... | {
"ground_truth": "0.25",
"style": "rule"
} | {
"index": "32705",
"split": "train"
} |
math | [
{
"content": "For a geometric sequence $\\{b_n\\}$ with a common ratio of $4$, if $T_n$ is the product of the first $n$ terms of the sequence $\\{b_n\\}$, then $\\frac{T_{20}}{T_{10}}$, $\\frac{T_{30}}{T_{20}}$, $\\frac{T_{40}}{T_{30}}$ also form a geometric sequence with a common ratio of $4^{100}$. Similarly,... | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "32706",
"split": "train"
} |
math | [
{
"content": "In a math competition, there are 5 problems, each with a different natural number score. The smaller the problem number, the lower its score (for example, the score for problem 1 is less than the score for problem 2). Xiao Ming solved all the problems correctly. The total score for the first 2 pro... | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "32707",
"split": "train"
} |
math | [
{
"content": "Let $a_n=6^{n}+8^{n}$. Determine the remainder upon dividing $a_ {83}$ by $49$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "35",
"style": "rule"
} | {
"index": "32708",
"split": "train"
} |
math | [
{
"content": "Let $g : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[g(x) g(y) - g(xy) = 2x + 2y\\]for all real numbers $x$ and $y.$\n\nCalculate the number $n$ of possible values of $g(2),$ and the sum $s$ of all possible values of $g(2),$ and find the product $n \\times s.$ Please output the final... | {
"ground_truth": "\\frac{28}{3}",
"style": "rule"
} | {
"index": "32709",
"split": "train"
} |
math | [
{
"content": "Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February. Please output the final answe... | {
"ground_truth": "142.00",
"style": "rule"
} | {
"index": "32710",
"split": "train"
} |
math | [
{
"content": "A hemisphere-shaped bowl with radius 1 foot is filled full with chocolate. All of the chocolate is then evenly distributed between 27 congruent, smaller hemisphere-shaped molds. What is the radius of each of the smaller molds, in feet? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "32711",
"split": "train"
} |
math | [
{
"content": "Given acute angles $\\alpha$ and $\\beta$ such that $\\sin \\alpha= \\frac { \\sqrt {5}}{5}$ and $\\sin(\\alpha-\\beta)=- \\frac { \\sqrt {10}}{10}$, determine the value of $\\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "32712",
"split": "train"
} |
math | [
{
"content": "Given that Ron has eight sticks with integer lengths, and he is unable to form a triangle using any three of these sticks as side lengths, determine the shortest possible length of the longest of the eight sticks. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "32713",
"split": "train"
} |
math | [
{
"content": "Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "32714",
"split": "train"
} |
math | [
{
"content": "A bamboo pole of length 24 meters is leaning against a wall with one end on the ground. If the vertical distance from a certain point on the pole to the wall and the vertical distance from that point to the ground are both 7 meters, find the vertical distance from the end of the pole touching the ... | {
"ground_truth": "16 - 4\\sqrt{2}",
"style": "rule"
} | {
"index": "32715",
"split": "train"
} |
math | [
{
"content": "If \\( a + x^2 = 2015 \\), \\( b + x^2 = 2016 \\), \\( c + x^2 = 2017 \\), and \\( abc = 24 \\), find the value of \\( \\frac{a}{bc} + \\frac{b}{ac} + \\frac{c}{ab} - \\frac{1}{a} - \\frac{1}{b} - \\frac{1}{c} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": "32716",
"split": "train"
} |
math | [
{
"content": "Given fifty distinct natural numbers, twenty-five of which do not exceed 50, and the remaining are greater than 50 but do not exceed 100. Additionally, no two of these numbers differ by exactly 50. Find the sum of these numbers. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "2525",
"style": "rule"
} | {
"index": "32717",
"split": "train"
} |
math | [
{
"content": "How many positive four-digit integers of the form $\\_\\_50$ are divisible by 50? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "32718",
"split": "train"
} |
math | [
{
"content": "Given the geometric series $6 - \\frac{12}{5} + \\frac{36}{25} - \\dots$, calculate the limiting sum of this series as the number of terms increases without bound. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{30}{7}",
"style": "rule"
} | {
"index": "32719",
"split": "train"
} |
math | [
{
"content": "A class has $25$ students. The teacher wants to stock $N$ candies, hold the Olympics and give away all $N$ candies for success in it (those who solve equally tasks should get equally, those who solve less get less, including, possibly, zero candies). At what smallest $N$ this will be poss... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "32720",
"split": "train"
} |
math | [
{
"content": "Suppose an amoeba is placed in a container one day, and on that day it splits into three amoebas. Each subsequent day, every surviving amoeba splits into three new amoebas. However, at the end of every second day starting from day two, only half of the amoebas survive. How many amoebas are in the ... | {
"ground_truth": "243",
"style": "rule"
} | {
"index": "32721",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression \\[ (a^2+b)^2 - (a^2-b)^2, \\]if $a=4$ and $b=1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "32722",
"split": "train"
} |
math | [
{
"content": "Let \\( S = \\{1, 2, 3, 4, \\ldots, 50\\} \\). A 3-element subset \\(\\{a, b, c\\}\\) of \\(S\\) is said to be good if \\(a + b + c\\) is divisible by 3. Determine the number of 3-element subsets of \\(S\\) which are good. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6544",
"style": "rule"
} | {
"index": "32723",
"split": "train"
} |
math | [
{
"content": "The graph of the function $y=\\sin (2x+\\varphi) (0 < \\varphi < \\pi)$ is shifted to the right by $\\frac{\\pi}{8}$ and then is symmetric about the $y$-axis. Determine the possible value(s) of $\\varphi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "32724",
"split": "train"
} |
math | [
{
"content": "There are 5 people standing in a row. If person A is not at the left end and not adjacent to person B, determine the number of possible arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "32725",
"split": "train"
} |
math | [
{
"content": "Given the parabola $C: y^2 = 16x$ with the focus $F$, and the line $l: x = -1$, if a point $A$ lies on $l$ and the line segment $AF$ intersects the parabola $C$ at point $B$ such that $\\overrightarrow{FA} = 5\\overrightarrow{FB}$, then find the length of $|AB|$. Please output the final answer wit... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "32726",
"split": "train"
} |
math | [
{
"content": "A digital clock shows the time $4:56$. How many minutes will pass until the clock next shows a time in which all of the digits are consecutive and are in increasing order? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "458",
"style": "rule"
} | {
"index": "32727",
"split": "train"
} |
math | [
{
"content": "The perimeter of a particular square and the circumference of a particular circle are equal. What is the ratio of the area of the square to the area of the circle? Express your answer as a common fraction in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "\\frac{\\pi}{4}",
"style": "rule"
} | {
"index": "32728",
"split": "train"
} |
math | [
{
"content": "Evaluate or simplify: \n1. $\\log_2 \\sqrt{2}+\\log_9 27+3^{\\log_3 16}$; \n2. $0.25^{-2}+\\left( \\frac{8}{27} \\right)^{-\\frac{1}{3}}-\\frac{1}{2}\\lg 16-2\\lg 5+\\left( \\frac{1}{2} \\right)^{0}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{33}{2}",
"style": "rule"
} | {
"index": "32729",
"split": "train"
} |
math | [
{
"content": "Given that an ellipse and a hyperbola $(x^{2}-y^{2}=1)$ share the same foci and the eccentricity is $\\frac{\\sqrt{2}}{2}$.\n(I) Find the standard equation of the ellipse;\n(II) A line passing through point $P(0,1)$ intersects the ellipse at points $A$ and $B$. $O$ is the origin. If $\\overrightar... | {
"ground_truth": "\\frac{\\sqrt{126}}{8}",
"style": "rule"
} | {
"index": "32730",
"split": "train"
} |
math | [
{
"content": "Find the last two digits of \\(\\left[(\\sqrt{29}+\\sqrt{21})^{1984}\\right]\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "32731",
"split": "train"
} |
math | [
{
"content": "Determine the product of the solutions to the quadratic equation $49 = -2x^2 - 8x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{49}{2}",
"style": "rule"
} | {
"index": "32732",
"split": "train"
} |
math | [
{
"content": "The left and right foci of the hyperbola $C$ are respectively $F_1$ and $F_2$, and $F_2$ coincides with the focus of the parabola $y^2=4x$. Let the point $A$ be an intersection of the hyperbola $C$ with the parabola, and suppose that $\\triangle AF_{1}F_{2}$ is an isosceles triangle with $AF_{1}$ ... | {
"ground_truth": "\\sqrt{2} + 1",
"style": "rule"
} | {
"index": "32733",
"split": "train"
} |
math | [
{
"content": "An entrepreneur invested $\\$20,\\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $\\$22,\\!446.40$. If... | {
"ground_truth": "7.840",
"style": "rule"
} | {
"index": "32734",
"split": "train"
} |
math | [
{
"content": "While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$ , there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$ . Given that $|a-b| = 2\\sqrt{3}$ , $|a| = 3\\sqrt... | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "32735",
"split": "train"
} |
math | [
{
"content": "Fill in the appropriate numbers in the parentheses:\n\n(1) 7÷9= $$\\frac {( )}{( )}$$\n\n(2) $$\\frac {12}{7}$$=\\_\\_\\_\\_÷\\_\\_\\_\\_\\_\n\n(3) 3 $$\\frac {5}{8}$$= $$\\frac {( )}{( )}$$\n\n(4) 6= $$\\frac {()}{11}$$ Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac {66}{11}",
"style": "rule"
} | {
"index": "32736",
"split": "train"
} |
math | [
{
"content": "School A and School B each have 3 teachers signed up for a teaching support program, with School A having 2 males and 1 female, and School B having 1 male and 2 females. \n(1) If one teacher is randomly selected from each of the schools, list all possible outcomes and calculate the probability th... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "32737",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$ and $CA$ are, respectively, $x+75^{\\circ} , 2x+25^{\\circ},3x-22^{\\circ}$. Then one interior angle of the triangle is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$61^{\\circ}$",
"style": "rule"
} | {
"index": "32738",
"split": "train"
} |
math | [
{
"content": "(1) In triangle $ABC$, the sides opposite angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively. If $2b\\cos B = a\\cos C + c\\cos A$, find $B$.\n\n(2) In an arithmetic sequence $\\{a_n\\}$, if $a_1 + a_7 + a_{13} = 6$, then find $S_{13}$, the sum of the first 13 terms.\n\n(3) In triangle... | {
"ground_truth": "\\frac{9}{4}",
"style": "rule"
} | {
"index": "32739",
"split": "train"
} |
math | [
{
"content": "In Zuminglish, words consist only of the letters $M, O,$ and $P$. $O$ is a vowel, while $M$ and $P$ are consonants. A valid Zuminglish word must have at least two consonants between any two $O's$. Determine the number of valid 7-letter words in Zuminglish and find the remainder when this number is... | {
"ground_truth": "912",
"style": "rule"
} | {
"index": "32740",
"split": "train"
} |
math | [
{
"content": "What is the smallest digit that is never found in the units place of an even number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "32741",
"split": "train"
} |
math | [
{
"content": "Two boxes of candies have a total of 176 pieces. If 16 pieces are taken out from the second box and put into the first box, the number of pieces in the first box is 31 more than m times the number of pieces in the second box (m is an integer greater than 1). Then, determine the number of pieces or... | {
"ground_truth": "131",
"style": "rule"
} | {
"index": "32742",
"split": "train"
} |
math | [
{
"content": "If I have a $5\\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that no row and no column contains more than one pawn? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": "32743",
"split": "train"
} |
math | [
{
"content": "Let $\\alpha$ and $\\beta$ be real numbers. Find the minimum value of\n\\[(2 \\cos \\alpha + 5 \\sin \\beta - 8)^2 + (2 \\sin \\alpha + 5 \\cos \\beta - 15)^2.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "32744",
"split": "train"
} |
math | [
{
"content": "Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \\[ {BC\\over PD}+{CA\\over PE}+{AB\\over PF}. \\] Please output the final an... | {
"ground_truth": "",
"style": "rule"
} | {
"index": "32745",
"split": "train"
} |
math | [
{
"content": "Given a regular 2017-sided polygon \\( A_{1} A_{2} \\cdots A_{2017} \\) inscribed in a unit circle \\(\\odot O\\), choose any two distinct vertices \\( A_{i} \\) and \\( A_{j} \\). What is the probability that \\( \\overrightarrow{O A_{i}} \\cdot \\overrightarrow{O A_{j}} > \\frac{1}{2} \\)? Pleas... | {
"ground_truth": "2/3",
"style": "rule"
} | {
"index": "32746",
"split": "train"
} |
math | [
{
"content": "Evaluate $|(4\\sqrt{2}-4i)(\\sqrt{3}+3i)|$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32747",
"split": "train"
} |
math | [
{
"content": "Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(j,i)$ do not both ap... | {
"ground_truth": "761",
"style": "rule"
} | {
"index": "32748",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid, one base is \\(40 \\text{ cm}\\) and the other is \\(24 \\text{ cm}\\). The diagonals of this trapezoid are mutually perpendicular. Find its area. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "32749",
"split": "train"
} |
math | [
{
"content": "Real numbers $a$ and $b$ satisfy the equations $3^a=81^{b+2}$ and $125^b=5^{a-3}$. What is $ab$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "32750",
"split": "train"
} |
math | [
{
"content": "A circle of diameter $1$ is removed from a $2\\times 3$ rectangle, as shown. Which whole number is closest to the area of the shaded region?\n[asy]\nfill((0,0)--(0,2)--(3,2)--(3,0)--cycle,gray);\ndraw((0,0)--(0,2)--(3,2)--(3,0)--cycle,linewidth(1));\nfill(circle((1,5/4),1/2),white);\ndraw(circle((... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32751",
"split": "train"
} |
math | [
{
"content": "There are eight different symbols designed on $n\\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\\leq k\\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even... | {
"ground_truth": "255",
"style": "rule"
} | {
"index": "32752",
"split": "train"
} |
math | [
{
"content": "Three people, A, B, and C, are playing a game: each person chooses a real number from the interval $[0,1]$. The person whose chosen number is between the numbers chosen by the other two wins. A randomly chooses a number from the interval $[0,1]$, B randomly chooses a number from the interval $\\le... | {
"ground_truth": "$\\frac{13}{24}$",
"style": "rule"
} | {
"index": "32753",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a triangle and $\\Gamma$ the $A$ - exscribed circle whose center is $J$ . Let $D$ and $E$ be the touchpoints of $\\Gamma$ with the lines $AB$ and $AC$ , respectively. Let $S$ be the area of the quadrilateral $ADJE$ , Find the maximum value that $\\frac{S}{AJ^2}$ ... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "32754",
"split": "train"
} |
math | [
{
"content": "The Fort Worth Zoo has a number of two-legged birds and a number of four-legged mammals. On one visit to the zoo, Margie counted 200 heads and 522 legs. How many of the animals that Margie counted were two-legged birds? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "139",
"style": "rule"
} | {
"index": "32755",
"split": "train"
} |
math | [
{
"content": "Suppose that $\\sec y - \\tan y = \\frac{15}{8}$ and that $\\csc y - \\cot y = \\frac{p}{q},$ where $\\frac{p}{q}$ is in lowest terms. Find $p+q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32756",
"split": "train"
} |
math | [
{
"content": "Calculate the value of the following expressions: \n(1) $$\\sqrt[4]{(3-\\pi )^{4}}$$ + (0.008) $$^{- \\frac {1}{3}}$$ - (0.25) $$^{ \\frac {1}{2}}$$ × ($$\\frac {1}{ \\sqrt {2}}$$)$$^{-4}$$; \n(2) $\\log_{3}$$\\sqrt {27}$$ - $\\log_{3}$$\\sqrt {3}$$ - $\\lg{625}$ - $\\lg{4}$ + $\\ln(e^{2})$ - $$... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32757",
"split": "train"
} |
math | [
{
"content": "Let \\(a\\), \\(b\\), and \\(c\\) be nonnegative real numbers such that \\(a^2 + b^2 + c^2 = 1\\). Find the maximum value of\n\\[2ab \\sqrt{3} + 2ac.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "32758",
"split": "train"
} |
math | [
{
"content": "A game uses a deck of $n$ different cards, where $n$ is an integer and $n \\geq 6.$ The number of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find $n.$\n ~ pi_is_3.14 Please output the final answer within \\boxed{}."... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "32759",
"split": "train"
} |
math | [
{
"content": "Three fair coins are tossed once. For each head that results, one fair die is rolled. Determine the probability that the sum of the results of the die rolls is odd. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{16}",
"style": "rule"
} | {
"index": "32760",
"split": "train"
} |
math | [
{
"content": "Given positive numbers \\(a, b, c, x, y, z\\) satisfying the equations \\(cy + bz = a\\), \\(az + cx = b\\), \\(bx + ay = c\\), find the minimum value of the function\n\\[ f(x, y, z) = \\frac{x^{2}}{1+x} + \\frac{y^{2}}{1+y} + \\frac{z^{2}}{1+z}. \\] Please output the final answer within \\boxed{}... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32761",
"split": "train"
} |
math | [
{
"content": "What is the value of \\(6102 - 2016\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4086",
"style": "rule"
} | {
"index": "32762",
"split": "train"
} |
math | [
{
"content": "Given that the function \\( f(x) \\) is an even function and has a period of 4, if the equation \\( f(x) = 0 \\) has exactly one root, which is 1, in the interval \\([0,2]\\), what is the sum of all the roots of \\( f(x) = 0 \\) in the interval \\([0,17]\\)? Please output the final answer within \... | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "32763",
"split": "train"
} |
math | [
{
"content": "Determine the sum of the following infinite series:\n\n$$\n1-\\frac{1}{a}+\\frac{1}{a^{2}}-\\frac{1}{a^{3}}+\\frac{1}{a^{4}}-\\frac{1}{a^{5}}+\\ldots\n$$\n\nif \\( a \\) is a positive number greater than 1. What will the sum be as the value of \\( a \\) approaches 1? Please output the final answer... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32764",
"split": "train"
} |
math | [
{
"content": "$8[x]$ represents the greatest integer not exceeding the real number $x$. Then, $\\left[\\log _{2} 1\\right]+\\left[\\log _{2} 2\\right]+\\left[\\log _{2} 3\\right]+\\cdots+\\left[\\log _{2} 2012\\right]=$ $\\qquad$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18084",
"style": "rule"
} | {
"index": "32765",
"split": "train"
} |
math | [
{
"content": "A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "\\frac{\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "32766",
"split": "train"
} |
math | [
{
"content": "At his usual rate a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is: Please output the final answer within \\boxe... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "32767",
"split": "train"
} |
math | [
{
"content": "How many six-digit numbers exist that have three even and three odd digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "281250",
"style": "rule"
} | {
"index": "32768",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the three sides $a, b, c$ form an arithmetic sequence, and $\\angle A = 3 \\angle C$. Find $\\cos \\angle C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1 + \\sqrt{33}}{8}",
"style": "rule"
} | {
"index": "32769",
"split": "train"
} |
math | [
{
"content": "Bao was given $\\$1,\\!000$ for his birthday. He decided to invest the money in a bank account that earns $10\\%$ interest, compounded annually. In dollars, how much total interest will Bao have earned 3 years later? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "331",
"style": "rule"
} | {
"index": "32770",
"split": "train"
} |
math | [
{
"content": "The degree measure of the sum of the interior angles of a convex polygon with $n$ sides is 1800. What is the degree measure of the sum of the interior angles of a convex polygon with $n + 2$ sides? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2160",
"style": "rule"
} | {
"index": "32771",
"split": "train"
} |
math | [
{
"content": "Martians measure angles in clerts. There are $500$ clerts in a full circle. How many clerts are there in a right angle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "32772",
"split": "train"
} |
math | [
{
"content": "Let the function be $$f(x)= \\frac {2^{x}}{2^{x}+ \\sqrt {2}}$$. Calculate the sum of $f(-2016) + f(-2015) + \\ldots + f(0) + f(1) + \\ldots + f(2017)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2017",
"style": "rule"
} | {
"index": "32773",
"split": "train"
} |
math | [
{
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$, respectively, and it satisfies $(2b-a) \\cdot \\cos C = c \\cdot \\cos A$.\n$(1)$ Find the size of angle $C$;\n$(2)$ Let $y = -4\\sqrt{3}\\sin^2\\frac{A}{2} + 2\\sin(C-B)$, find the maximum value of $y$ an... | {
"ground_truth": "2-2 \\sqrt {3}",
"style": "rule"
} | {
"index": "32774",
"split": "train"
} |
math | [
{
"content": "For an integer $x \\geq 1$, let $p(x)$ be the least prime that does not divide $x$, and define $q(x)$ to be the product of all primes less than $p(x)$. In particular, $p(1) = 2.$ For $x$ having $p(x) = 2$, define $q(x) = 1$. Consider the sequence $x_0, x_1, x_2, \\ldots$ defined by $x_0 = 1$ and \... | {
"ground_truth": "142",
"style": "rule"
} | {
"index": "32775",
"split": "train"
} |
math | [
{
"content": "Compute the smallest positive integer $n$ for which $$0<\\sqrt[4]{n}-\\lfloor\\sqrt[4]{n}\\rfloor<\\frac{1}{2015}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4097",
"style": "rule"
} | {
"index": "32776",
"split": "train"
} |
math | [
{
"content": "Let $\\{a_n\\}$ be a geometric sequence, $\\{b_n\\}$ be an arithmetic sequence, and $b_1=0$, $c_n=a_n+b_n$. If $\\{c_n\\}$ is $1$, $1$, $2$, $\\ldots$, find the sum of the first $10$ terms of the sequence $\\{c_n\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "978",
"style": "rule"
} | {
"index": "32777",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $a,b$ such that $b^{619}$ divides $a^{1000}+1$ and $a^{619}$ divides $b^{1000}+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1, 1)",
"style": "rule"
} | {
"index": "32778",
"split": "train"
} |
math | [
{
"content": "We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "382",
"style": "rule"
} | {
"index": "32779",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{M} = \\begin{pmatrix} 2 & 0 \\\\ 1 & -3 \\end{pmatrix}.$ Find constants $a$ and $b$ so that\n\\[\\mathbf{M}^{-1} = a \\mathbf{M} + b \\mathbf{I}.\\]Enter the ordered pair $(a,b).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( \\frac{1}{6}, \\frac{1}{6} \\right)",
"style": "rule"
} | {
"index": "32780",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ that satisfy $|\\overrightarrow{a}| = \\sqrt{3}$, $|\\overrightarrow{b}| = 2$, and $|\\overrightarrow{a} + 2\\overrightarrow{b}| = \\sqrt{7}$.\n(I) Find $\\overrightarrow{a} \\cdot \\overrightarrow{b}$;\n(II) If vector $λ\\overrightarro... | {
"ground_truth": "\\frac{20}{9}",
"style": "rule"
} | {
"index": "32781",
"split": "train"
} |
math | [
{
"content": "Compute $9 \\cdot (-5) - (7 \\cdot -2) + (8 \\cdot -6)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-79",
"style": "rule"
} | {
"index": "32782",
"split": "train"
} |
math | [
{
"content": "Determine the number of possible values for \\( m \\) such that the lengths of the sides of a triangle are \\( \\ln 20 \\), \\( \\ln 60 \\), and \\( \\ln m \\), and the triangle has a positive area. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1196",
"style": "rule"
} | {
"index": "32783",
"split": "train"
} |
math | [
{
"content": "The figure below shows a ring made of six small sections which you are to paint on a wall. You have four paint colors available and you will paint each of the six sections a solid color. Find the number of ways you can choose to paint the sections if no two adjacent sections can be painted with th... | {
"ground_truth": "732",
"style": "rule"
} | {
"index": "32784",
"split": "train"
} |
math | [
{
"content": "The function $f$ is given by the table\n\\[\\begin{tabular}{|c||c|c|c|c|c|} \\hline x & 1 & 2 & 3 & 4 & 5 \\ \\hline f(x) & 4 & 1 & 3 & 5 & 2 \\ \\hline \\end{tabular}\\]\nIf $u_0=4$ and $u_{n+1} = f(u_n)$ for $n \\ge 0$, find $u_{2002}$ Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "32785",
"split": "train"
} |
math | [
{
"content": "Let $\\underline{xyz}$ represent the three-digit number with hundreds digit $x$ , tens digit $y$ , and units digit $z$ , and similarly let $\\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$ . How many three-digit numbers $\\underline{abc}$ , none o... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "32786",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number that starts with the digit five, which becomes four times smaller if this five is removed from the beginning of its decimal representation and appended to the end. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "512820",
"style": "rule"
} | {
"index": "32787",
"split": "train"
} |
math | [
{
"content": "The integers $1,2, \\ldots, 64$ are written in the squares of a $8 \\times 8$ chess board, such that for each $1 \\leq i<64$, the numbers $i$ and $i+1$ are in squares that share an edge. What is the largest possible sum that can appear along one of the diagonals? Please output the final answer wit... | {
"ground_truth": "432",
"style": "rule"
} | {
"index": "32788",
"split": "train"
} |
math | [
{
"content": "Three real numbers $a,b,$ and $c$ satisfy the equations $a+b+c=2$, $ab+ac+bc=-7$ and $abc=-14$. What is the largest of the three numbers? Express your answer in simplest radical form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "32789",
"split": "train"
} |
math | [
{
"content": "For integers $a$, $b$, $c$, and $d$, $(x^2+ax+b)(x^2+cx+d)=x^4+x^3-2x^2+17x-5$. What is the value of $a+b+c+d$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "32790",
"split": "train"
} |
math | [
{
"content": "Given the function $$f(x)= \\begin{cases} 2\\cos \\frac {\\pi }{3}x & x\\leq 2000 \\\\ x-100 & x>2000\\end{cases}$$, then $f[f(2010)]$ equals \\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32791",
"split": "train"
} |
math | [
{
"content": "The four positive integers $a,$ $b,$ $c,$ $d$ satisfy\n\\[a \\times b \\times c \\times d = 10!.\\]Find the smallest possible value of $a + b + c + d.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "175",
"style": "rule"
} | {
"index": "32792",
"split": "train"
} |
math | [
{
"content": "Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\\sqrt{89}$ units and diagonals that differ by 6 units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "32793",
"split": "train"
} |
math | [
{
"content": "Extend a rectangular pattern of 12 black and 18 white square tiles by attaching a border of white tiles around the rectangle. The original rectangle is 5x6 tiles and the border adds one tile to each side. Calculate the ratio of black tiles to white tiles in the extended pattern. Please output the ... | {
"ground_truth": "\\frac{12}{44}",
"style": "rule"
} | {
"index": "32794",
"split": "train"
} |
math | [
{
"content": "The sequence $\\{a\\_n\\}$ satisfies $a\\_1=1$, $a\\_2=3$, and $a_{n+2}=|a_{n+1}|-a_{n}$, where $n∈N^{*}$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence. Find $S_{100}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "32795",
"split": "train"
} |
math | [
{
"content": "Find the greatest four-digit number where all digits are distinct, and which is divisible by each of its digits. Zero cannot be used. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9864",
"style": "rule"
} | {
"index": "32796",
"split": "train"
} |
math | [
{
"content": "Find the minimum value, for \\(a, b > 0\\), of the expression\n\n\\[\n\\frac{|a + 3b - b(a + 9b)| + |3b - a + 3b(a - b)|}{\\sqrt{a^{2} + 9b^{2}}}\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "32797",
"split": "train"
} |
math | [
{
"content": "Four red beads, two white beads, and one green bead are placed in a line in random order. What is the probability that no two neighboring beads are the same color?\nA) $\\frac{1}{15}$\nB) $\\frac{2}{15}$\nC) $\\frac{1}{7}$\nD) $\\frac{1}{30}$\nE) $\\frac{1}{21}$ Please output the final answer with... | {
"ground_truth": "\\frac{2}{15}",
"style": "rule"
} | {
"index": "32798",
"split": "train"
} |
math | [
{
"content": "How many distinct triangles can be constructed by connecting three different vertices of a cube? (Two triangles are distinct if they have different locations in space.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "56",
"style": "rule"
} | {
"index": "32799",
"split": "train"
} |
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