data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Given $sin( \\frac {\\pi}{6}-\\alpha)-cos\\alpha= \\frac {1}{3}$, find $cos(2\\alpha+ \\frac {\\pi}{3})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {7}{9}",
"style": "rule"
} | {
"index": "2800",
"split": "train"
} |
math | [
{
"content": "Given that vectors $a$ and $b$ satisfy $(2a+3b) \\perp b$, and $|b|=2\\sqrt{2}$, find the projection of vector $a$ onto the direction of $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3\\sqrt{2}",
"style": "rule"
} | {
"index": "2801",
"split": "train"
} |
math | [
{
"content": "Given that \\(9^{-1} \\equiv 90 \\pmod{101}\\), find \\(81^{-1} \\pmod{101}\\), as a residue modulo 101. (Give an answer between 0 and 100, inclusive.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2802",
"split": "train"
} |
math | [
{
"content": "Given that the line $y=kx+b$ is a common tangent to the curves $y=\\ln \\left(1+x\\right)$ and $y=2+\\ln x$, find the value of $k+b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3-\\ln 2",
"style": "rule"
} | {
"index": "2803",
"split": "train"
} |
math | [
{
"content": "Let the sequence $a_{i}$ be defined as $a_{i+1}=2^{a_{i}}$. Find the number of integers $1 \\leq n \\leq 1000$ such that if $a_{0}=n$, then 100 divides $a_{1000}-a_{1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "2804",
"split": "train"
} |
math | [
{
"content": "$AB$ and $AC$ are tangents to a circle, $\\angle BAC = 60^{\\circ}$, the length of the broken line $BAC$ is 1. Find the distance between the points of tangency $B$ and $C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "2805",
"split": "train"
} |
math | [
{
"content": "Given that \\( 990 \\times 991 \\times 992 \\times 993 = \\overline{966428 A 91 B 40} \\), find the values of \\( \\overline{A B} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "2806",
"split": "train"
} |
math | [
{
"content": "Six men and their wives are sitting at a round table with 12 seats. These men and women are very jealous - no man will allow his wife to sit next to any man except for himself, and no woman will allow her husband to sit next to any woman except for herself. In how many distinct ways can these 12 p... | {
"ground_truth": "288000",
"style": "rule"
} | {
"index": "2807",
"split": "train"
} |
math | [
{
"content": "To calculate $41^2$, Tom mentally computes $40^2$ and adds a number. To find $39^2$, Tom subtracts a number from $40^2$. What number does he add to calculate $41^2$ and subtract to calculate $39^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "2808",
"split": "train"
} |
math | [
{
"content": "For all $x \\in (0, +\\infty)$, the inequality $(2x - 2a + \\ln \\frac{x}{a})(-2x^{2} + ax + 5) \\leq 0$ always holds. Determine the range of values for the real number $a$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left\\{ \\sqrt{5} \\right\\}",
"style": "rule"
} | {
"index": "2809",
"split": "train"
} |
math | [
{
"content": "The midpoints of the sides of a regular hexagon $ABCDEF$ are joined to form a smaller hexagon. What fraction of the area of $ABCDEF$ is enclosed by the smaller hexagon? [asy]\nimport olympiad; import geometry; size(150); defaultpen(linewidth(0.8)); dotfactor=4;\npair[] bigHexagon = new pair[6];\n... | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "2810",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha$ is a root of the equation $5x^{2}-7x-6=0$, calculate the value of $\\frac {\\sin (-\\alpha- \\frac {3}{2}\\pi)\\cdot \\sin ( \\frac {3}{2}\\pi-\\alpha)\\cdot \\tan ^{2}(2\\pi-\\alpha)}{\\cos ( \\frac {\\pi}{2}-\\alpha)\\cdot \\cos ( \\frac {\\pi}{2}+\\alpha)\\cdot \\cos ... | {
"ground_truth": "\\frac{25}{16}",
"style": "rule"
} | {
"index": "2811",
"split": "train"
} |
math | [
{
"content": "The number of gallons of coffee a mathematician drinks on any given day is inversely proportional to how much sleep he gets the night before. On Monday, he got 9 hours of sleep and drank 2 gallons of coffee. On Tuesday, he got 6 hours of sleep. How many gallons of coffee did he drink? Please outpu... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "2812",
"split": "train"
} |
math | [
{
"content": "Suppose that $p$ and $q$ are positive numbers for which $\\log_{9}(p) = \\log_{12}(q) = \\log_{16}(p+q)$. What is the value of $\\frac{q}{p}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1+\\sqrt{5}}{2}",
"style": "rule"
} | {
"index": "2813",
"split": "train"
} |
math | [
{
"content": "Given the ellipse \\(3x^{2} + y^{2} = 6\\) and the point \\(P\\) with coordinates \\((1, \\sqrt{3})\\). Find the maximum area of triangle \\(PAB\\) formed by point \\(P\\) and two points \\(A\\) and \\(B\\) on the ellipse. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "2814",
"split": "train"
} |
math | [
{
"content": "The integer 2019 can be formed by placing two consecutive two-digit positive integers, 19 and 20, in decreasing order. What is the sum of all four-digit positive integers greater than 2019 that can be formed in this way? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "478661",
"style": "rule"
} | {
"index": "2815",
"split": "train"
} |
math | [
{
"content": "The base of the pyramid is an isosceles right triangle, where each leg measures 8. Each of the pyramid's lateral edges is 9. Find the volume of the pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "224/3",
"style": "rule"
} | {
"index": "2816",
"split": "train"
} |
math | [
{
"content": "Let the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ ($a<0$, $b<0$, $a \\neq b$) have one common root. Let the other two roots be $x_1$ and $x_2$.\n\n(1) Find the value of $x_1+x_2$.\n\n(2) Find the maximum value of $x_1x_2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "2817",
"split": "train"
} |
math | [
{
"content": "Given that one of the roots of the function $f(x)=ax+b$ is $2$, find the roots of the function $g(x)=bx^{2}-ax$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "2818",
"split": "train"
} |
math | [
{
"content": "Find all functions $f:\\mathbb{R}\\to\\mathbb{R}$ such that $f(x)f(y)+f(x+y)=xy$ for all real numbers $x$ and $y$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "f(x) = x - 1 \\text{ or } f(x) = -x - 1.",
"style": "rule"
} | {
"index": "2819",
"split": "train"
} |
math | [
{
"content": "Convert $645_{10}$ to base 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10400_5",
"style": "rule"
} | {
"index": "2820",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\left\\{a_{n}\\right\\}$ that satisfies\n$$\na_{n+1} = -\\frac{1}{2} a_{n} + \\frac{1}{3^{n}}\\quad (n \\in \\mathbf{Z}_{+}),\n$$\nfind all values of $a_{1}$ such that the sequence $\\left\\{a_{n}\\right\\}$ is monotonic, i.e., $\\left\\{a_{n}\\right\\}$ is either increasing o... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "2821",
"split": "train"
} |
math | [
{
"content": "Let $\\eta(m)$ be the product of all positive integers that divide $m$ , including $1$ and $m$ . If $\\eta(\\eta(\\eta(10))) = 10^n$ , compute $n$ .\n\n*Proposed by Kevin Sun* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "450",
"style": "rule"
} | {
"index": "2822",
"split": "train"
} |
math | [
{
"content": "A pyramid is formed on a $6\\times 8$ rectangular base. The four edges joining the apex to the corners of the rectangular base each have length $13$. What is the volume of the pyramid? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "2823",
"split": "train"
} |
math | [
{
"content": "Define a **valid sequence** as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediately followed by $D$,... | {
"ground_truth": "8748",
"style": "rule"
} | {
"index": "2824",
"split": "train"
} |
math | [
{
"content": "According to the chart shown, what was the average daily high temperature in Addington from September 15th, 2008 through September 19th, 2008, inclusive? Express your answer as a decimal to the nearest tenth. [asy]\nsize(300);\ndefaultpen(linewidth(.7pt)+fontsize(6pt));\n\nint[] highs={49,62,58,57... | {
"ground_truth": "54.4",
"style": "rule"
} | {
"index": "2825",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $\\frac{{m^2-4m+4}}{{m-1}}÷(\\frac{3}{{m-1}}-m-1)$, where $m=\\sqrt{3}-2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{-3+4\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "2826",
"split": "train"
} |
math | [
{
"content": "If $x, y, z$ are real numbers such that $xy=6, x-z=2$, and $x+y+z=9$, compute $\\frac{x}{y}-\\frac{z}{x}-\\frac{z^{2}}{xy}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "2827",
"split": "train"
} |
math | [
{
"content": "The line $y = 3$ intersects the graph of $y = 4x^2 + x - 1$ at the points $A$ and $B$. The distance between $A$ and $B$ can be written as $\\frac{\\sqrt{m}}{n}$, where $m$ and $n$ are positive integers that do not share any factors other than one. Find the value of $m - n$. Please output the final... | {
"ground_truth": "61",
"style": "rule"
} | {
"index": "2828",
"split": "train"
} |
math | [
{
"content": "The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$? Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "2829",
"split": "train"
} |
math | [
{
"content": "Convert the following radians to degrees and degrees to radians: \n(1) $$\\frac {\\pi}{12}$$ = \\_\\_\\_\\_\\_\\_ ; (2) $$\\frac {13\\pi}{6}$$ = \\_\\_\\_\\_\\_\\_ ; (3) -$$\\frac {5}{12}$$π = \\_\\_\\_\\_\\_\\_ . \n(4) 36° = \\_\\_\\_\\_\\_\\_ rad; (5) -105° = \\_\\_\\_\\_\\_\\_ rad. Please out... | {
"ground_truth": "- \\frac {7}{12}\\pi",
"style": "rule"
} | {
"index": "2830",
"split": "train"
} |
math | [
{
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$, and the sequence $\\{b_n\\}$ is a geometric sequence, satisfying $a_1=3$, $b_1=1$, $b_2+S_2=10$, and $a_5-2b_2=a_3$. The sum of the first $n$ terms of the sequence $\\left\\{ \\frac{a_n}{b_n} \\right\\}$ is $T_n$. If $T... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2831",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ such that $\\frac{n}{n+110}$ is equal to a terminating decimal? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "2832",
"split": "train"
} |
math | [
{
"content": "What is the largest value among $\\operatorname{lcm}[12,2],$ $\\operatorname{lcm}[12,4],$ $\\operatorname{lcm}[12,6],$ $\\operatorname{lcm}[12,8],$ $\\operatorname{lcm}[12,10],$ and $\\operatorname{lcm}[12,12]?$ Express your answer as an integer. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "2833",
"split": "train"
} |
math | [
{
"content": "Compute the smallest positive value of $x,$ in degrees, for which the function\n\\[f(x) = \\sin \\frac{x}{3} + \\sin \\frac{x}{11}\\]achieves its maximum value. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8910^\\circ",
"style": "rule"
} | {
"index": "2834",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin 2α = - \\frac{24}{25}$, and $α ∈ (\\frac{3π}{4},π)$, find the value of $\\sin α$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "2835",
"split": "train"
} |
math | [
{
"content": "Evaluate the expression $\\frac{\\sqrt{3}\\tan 12^\\circ - 3}{\\sin 12^\\circ (4\\cos^2 12^\\circ - 2)}=\\_\\_\\_\\_\\_\\_\\_\\_.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4\\sqrt{3}",
"style": "rule"
} | {
"index": "2836",
"split": "train"
} |
math | [
{
"content": "The function \\[f(x) = \\left\\{ \\begin{aligned} x-3 & \\quad \\text{ if } x < 5 \\\\ \\sqrt{x} & \\quad \\text{ if } x \\ge 5 \\end{aligned} \\right.\\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-6) + f^{-1}(-5) + \\dots + f^{-1}(5) + f^{-1}(6).$ Please output the final answer within \\b... | {
"ground_truth": "94",
"style": "rule"
} | {
"index": "2837",
"split": "train"
} |
math | [
{
"content": "Given $\\theta∈(0,\\frac{π}{2}$, $\\sin \\theta - \\cos \\theta = \\frac{\\sqrt{5}}{5}$, find the value of $\\tan 2\\theta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"index": "2838",
"split": "train"
} |
math | [
{
"content": "Given that $x = \\frac{5}{7}$ is a solution to the equation $56 x^2 + 27 = 89x - 8,$ what is the other value of $x$ that will solve the equation? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{8}",
"style": "rule"
} | {
"index": "2839",
"split": "train"
} |
math | [
{
"content": "In the Cartesian plane, let $A=(0,0), B=(200,100)$, and $C=(30,330)$. Compute the number of ordered pairs $(x, y)$ of integers so that $\\left(x+\\frac{1}{2}, y+\\frac{1}{2}\\right)$ is in the interior of triangle $A B C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31480",
"style": "rule"
} | {
"index": "2840",
"split": "train"
} |
math | [
{
"content": "How many five-character license plates can be made if the sequence consists of a consonant, followed by a vowel, followed by a consonant, followed by a digit, and finally any letter? (For this problem, consider Y a vowel.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "624,000",
"style": "rule"
} | {
"index": "2841",
"split": "train"
} |
math | [
{
"content": "Given that point P(x, y) satisfies the equation (x-4 cos θ)^{2} + (y-4 sin θ)^{2} = 4, where θ ∈ R, find the area of the region that point P occupies. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "32 \\pi",
"style": "rule"
} | {
"index": "2842",
"split": "train"
} |
math | [
{
"content": "Let $\\{x\\}$ denote the smallest integer not less than the real number \\(x\\). Find the value of the expression $\\left\\{\\log _{2} 1\\right\\}+\\left\\{\\log _{2} 2\\right\\}+\\left\\{\\log _{2} 3\\right\\}+\\cdots+\\left\\{\\log _{2} 1991\\right\\}$. Please output the final answer within \\bo... | {
"ground_truth": "19854",
"style": "rule"
} | {
"index": "2843",
"split": "train"
} |
math | [
{
"content": "Find a number such that when it is multiplied by its reverse, the product is 78445. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "145",
"style": "rule"
} | {
"index": "2844",
"split": "train"
} |
math | [
{
"content": "A semicircle and a circle each have a radius of 5 units. A square is inscribed in each. Calculate the ratio of the perimeter of the square inscribed in the semicircle to the perimeter of the square inscribed in the circle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{10}}{5}",
"style": "rule"
} | {
"index": "2845",
"split": "train"
} |
math | [
{
"content": "A two-digit positive integer $x$ has the property that when 109 is divided by $x$, the remainder is 4. What is the sum of all such two-digit positive integers $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "71",
"style": "rule"
} | {
"index": "2846",
"split": "train"
} |
math | [
{
"content": "There are three cards with numbers on both the front and back sides: one with 0 and 1, another with 2 and 3, and a third with 4 and 5. A student uses these cards to form a three-digit even number. How many different three-digit even numbers can the student make? Please output the final answer with... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "2847",
"split": "train"
} |
math | [
{
"content": "How many three-digit natural numbers are there such that the sum of their digits is equal to 24? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2848",
"split": "train"
} |
math | [
{
"content": "A pair of standard $6$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? Please output the final answer within \\bo... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "2849",
"split": "train"
} |
math | [
{
"content": "What is the smallest possible value of the sum $\\lvert x + 2\\rvert + \\lvert x + 4\\rvert + \\lvert x + 5\\rvert$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "2850",
"split": "train"
} |
math | [
{
"content": "The remainder when \\( 104^{2006} \\) is divided by 29 is ( ) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "2851",
"split": "train"
} |
math | [
{
"content": "Let $n$ be an integer of the form $a^{2}+b^{2}$, where $a$ and $b$ are relatively prime integers and such that if $p$ is a prime, $p \\leq \\sqrt{n}$, then $p$ divides $a b$. Determine all such $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "n = 2, 5, 13",
"style": "rule"
} | {
"index": "2852",
"split": "train"
} |
math | [
{
"content": "You are trapped in ancient Japan, and a giant enemy crab is approaching! You must defeat it by cutting off its two claws and six legs and attacking its weak point for massive damage. You cannot cut off any of its claws until you cut off at least three of its legs, and you cannot attack its weak po... | {
"ground_truth": "14400",
"style": "rule"
} | {
"index": "2853",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be integers such that $ab = 144.$ Find the minimum value of $a + b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-145",
"style": "rule"
} | {
"index": "2854",
"split": "train"
} |
math | [
{
"content": "Each of the letters $W$, $X$, $Y$, and $Z$ represents a different integer in the set $\\{ 1,2,3,4\\}$, but not necessarily in that order. If $\\frac{W}{X} - \\frac{Y}{Z}=1$, then the sum of $W$ and $Y$ is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "2855",
"split": "train"
} |
math | [
{
"content": "Five friends were comparing how much scrap iron they brought to the collection. On average, it was $55 \\mathrm{~kg}$, but Ivan brought only $43 \\mathrm{~kg}$.\n\nWhat is the average amount of iron brought without Ivan?\n\n(Note: By how many kilograms does Ivan's contribution differ from the aver... | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "2856",
"split": "train"
} |
math | [
{
"content": "Calculate the value of \\[\\cot(\\cot^{-1}5 + \\cot^{-1}11 + \\cot^{-1}17 + \\cot^{-1}23).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{97}{40}",
"style": "rule"
} | {
"index": "2857",
"split": "train"
} |
math | [
{
"content": "If $a$ is three times $b$, $b$ is twice $c$, and $c$ is four times $d$, what is the value of $\\frac{a\\cdot c}{b\\cdot d}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "2858",
"split": "train"
} |
math | [
{
"content": "Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to ru... | {
"ground_truth": "\\frac{3}{16}",
"style": "rule"
} | {
"index": "2859",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a\\_n\\}$, where $a\\_1=a\\_2=1$ and $a_{n+2}-a_{n}=1$, find the sum of the first 100 terms of the sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2550",
"style": "rule"
} | {
"index": "2860",
"split": "train"
} |
math | [
{
"content": "In a sequence, if for all $n \\in \\mathbb{N}^0$, we have $a_na_{n+1}a_{n+2} = k$ (where $k$ is a constant), then this sequence is called a geometric sequence, and $k$ is called the common product. Given that the sequence $a_n$ is a geometric sequence, and $a_1 = 1$, $a_2 = 2$, with a common produ... | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "2861",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be an isosceles trapezoid having parallel bases $\\overline{AB}$ and $\\overline{CD}$ with $AB>CD.$ Line segments from a point inside $ABCD$ to the vertices divide the trapezoid into four triangles whose areas are $2, 3, 4,$ and $5$ starting with the triangle with base $\\overline{CD}$ ... | {
"ground_truth": "2+\\sqrt{2}",
"style": "rule"
} | {
"index": "2862",
"split": "train"
} |
math | [
{
"content": "In triangle \\(ABC\\), the perpendicular bisectors of sides \\(AB\\) and \\(AC\\) are drawn, intersecting lines \\(AC\\) and \\(AB\\) at points \\(N\\) and \\(M\\) respectively. The length of segment \\(NM\\) is equal to the length of side \\(BC\\) of the triangle. The angle at vertex \\(C\\) of t... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "2863",
"split": "train"
} |
math | [
{
"content": "Given that \\( A \\) and \\( B \\) are two distinct points on the parabola \\( y = 3 - x^2 \\) that are symmetric with respect to the line \\( x + y = 0 \\), calculate the distance |AB|. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{2}",
"style": "rule"
} | {
"index": "2864",
"split": "train"
} |
math | [
{
"content": "Observe the following set of equations, and based on the equations above, guess that ${{S}_{2n-1}}=\\left( 4n-3 \\right)\\left( an+b \\right)$, then ${{a}^{2}}+{{b}^{2}}=$ ____.\n\n${{S}_{1}}=1$\n\n${{S}_{2}}=2+3+4=9$,\n\n${{S}_{3}}=3+4+5+6+7=25$,\n\n${{S}_{4}}=4+5+6+7+8+9+10=49$, (......) Please ... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "2865",
"split": "train"
} |
math | [
{
"content": "For some constants $a$ and $c,$\n\\[\\begin{pmatrix} a \\\\ -1 \\\\ c \\end{pmatrix} \\times \\begin{pmatrix} 7 \\\\ 3 \\\\ 5 \\end{pmatrix} = \\begin{pmatrix} -11 \\\\ -16 \\\\ 25 \\end{pmatrix}.\\]Enter the ordered pair $(a,c).$ Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "(6,2)",
"style": "rule"
} | {
"index": "2866",
"split": "train"
} |
math | [
{
"content": "For the quadratic equation in one variable $x$, $x^{2}+mx+n=0$ always has two real roots $x_{1}$ and $x_{2}$. \n$(1)$ When $n=3-m$ and both roots are negative, find the range of real number $m$. \n$(2)$ The inequality $t\\leqslant \\left(m-1\\right)^{2}+\\left(n-1\\right)^{2}+\\left(m-n\\right)^... | {
"ground_truth": "\\frac{9}{8}",
"style": "rule"
} | {
"index": "2867",
"split": "train"
} |
math | [
{
"content": "Distribute 5 students into 3 groups: Group A, Group B, and Group C, with Group A having at least two people, and Groups B and C having at least one person each, and calculate the number of different distribution schemes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "2868",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be the real roots of\n\\[x^4 - 4x - 1 = 0.\\]Find $ab + a + b.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "2869",
"split": "train"
} |
math | [
{
"content": "How many triangles can be formed using the vertices of a regular pentadecagon (a 15-sided polygon), if no side of the triangle can be a side of the pentadecagon? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "440",
"style": "rule"
} | {
"index": "2870",
"split": "train"
} |
math | [
{
"content": "In the addition shown, each of \\( P \\), \\( Q \\), and \\( R \\) is a digit.\n\n\\[\n\\begin{array}{r}\nP Q R \\\\\n+\\quad Q R \\\\\n\\hline 1012\n\\end{array}\n\\]\n\nWhat is the value of \\( P + Q + R \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "2871",
"split": "train"
} |
math | [
{
"content": "Let $a_1, a_2, \\dots$ be a sequence defined by $a_1 = a_2=1$ and $a_{n+2}=a_{n+1}+a_n$ for $n\\geq 1$. Find \\[\n\\sum_{n=1}^\\infty \\frac{a_n}{4^{n+1}}.\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{11}",
"style": "rule"
} | {
"index": "2872",
"split": "train"
} |
math | [
{
"content": "A square piece of paper is modified by cutting out one smaller right triangle from each corner, each having one leg along a side of the square facing outside. The remaining form is a rectangle. If each triangle has legs of lengths $4$ units and $3$ units, and the original square had a side length ... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "2873",
"split": "train"
} |
math | [
{
"content": "Compute $\\sum_{n=2009}^{\\infty} \\frac{1}{\\binom{n}{2009}}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2009}{2008}",
"style": "rule"
} | {
"index": "2874",
"split": "train"
} |
math | [
{
"content": "Find \\( g\\left(\\frac{1}{1996}\\right) + g\\left(\\frac{2}{1996}\\right) + g\\left(\\frac{3}{1996}\\right) + \\cdots + g\\left(\\frac{1995}{1996}\\right) \\) where \\( g(x) = \\frac{9x}{3 + 9x} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "997",
"style": "rule"
} | {
"index": "2875",
"split": "train"
} |
math | [
{
"content": "Find the total area of the region outside of an equilateral triangle but inside three circles each with radius 1, centered at the vertices of the triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2 \\pi-\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "2876",
"split": "train"
} |
math | [
{
"content": "Wei decides to modify the design of his logo by using a larger square and three tangent circles instead. Each circle remains tangent to two sides of the square and to one adjacent circle. If each side of the square is now 24 inches, calculate the number of square inches that will be shaded. Please... | {
"ground_truth": "576 - 108\\pi",
"style": "rule"
} | {
"index": "2877",
"split": "train"
} |
math | [
{
"content": "Two unit squares $S_{1}$ and $S_{2}$ have horizontal and vertical sides. Let $x$ be the minimum distance between a point in $S_{1}$ and a point in $S_{2}$, and let $y$ be the maximum distance between a point in $S_{1}$ and a point in $S_{2}$. Given that $x=5$, the difference between the maximum an... | {
"ground_truth": "472",
"style": "rule"
} | {
"index": "2878",
"split": "train"
} |
math | [
{
"content": "In the diagram, each of \\( \\triangle W X Z \\) and \\( \\triangle X Y Z \\) is an isosceles right-angled triangle. The length of \\( W X \\) is \\( 6 \\sqrt{2} \\). The perimeter of quadrilateral \\( W X Y Z \\) is closest to Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "23",
"style": "rule"
} | {
"index": "2879",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has side lengths $AB=120,BC=220$, and $AC=180$. Lines $\\ell_A,\\ell_B$, and $\\ell_C$ are drawn parallel to $\\overline{BC},\\overline{AC}$, and $\\overline{AB}$, respectively, such that the intersections of $\\ell_A,\\ell_B$, and $\\ell_C$ with the interior of $\\triangle ABC$ are... | {
"ground_truth": "715",
"style": "rule"
} | {
"index": "2880",
"split": "train"
} |
math | [
{
"content": "We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{7}{128}",
"style": "rule"
} | {
"index": "2881",
"split": "train"
} |
math | [
{
"content": "Point \\( M \\) divides the side \\( BC \\) of parallelogram \\( ABCD \\) in the ratio \\( BM : MC = 3 \\). Line \\( AM \\) intersects the diagonal \\( BD \\) at point \\( K \\). Find the area of the quadrilateral \\( CMKD \\) if the area of parallelogram \\( ABCD \\) is 1. Please output the final... | {
"ground_truth": "19/56",
"style": "rule"
} | {
"index": "2882",
"split": "train"
} |
math | [
{
"content": "The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\\circ}$ and $ SB \\equal{} SC \\equal{} 1$ . Find the volume of the tetrahedron. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": "2883",
"split": "train"
} |
math | [
{
"content": "Suppose the estimated €25 billion (Euros) cost to send a person to the planet Mars is shared equally by the 300 million people in a consortium of countries. Given the exchange rate of 1 Euro = 1.2 dollars, calculate each person's share in dollars. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "2884",
"split": "train"
} |
math | [
{
"content": "Let the sides opposite to the internal angles $A$, $B$, and $C$ of triangle $\\triangle ABC$ be $a$, $b$, and $c$ respectively. Given that $\\sin B - \\sin C = \\sin (A-C)$. \n$(1)$ Find the value of $A$. \n$(2)$ If $AB=2$, $AC=5$, and the medians $AM$ and $BN$ on sides $BC$ and $AC$ intersect a... | {
"ground_truth": "\\frac{4\\sqrt{91}}{91}",
"style": "rule"
} | {
"index": "2885",
"split": "train"
} |
math | [
{
"content": "A line of soldiers 1 mile long is jogging. The drill sergeant, in a car, moving at twice their speed, repeatedly drives from the back of the line to the front of the line and back again. When each soldier has marched 15 miles, how much mileage has been added to the car, to the nearest mile? Please... | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "2886",
"split": "train"
} |
math | [
{
"content": "How many distinct four-digit numbers are divisible by 5 and have 45 as their last two digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "2887",
"split": "train"
} |
math | [
{
"content": "Let $(1+x+x^2)^n=a_0 + a_1x+a_2x^2+ \\cdots + a_{2n}x^{2n}$ be an identity in $x$. If we let $s=a_0+a_2+a_4+\\cdots +a_{2n}$, then $s$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3^n+1}{2}",
"style": "rule"
} | {
"index": "2888",
"split": "train"
} |
math | [
{
"content": "A father and son were walking one after the other along a snow-covered road. The father's step length is $80 \\mathrm{~cm}$, and the son's step length is $60 \\mathrm{~cm}$. Their steps coincided 601 times, including at the very beginning and at the end of the journey. What distance did they trave... | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": "2889",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the sides corresponding to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $B= \\frac{\\pi}{4}$ and $\\cos A-\\cos 2A=0$.\n\n$(1)$ Find the angle $C$.\n\n$(2)$ If $b^{2}+c^{2}=a-bc+2$, find the area of $\\triangle ABC$. Please output th... | {
"ground_truth": "1- \\frac{ \\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "2890",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ is inscribed in a circle of radius 2 with $\\angle ABC \\geq 90^\\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "$x = -\\frac{1}{2} (\\sqrt6 \\pm \\sqrt 2)$",
"style": "rule"
} | {
"index": "2891",
"split": "train"
} |
math | [
{
"content": "Given the function $f$ mapping from set $M$ to set $N$, where $M=\\{a, b, c\\}$ and $N=\\{-3, -2, -1, 0, 1, 2, 3\\}$, calculate the number of mappings $f$ that satisfy the condition $f(a) + f(b) + f(c) = 0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "37",
"style": "rule"
} | {
"index": "2892",
"split": "train"
} |
math | [
{
"content": "In a village, a plot of land is shaped as a right triangle, where one of the legs measures 5 units and another measures 12 units. Farmer Euclid decides to leave a small unplanted square at the vertex of this right angle, and the shortest distance from this unplanted square to the hypotenuse is 3 u... | {
"ground_truth": "\\frac{412}{1000}",
"style": "rule"
} | {
"index": "2893",
"split": "train"
} |
math | [
{
"content": "Let $p,q,r$ be the roots of $x^3 - 6x^2 + 8x - 1 = 0$, and let $t = \\sqrt{p} + \\sqrt{q} + \\sqrt{r}$. Find $t^4 - 12t^2 - 8t$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "2894",
"split": "train"
} |
math | [
{
"content": "A triangle has sides of length $48$ , $55$ , and $73$ . A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relati... | {
"ground_truth": "200689",
"style": "rule"
} | {
"index": "2895",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\overrightarrow{m} \\cdot \\overrightarrow{n} - \\frac{1}{2}$, where $\\overrightarrow{m}=( \\sqrt{3}\\sin x,\\cos x)$ and $\\overrightarrow{n}=(\\cos x,-\\cos x)$.\n1. Find the range of the function $y=f(x)$ when $x\\in[0, \\frac{\\pi}{2}]$.\n2. In $\\triangle ABC$, the... | {
"ground_truth": "\\sqrt{7}",
"style": "rule"
} | {
"index": "2896",
"split": "train"
} |
math | [
{
"content": "There is a heads up coin on every integer of the number line. Lucky is initially standing on the zero point of the number line facing in the positive direction. Lucky performs the following procedure: he looks at the coin (or lack thereof) underneath him, and then, - If the coin is heads up, Lucky... | {
"ground_truth": "6098",
"style": "rule"
} | {
"index": "2897",
"split": "train"
} |
math | [
{
"content": "How many complex numbers $z$ such that $\\left| z \\right| < 30$ satisfy the equation\n\\[\n e^z = \\frac{z - 1}{z + 1} \\, ?\n\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "2898",
"split": "train"
} |
math | [
{
"content": "A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "2899",
"split": "train"
} |
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