data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Let $P = (\\sqrt{2007}+\\sqrt{2008}),$ $Q = (-\\sqrt{2007}-\\sqrt{2008}),$ $R = (\\sqrt{2007}-\\sqrt{2008}),$ and $S = (-\\sqrt{2008}+\\sqrt{2007}).$ Calculate $PQRS.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2700",
"split": "train"
} |
math | [
{
"content": "The product of the digits of a 5 -digit number is 180 . How many such numbers exist? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "360",
"style": "rule"
} | {
"index": "2701",
"split": "train"
} |
math | [
{
"content": "A trapezoid \\(ABCD\\) is inscribed in a circle, with bases \\(AB = 1\\) and \\(DC = 2\\). Let \\(F\\) denote the intersection point of the diagonals of this trapezoid. Find the ratio of the sum of the areas of triangles \\(ABF\\) and \\(CDF\\) to the sum of the areas of triangles \\(AFD\\) and \\... | {
"ground_truth": "5/4",
"style": "rule"
} | {
"index": "2702",
"split": "train"
} |
math | [
{
"content": "Consider a machine that when a positive integer $N$ is entered, the machine's processing rule is:\n\n- If $N$ is odd, output $4N + 2$.\n- If $N$ is even, output $N / 2$.\n\nUsing the above rule, if starting with an input of $N = 9$, after following the machine's process for six times the output is... | {
"ground_truth": "640",
"style": "rule"
} | {
"index": "2703",
"split": "train"
} |
math | [
{
"content": "In an arithmetic sequence \\(\\{a_{n}\\}\\), if \\(\\frac{a_{11}}{a_{10}} < -1\\) and the sum of the first \\(n\\) terms \\(S_{n}\\) has a maximum value, then the value of \\(n\\) when \\(S_{n}\\) attains its smallest positive value is \\(\\qquad\\). Please output the final answer within \\boxed{}... | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "2704",
"split": "train"
} |
math | [
{
"content": "What is the smallest four-digit palindrome that is divisible by 4? (A palindrome is a number that reads the same forwards and backwards, like 61216.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2112",
"style": "rule"
} | {
"index": "2705",
"split": "train"
} |
math | [
{
"content": "Given the function \\( f(x) = x^3 + 3x^2 + 6x + 14 \\), and the conditions \\( f(a) = 1 \\) and \\( f(b) = 19 \\), find \\( a + b \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "2706",
"split": "train"
} |
math | [
{
"content": "A box contains 5 white balls and 6 black balls. Two balls are drawn out of the box at random. What is the probability that they both are white? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{2}{11}",
"style": "rule"
} | {
"index": "2707",
"split": "train"
} |
math | [
{
"content": "The mean of one set of five numbers is 13, and the mean of a separate set of six numbers is 24. What is the mean of the set of all eleven numbers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "19",
"style": "rule"
} | {
"index": "2708",
"split": "train"
} |
math | [
{
"content": "Among the natural numbers from 1 to 100, find the total number of numbers that contain a digit 7 or are multiples of 7. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "2709",
"split": "train"
} |
math | [
{
"content": "Compute the sum:\n\\[\\sum_{1 \\le a < b < c} \\frac{1}{3^a 5^b 7^c}.\\]\n(The sum is taken over all triples $(a,b,c)$ of positive integers such that $1 \\le a < b < c.$) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{21216}",
"style": "rule"
} | {
"index": "2710",
"split": "train"
} |
math | [
{
"content": "Given a circle with radius $4$, find the area of the region formed by all line segments of length $4$ that are tangent to this circle at their midpoints.\nA) $2\\pi$\nB) $4\\pi$\nC) $8\\pi$\nD) $16\\pi$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "2711",
"split": "train"
} |
math | [
{
"content": "Given that the coefficient of $x^2$ in the expansion of $(1+ax)(1+x)^5$ is $5$, determine the value of $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2712",
"split": "train"
} |
math | [
{
"content": "Given that $0 < \\alpha < \\frac{\\pi}{2}$ and $0 < \\beta < \\frac{\\pi}{2}$, if $\\sin\\left(\\frac{\\pi}{3}-\\alpha\\right) = \\frac{3}{5}$ and $\\cos\\left(\\frac{\\beta}{2} - \\frac{\\pi}{3}\\right) = \\frac{2\\sqrt{5}}{5}$, \n(I) find the value of $\\sin \\alpha$; \n(II) find the value of... | {
"ground_truth": "\\frac{11\\sqrt{5}}{25}",
"style": "rule"
} | {
"index": "2713",
"split": "train"
} |
math | [
{
"content": "The value of \\(987 + 113 - 1000\\) is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "2714",
"split": "train"
} |
math | [
{
"content": "A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting pieces can be used as connections to gas stoves, given that regulations require a stove to be at least 1 meter away from the main gas pipeline. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": "2715",
"split": "train"
} |
math | [
{
"content": "If a podcast series that lasts for 837 minutes needs to be stored on CDs and each CD can hold up to 75 minutes of audio, determine the number of minutes of audio that each CD will contain. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "69.75",
"style": "rule"
} | {
"index": "2716",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer with exactly 12 positive integer divisors? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "288",
"style": "rule"
} | {
"index": "2717",
"split": "train"
} |
math | [
{
"content": "In the center of a circular field, there is a geologists' house. Six straight roads radiate from it, dividing the field into six equal sectors. Two geologists set out on a journey from their house at a speed of 5 km/h along randomly chosen roads. Determine the probability that the distance between... | {
"ground_truth": "0.5",
"style": "rule"
} | {
"index": "2718",
"split": "train"
} |
math | [
{
"content": "What is the minimum number of planes determined by $6$ points in space which are not all coplanar, and among which no three are collinear? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "2719",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=(ax^{2}+bx+c)e^{x}$ $(a > 0)$, the derivative $y=f′(x)$ has two zeros at $-3$ and $0$. \n(Ⅰ) Determine the intervals of monotonicity for $f(x)$. \n(Ⅱ) If the minimum value of $f(x)$ is $-1$, find the maximum value of $f(x)$. Please output the final answer within \\boxed{... | {
"ground_truth": "\\dfrac {5}{e^{3}}",
"style": "rule"
} | {
"index": "2720",
"split": "train"
} |
math | [
{
"content": "A container with a capacity of 100 liters is filled with pure alcohol. After pouring out a portion of the alcohol, the container is filled with water. The mixture is then stirred thoroughly, and an amount of liquid equal to the first portion poured out is poured out again. The container is filled ... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "2721",
"split": "train"
} |
math | [
{
"content": "There is a prime number which is a factor of every sum of three consecutive integers. What is the number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "2722",
"split": "train"
} |
math | [
{
"content": "Let $A$, $M$, and $C$ be nonnegative integers such that $A+M+C=12$. What is the maximum value of \\[A\\cdot M\\cdot C+A\\cdot M+M\\cdot\nC+C\\cdot A?\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "112",
"style": "rule"
} | {
"index": "2723",
"split": "train"
} |
math | [
{
"content": "Let $x_1\\leq x_2\\leq \\cdots\\leq x_{100}$ be real numbers such that $|x_1| + |x_2| + \\cdots + |x_{100}| = 1$ and $x_1 + x_2 + \\cdots + x_{100} = 0$. Among all such $100$-tuples of numbers, the greatest value that $x_{76} - x_{16}$ can achieve is $\\tfrac mn$, where $m$ and $n$ are relatively ... | {
"ground_truth": "841",
"style": "rule"
} | {
"index": "2724",
"split": "train"
} |
math | [
{
"content": "For environmental protection, Wuyang Mineral Water recycles empty bottles. Consumers can exchange 4 empty bottles for 1 bottle of mineral water (if there are fewer than 4 empty bottles, they cannot be exchanged). Huacheng Middle School bought 1999 bottles of Wuyang brand mineral water. If they exc... | {
"ground_truth": "2345",
"style": "rule"
} | {
"index": "2725",
"split": "train"
} |
math | [
{
"content": "If $2^{11} \\times 6^{5}=4^{x} \\times 3^{y}$ for some positive integers $x$ and $y$, what is the value of $x+y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "2726",
"split": "train"
} |
math | [
{
"content": "What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 75 and 100 units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "2727",
"split": "train"
} |
math | [
{
"content": "Anne-Marie has a deck of 16 cards, each with a distinct positive factor of 2002 written on it. She shuffles the deck and begins to draw cards from the deck without replacement. She stops when there exists a nonempty subset of the cards in her hand whose numbers multiply to a perfect square. What i... | {
"ground_truth": "\\frac{837}{208}",
"style": "rule"
} | {
"index": "2728",
"split": "train"
} |
math | [
{
"content": "A cube with side length $2$ is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "2729",
"split": "train"
} |
math | [
{
"content": "The average age of 6 people in a room is 25 years. A 20-year-old person leaves the room and a new person aged 30 years enters the room. Find the new average age of the people in the room. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{80}{3}",
"style": "rule"
} | {
"index": "2730",
"split": "train"
} |
math | [
{
"content": "In a round-robin tournament with 6 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won. What is the maximum number of teams that could be tied for the m... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "2731",
"split": "train"
} |
math | [
{
"content": "The poetry lecture lasted 2 hours and $m$ minutes. The positions of the hour and minute hands on the clock at the end of the lecture are exactly swapped from their positions at the beginning of the lecture. If $[x]$ denotes the integer part of the decimal number $x$, find $[m]=$ $\\qquad$ . Please... | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "2732",
"split": "train"
} |
math | [
{
"content": "Let $T$ be the set of all rational numbers $r$, $0<r<1$, that have a repeating decimal expansion in the form $0.efghefgh\\ldots=0.\\overline{efgh}$, where the digits $e$, $f$, $g$, and $h$ are not necessarily distinct. To write the elements of $T$ as fractions in lowest terms, how many different n... | {
"ground_truth": "6000",
"style": "rule"
} | {
"index": "2733",
"split": "train"
} |
math | [
{
"content": "Given the function $f\\left( x \\right)=\\frac{{{e}^{x}}-a}{{{e}^{x}}+1}\\left( a\\in R \\right)$ defined on $R$ as an odd function.\n\n(1) Find the range of the function $y=f\\left( x \\right)$;\n\n(2) When ${{x}_{1}},{{x}_{2}}\\in \\left[ \\ln \\frac{1}{2},\\ln 2 \\right]$, the inequality $\\lef... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2734",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\left\\lceil-\\sqrt{\\frac{49}{4}}\\right\\rceil$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "2735",
"split": "train"
} |
math | [
{
"content": "Let $A B C D$ be a parallelogram with $A B=480, A D=200$, and $B D=625$. The angle bisector of $\\angle B A D$ meets side $C D$ at point $E$. Find $C E$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "280",
"style": "rule"
} | {
"index": "2736",
"split": "train"
} |
math | [
{
"content": "Find the number of ways that 2010 can be written as a sum of one or more positive integers in non-decreasing order such that the difference between the last term and the first term is at most 1. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2010",
"style": "rule"
} | {
"index": "2737",
"split": "train"
} |
math | [
{
"content": "Given that angle $\\alpha$ is in the fourth quadrant, and the x-coordinate of the point where its terminal side intersects the unit circle is $\\frac{1}{3}$.\n(1) Find the value of $\\tan \\alpha$;\n(2) Find the value of $\\frac{\\sin^{2}\\alpha - \\sqrt{2}\\sin \\alpha\\cos \\alpha}{1 + \\cos^{2}... | {
"ground_truth": "\\frac{6}{5}",
"style": "rule"
} | {
"index": "2738",
"split": "train"
} |
math | [
{
"content": "Let \\( a, b, c \\) be pairwise distinct positive integers such that \\( a+b, b+c \\) and \\( c+a \\) are all square numbers. Find the smallest possible value of \\( a+b+c \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "2739",
"split": "train"
} |
math | [
{
"content": "A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. What is the sum when every second term is added up, starting with the first term and ending with the second last term? Please output the final answer within \\boxed{}... | {
"ground_truth": "2151",
"style": "rule"
} | {
"index": "2740",
"split": "train"
} |
math | [
{
"content": "What is the sum of the prime numbers between 10 and 20? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "2741",
"split": "train"
} |
math | [
{
"content": "A child builds towers using identically shaped cubes of different colors. How many different towers with a height 8 cubes can the child build with 2 red cubes, 3 blue cubes, and 4 green cubes? (One cube will be left out.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1260",
"style": "rule"
} | {
"index": "2742",
"split": "train"
} |
math | [
{
"content": "Find the real root of the equation \\[\\sqrt{x} + \\sqrt{x+2} = 10.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2401}{100}",
"style": "rule"
} | {
"index": "2743",
"split": "train"
} |
math | [
{
"content": "When $\\frac{1}{2222}$ is expressed as a decimal, what is the sum of the first 60 digits after the decimal point? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "2744",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 1.$ Find the maximum value of $x + y^2 + z^3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "2745",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_{n}\\}$ with the sum of its first $n$ terms denoted as $A_{n}$, and $a_{1}+a_{2}=3$, $A_{5}=15$, a sequence $\\{b_{n}\\}$ satisfies ${b}_{n}={a}_{n}•[1+{(-1)}^{n}n]$ for $n\\in{N}^{*}$.<br/>$(1)$ Find the general formula for the sequence $\\{a_{n}\\}$;<br/>$(2)$ ... | {
"ground_truth": "2520",
"style": "rule"
} | {
"index": "2746",
"split": "train"
} |
math | [
{
"content": "Let $P(z)=x^3+ax^2+bx+c$, where $a,$ $b,$ and $c$ are real. There exists a complex number $w$ such that the three roots of $P(z)$ are $w+3i$, $w+9i$, and $2w-4$, where $i^2=-1$. Find $a+b+c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-136",
"style": "rule"
} | {
"index": "2747",
"split": "train"
} |
math | [
{
"content": "Find the angle of inclination of the tangent line to the curve $y= \\frac {1}{2}x^{2}-2x$ at the point $(1,- \\frac {3}{2})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\pi}{4}",
"style": "rule"
} | {
"index": "2748",
"split": "train"
} |
math | [
{
"content": "What is the sum of all positive integers $n$ such that $\\text{lcm}(2n, n^2) = 14n - 24$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "2749",
"split": "train"
} |
math | [
{
"content": "For every $3^\\circ$ rise in temperature, the volume of a certain gas expands by $4$ cubic centimeters. If the volume of the gas is $24$ cubic centimeters when the temperature is $32^\\circ$, what was the volume of the gas in cubic centimeters when the temperature was $20^\\circ$? Please output t... | {
"ground_truth": "8",
"style": "rule"
} | {
"index": "2750",
"split": "train"
} |
math | [
{
"content": "If $x$ is a real number, let $\\lfloor x \\rfloor$ be the greatest integer that is less than or equal to $x$ . If $n$ is a positive integer, let $S(n)$ be defined by\n\\[\n S(n) \n = \\left\\lfloor \\frac{n}{10^{\\lfloor \\log n \\rfloor}} \\right\\rfloor\n + 10 \\left( n - 10... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "2751",
"split": "train"
} |
math | [
{
"content": "Determine the probability that a point $P$ randomly selected within the plane region $N$ defined by the system of inequalities $\\begin{cases} 0 \\leq x \\leq 1 \\\\ 0 \\leq y \\leq e \\end{cases}$ also lies within the plane region $M$ defined by the system of inequalities $\\begin{cases} x + y \\... | {
"ground_truth": "1 - \\frac{3}{2e}",
"style": "rule"
} | {
"index": "2752",
"split": "train"
} |
math | [
{
"content": "If the graph of the linear function $y=-3x+m$ (where $m$ is a constant) passes through the second, third, and fourth quadrants, then the possible values of $m$ are _____. (Write down one possible value) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "2753",
"split": "train"
} |
math | [
{
"content": "Brent rolls a fair dodecahedral die with numbers $1,2,3,...,12$ on its faces. What is the expected number of digits in the number he obtains? Express your answer in a decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.25",
"style": "rule"
} | {
"index": "2754",
"split": "train"
} |
math | [
{
"content": "Let $\\mathcal{T}$ be the set of real numbers that can be represented as repeating decimals of the form $0.\\overline{ab}$ where $a$ and $b$ are distinct digits. Find the sum of the elements of $\\mathcal{T}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "413.5",
"style": "rule"
} | {
"index": "2755",
"split": "train"
} |
math | [
{
"content": "Square $EFGH$ has a side length of $40$. Point $Q$ lies inside the square such that $EQ = 15$ and $FQ = 34$. The centroids of $\\triangle{EFQ}$, $\\triangle{FGQ}$, $\\triangle{GHQ}$, and $\\triangle{HEQ}$ are the vertices of a convex quadrilateral. What is the area of that quadrilateral? Please ou... | {
"ground_truth": "\\frac{1600}{9}",
"style": "rule"
} | {
"index": "2756",
"split": "train"
} |
math | [
{
"content": "Harry Potter is creating an enhanced magical potion called \"Elixir of Life\" (this is a very potent sleeping potion composed of powdered daffodil root and wormwood infusion. The concentration of the \"Elixir of Life\" is the percentage of daffodil root powder in the entire potion). He first adds ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "2757",
"split": "train"
} |
math | [
{
"content": "An isosceles trapezoid \\(ABCD\\) is circumscribed around a circle. The lateral sides \\(AB\\) and \\(CD\\) are tangent to the circle at points \\(M\\) and \\(N\\), respectively, and \\(K\\) is the midpoint of \\(AD\\). In what ratio does the line \\(BK\\) divide the segment \\(MN\\)? Please outpu... | {
"ground_truth": "1:3",
"style": "rule"
} | {
"index": "2758",
"split": "train"
} |
math | [
{
"content": "A line with a slope of 2 and a line with a slope of -4 each have a $y$-intercept of 6. What is the distance between the $x$-intercepts of these lines? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{2}",
"style": "rule"
} | {
"index": "2759",
"split": "train"
} |
math | [
{
"content": "Suppose an integer $x$ , a natural number $n$ and a prime number $p$ satisfy the equation $7x^2-44x+12=p^n$ . Find the largest value of $p$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "47",
"style": "rule"
} | {
"index": "2760",
"split": "train"
} |
math | [
{
"content": "The simple quadrilateral \\(ABCD\\) has sides \\(AB\\), \\(BC\\), and \\(CD\\) with lengths 4, 5, and 20, respectively. If the angles \\(B\\) and \\(C\\) are obtuse, and \\(\\sin C = -\\cos B = \\frac{3}{5}\\), calculate the length of the side \\(AD\\). Please output the final answer within \\boxe... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "2761",
"split": "train"
} |
math | [
{
"content": "Wanda is trying to locate the Fermat point $P$ of $\\triangle ABC$, where $A$ is at the origin, $B$ is at $(10,0)$, and $C$ is at $(3,5)$ (the Fermat point is the point such that the sum of its distances from the vertices of a triangle is minimized). She guesses that the point is at $P = (4,2)$, a... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "2762",
"split": "train"
} |
math | [
{
"content": "Let $A=H_{1}, B=H_{6}+1$. A real number $x$ is chosen randomly and uniformly in the interval $[A, B]$. Find the probability that $x^{2}>x^{3}>x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "2763",
"split": "train"
} |
math | [
{
"content": "Elmo bakes cookies at a rate of one per 5 minutes. Big Bird bakes cookies at a rate of one per 6 minutes. Cookie Monster *consumes* cookies at a rate of one per 4 minutes. Together Elmo, Big Bird, Cookie Monster, and Oscar the Grouch produce cookies at a net rate of one per 8 minutes. How many min... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "2764",
"split": "train"
} |
math | [
{
"content": "Among the non-empty subsets of the set \\( A = \\{1, 2, \\cdots, 10\\} \\), how many subsets have the sum of their elements being a multiple of 10? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "103",
"style": "rule"
} | {
"index": "2765",
"split": "train"
} |
math | [
{
"content": "Congcong performs a math magic trick by writing the numbers $1, 2, 3, 4, 5, 6, 7$ on the blackboard and lets others select 5 of these numbers. The product of these 5 numbers is then calculated and told to Congcong, who guesses the chosen numbers. If when it is Benben's turn to select, Congcong can... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "2766",
"split": "train"
} |
math | [
{
"content": "Given \\\\(f(x)=2x^{5}+3x^{3}-2x^{2}+x-1\\\\), when calculating the value of the function at \\\\(x=2\\\\) using the Horner's method, find \\\\(v_{3}=\\\\) \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2767",
"split": "train"
} |
math | [
{
"content": "If $f (x) = x^2 - 1$, what is the value of $f (-1)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "2768",
"split": "train"
} |
math | [
{
"content": "Little Wang has three ballpoint pens of the same style but different colors. Each pen has a cap that matches its color. Normally, Wang keeps the pen and cap of the same color together, but sometimes he mixes and matches the pens and caps. If Wang randomly pairs the pens and caps, calculate the pro... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2769",
"split": "train"
} |
math | [
{
"content": "**p1.** Triangle $ABC$ has side lengths $AB = 3^2$ and $BC = 4^2$ . Given that $\\angle ABC$ is a right angle, determine the length of $AC$ .**p2.** Suppose $m$ and $n$ are integers such that $m^2+n^2 = 65$ . Find the largest possible value of $m-n$ .**p3.** Six middle school student... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "2770",
"split": "train"
} |
math | [
{
"content": "In the xy-plane with a rectangular coordinate system, the terminal sides of angles $\\alpha$ and $\\beta$ intersect the unit circle at points $A$ and $B$, respectively.\n1. If point $A$ is in the first quadrant with a horizontal coordinate of $\\frac{3}{5}$ and point $B$ has a vertical coordinate ... | {
"ground_truth": "\\frac{\\sqrt{63}}{8}",
"style": "rule"
} | {
"index": "2771",
"split": "train"
} |
math | [
{
"content": "What is the volume of the pyramid whose net is shown, if the base is a square with a side length of $1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{3}}{6}",
"style": "rule"
} | {
"index": "2772",
"split": "train"
} |
math | [
{
"content": "Determine the value of $x$ for which $(2010 + x)^2 = 4x^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-670",
"style": "rule"
} | {
"index": "2773",
"split": "train"
} |
math | [
{
"content": "The function defined on the set of real numbers, \\(f(x)\\), satisfies \\(f(x-1) = \\frac{1 + f(x+1)}{1 - f(x+1)}\\). Find the value of \\(f(1) \\cdot f(2) \\cdot f(3) \\cdots f(2008) + 2008\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2009",
"style": "rule"
} | {
"index": "2774",
"split": "train"
} |
math | [
{
"content": "Given that $F\\_1$ and $F\\_2$ are the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 (a > 0, b > 0)$, a line parallel to one of the hyperbola's asymptotes passes through $F\\_2$ and intersects the hyperbola at point $P$. If $|PF\\_1| = 3|PF\\_2|$, find the eccentric... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "2775",
"split": "train"
} |
math | [
{
"content": "A school has 100 students and 5 teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are 50, 20, 20, 5, and 5. Let be the average value obtained if a teacher is picked at random and the number of students in their ... | {
"ground_truth": "-13.5",
"style": "rule"
} | {
"index": "2776",
"split": "train"
} |
math | [
{
"content": "Given a finite arithmetic sequence \\( a_{1}, a_{2}, \\cdots a_{k} \\), and the conditions:\n$$\n\\begin{array}{c}\na_{4}+a_{7}+a_{10}=17, \\\\\na_{4}+a_{5}+a_{6}+a_{7}+a_{8}+a_{9}+a_{10}+a_{11}+a_{12}+a_{13}+a_{14}=77 .\n\\end{array}\n$$\nIf \\( a_{k}=13 \\), then \\( k \\) is Please output the f... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "2777",
"split": "train"
} |
math | [
{
"content": "Given $f\\left(\\alpha \\right)=\\frac{\\mathrm{sin}\\left(\\pi -\\alpha \\right)\\mathrm{cos}\\left(2\\pi -\\alpha \\right)\\mathrm{cos}\\left(-\\alpha +\\frac{3\\pi }{2}\\right)}{\\mathrm{cos}\\left(\\frac{\\pi }{2}-\\alpha \\right)\\mathrm{sin}\\left(-\\pi -\\alpha \\right)}$.\n\n(1) Simplify $... | {
"ground_truth": "\\frac{2\\sqrt{6}}{5}",
"style": "rule"
} | {
"index": "2778",
"split": "train"
} |
math | [
{
"content": "What is the least five-digit positive integer which is congruent to 7 (mod 12)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10,003",
"style": "rule"
} | {
"index": "2779",
"split": "train"
} |
math | [
{
"content": "A person starting with $64$ and making $6$ bets, wins three times and loses three times, the wins and losses occurring in random order. The chance for a win is equal to the chance for a loss. If each wager is for half the money remaining at the time of the bet, then the final result is: Please out... | {
"ground_truth": "$37$",
"style": "rule"
} | {
"index": "2780",
"split": "train"
} |
math | [
{
"content": "Given that the terminal side of angle \\\\(\\alpha\\\\) passes through the point \\\\(P(m,2\\sqrt{2})\\\\), \\\\(\\sin \\alpha= \\frac{2\\sqrt{2}}{3}\\\\) and \\\\(\\alpha\\\\) is in the second quadrant. \n\\\\((1)\\\\) Find the value of \\\\(m\\\\); \n\\\\((2)\\\\) If \\\\(\\tan \\beta= \\sqrt{... | {
"ground_truth": "\\frac{\\sqrt{2}}{11}",
"style": "rule"
} | {
"index": "2781",
"split": "train"
} |
math | [
{
"content": "How many two-digit positive integers have at least one 7 as a digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "2782",
"split": "train"
} |
math | [
{
"content": "In the diagram, three circles of radius 2 with centers $P$, $Q$, and $R$ are tangent to one another and to two sides of $\\triangle ABC$, as shown. Assume the centers $P$, $Q$, and $R$ form a right triangle, with $PQ$ as the hypotenuse.\n\nFind the perimeter of triangle $ABC$. Please output the fi... | {
"ground_truth": "8 + 4\\sqrt{2}",
"style": "rule"
} | {
"index": "2783",
"split": "train"
} |
math | [
{
"content": "What is the least five-digit positive integer which is congruent to 5 (mod 15)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10,\\!010",
"style": "rule"
} | {
"index": "2784",
"split": "train"
} |
math | [
{
"content": "How many three-digit numbers exist that are 5 times the product of their digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "175",
"style": "rule"
} | {
"index": "2785",
"split": "train"
} |
math | [
{
"content": "Given a right triangle \\( ABC \\) with hypotenuse \\( AB \\). One leg \\( AC = 15 \\) and the altitude from \\( C \\) to \\( AB \\) divides \\( AB \\) into segments \\( AH \\) and \\( HB \\) with \\( HB = 16 \\). What is the area of triangle \\( ABC \\)? Please output the final answer within \\bo... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "2786",
"split": "train"
} |
math | [
{
"content": "There are enough cuboids with side lengths of 2, 3, and 5. They are neatly arranged in the same direction to completely fill a cube with a side length of 90. The number of cuboids a diagonal of the cube passes through is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "2787",
"split": "train"
} |
math | [
{
"content": "If the line \\( x = \\frac{\\pi}{4} \\) intercepts the curve \\( C: (x - \\arcsin a)(x - \\arccos a) + (y - \\arcsin a)(y + \\arccos a) = 0 \\) at a chord of length \\( d \\), find the minimum value of \\( d \\) as \\( a \\) varies. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "2788",
"split": "train"
} |
math | [
{
"content": "Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$''. Moreover, both these two-digit primes are greater than 50 but less than 100. Find the total number of possible house numbers for Dr. Math. Please output the fina... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "2789",
"split": "train"
} |
math | [
{
"content": "The following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2)); for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (in... | {
"ground_truth": "608",
"style": "rule"
} | {
"index": "2790",
"split": "train"
} |
math | [
{
"content": "A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two equal angles as indi... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2791",
"split": "train"
} |
math | [
{
"content": "Find the smallest constant $N$, such that for any triangle with sides $a, b,$ and $c$, and perimeter $p = a + b + c$, the inequality holds:\n\\[\n\\frac{a^2 + b^2 + k}{c^2} > N\n\\]\nwhere $k$ is a constant. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2792",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, $C$ are respectively $a$, $b$, $c$. It is known that $2a\\cos A=c\\cos B+b\\cos C$.\n(Ⅰ) Find the value of $\\cos A$;\n(Ⅱ) If $a=1$ and $\\cos^2 \\frac{B}{2}+\\cos^2 \\frac{C}{2}=1+ \\frac{\\sqrt{3}}{4}$, find the value of side $c$. Please ... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "2793",
"split": "train"
} |
math | [
{
"content": "Each of the $100$ students in a certain summer camp can either sing, dance, or act. Some students have more than one talent, but no student has all three talents. There are $42$ students who cannot sing, $65$ students who cannot dance, and $29$ students who cannot act. How many students have two o... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "2794",
"split": "train"
} |
math | [
{
"content": "A charity sells $140$ benefit tickets for a total of $2001$. Some tickets sell for full price (a whole dollar amount), and the rest sells for half price. How much money is raised by the full-price tickets? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$782",
"style": "rule"
} | {
"index": "2795",
"split": "train"
} |
math | [
{
"content": "Alice and Bob are playing a game where Alice declares, \"My number is 36.\" Bob has to choose a number such that all the prime factors of Alice's number are also prime factors of his, but with the condition that the exponent of at least one prime factor in Bob's number is strictly greater than in ... | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "2796",
"split": "train"
} |
math | [
{
"content": "Given the ellipse C: $$\\frac {x^{2}}{4}+y^{2}=1,$$ \n(1) If a line $l$ passes through point Q $(1, \\frac {1}{2})$ and intersects the ellipse C at points A and B, find the equation of line $l$ such that Q is the midpoint of AB; \n(2) Given a fixed point M $(0, 2)$, and P is any point on the ell... | {
"ground_truth": "\\frac {2 \\sqrt {21}}{3}",
"style": "rule"
} | {
"index": "2797",
"split": "train"
} |
math | [
{
"content": "Positive real numbers $b \\not= 1$ and $n$ satisfy the equations \\[\\sqrt{\\log_b n} = \\log_b \\sqrt{n} \\qquad \\text{and} \\qquad b \\cdot \\log_b n = \\log_b (bn).\\] The value of $n$ is $\\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$ Please output the fi... | {
"ground_truth": "881",
"style": "rule"
} | {
"index": "2798",
"split": "train"
} |
math | [
{
"content": "Find the product 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": "38",
"style": "rule"
} | {
"index": "2799",
"split": "train"
} |
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