data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "For each even positive integer $x$, let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square. Please outpu... | {
"ground_truth": "899",
"style": "rule"
} | {
"index": "32000",
"split": "train"
} |
math | [
{
"content": "Let $ABCD$ be an isosceles trapezoid with $AD=BC$ and $AB<CD.$ Suppose that the distances from $A$ to the lines $BC,CD,$ and $BD$ are $15,18,$ and $10,$ respectively. Let $K$ be the area of $ABCD.$ Find $\\sqrt2 \\cdot K.$\nDiagram\n[asy] /* Made by MRENTHUSIASM */ size(250); pair A, B, C, D, E, F... | {
"ground_truth": "567",
"style": "rule"
} | {
"index": "32001",
"split": "train"
} |
math | [
{
"content": "Given $\\tan 2\\alpha= \\frac {3}{4}$, $\\alpha\\in(- \\frac {\\pi}{2}, \\frac {\\pi}{2})$, $f(x)=\\sin (x+\\alpha)+\\sin (\\alpha-x)-2\\sin \\alpha$, and for any $x\\in\\mathbb{R}$, it always holds that $f(x)\\geqslant 0$, find the value of $\\sin (\\alpha- \\frac {\\pi}{4})$. Please output the f... | {
"ground_truth": "- \\frac {2 \\sqrt {5}}{5}",
"style": "rule"
} | {
"index": "32002",
"split": "train"
} |
math | [
{
"content": "A parallelogram has side lengths of 10, $12x-2$, $5y+5$, and $4$. What is the value of $x+y$?\n\n[asy]draw((0,0)--(24,0)--(30,20)--(6,20)--cycle);\nlabel(\"$12x-2$\",(15,0),S);\nlabel(\"10\",(3,10),W);\nlabel(\"$5y+5$\",(18,20),N);\nlabel(\"4\",(27,10),E);\n[/asy] Please output the final answer wi... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "32003",
"split": "train"
} |
math | [
{
"content": "When two fair dice are rolled once each, what is the probability that one of the upward-facing numbers is 2, given that the two numbers are not the same? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "32004",
"split": "train"
} |
math | [
{
"content": "A modified deck of cards contains 60 cards consisting of 15 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, Prince, and Princess) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, $\\clubsuit$). Hence, each suit now has 15 cards. Four suits are still divided into two colors wi... | {
"ground_truth": "\\frac{7}{20}",
"style": "rule"
} | {
"index": "32005",
"split": "train"
} |
math | [
{
"content": "Handy Aaron helped a neighbor $1 \\frac{1}{4}$ hours on Monday, $50$ minutes on Tuesday, from 8:20 to 10:45 on Wednesday morning, and a half-hour on Friday. He is paid $\\textdollar 3$ per hour. How much did he earn for the week? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\$15",
"style": "rule"
} | {
"index": "32006",
"split": "train"
} |
math | [
{
"content": "For any integer $n \\ge2$, we define $ A_n$ to be the number of positive integers $ m$ with the following property: the distance from $n$ to the nearest multiple of $m$ is equal to the distance from $n^3$ to the nearest multiple of $ m$. Find all integers $n \\ge 2 $ for which $ A_n$ is odd. (Note... | {
"ground_truth": "$\\boxed{n=(2k)^2}$",
"style": "rule"
} | {
"index": "32007",
"split": "train"
} |
math | [
{
"content": "To estimate the consumption of disposable wooden chopsticks, in 1999, a sample of 10 restaurants from a total of 600 high, medium, and low-grade restaurants in a certain county was taken. The daily consumption of disposable chopstick boxes in these restaurants was as follows: \n0.6, 3.7, 2.2, 1.5... | {
"ground_truth": "7260",
"style": "rule"
} | {
"index": "32008",
"split": "train"
} |
math | [
{
"content": "A lucky integer is a positive integer which is divisible by the sum of its digits. What is the least positive multiple of 6 that is not a lucky integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "114",
"style": "rule"
} | {
"index": "32009",
"split": "train"
} |
math | [
{
"content": "A basketball team consists of 12 players, including two pairs of twins, Alex and Brian, and Chloe and Diana. In how many ways can we choose a team of 5 players if no pair of twins can both be in the team lineup simultaneously? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "560",
"style": "rule"
} | {
"index": "32010",
"split": "train"
} |
math | [
{
"content": "$[x-(y-z)] - [(x-y) - z] = $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2z",
"style": "rule"
} | {
"index": "32011",
"split": "train"
} |
math | [
{
"content": "Complex numbers $p,$ $q,$ and $r$ are zeros of a polynomial $P(z) = z^3 + sz + t,$ and $|p|^2 + |q|^2 + |r|^2 = 325.$ The points corresponding to $p,$ $q,$ and $r$ in the complex plane form a right triangle with right angle at $q.$ Find the square of the hypotenuse, $h^2,$ of this triangle. Please... | {
"ground_truth": "487.5",
"style": "rule"
} | {
"index": "32012",
"split": "train"
} |
math | [
{
"content": "The total number of matches played in the 2006 World Cup competition can be calculated by summing the number of matches determined at each stage of the competition. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "32013",
"split": "train"
} |
math | [
{
"content": "Given positive integers $x$ and $y$ such that $x\\neq y$ and $\\frac{1}{x} + \\frac{1}{y} = \\frac{1}{18}$, what is the smallest possible value for $x + y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "32014",
"split": "train"
} |
math | [
{
"content": "Given $x \\gt -1$, $y \\gt 0$, and $x+2y=1$, find the minimum value of $\\frac{1}{x+1}+\\frac{1}{y}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3+2\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "32015",
"split": "train"
} |
math | [
{
"content": "On a board the following six vectors are written: \\((1,0,0), \\quad(-1,0,0), \\quad(0,1,0), \\quad(0,-1,0), \\quad(0,0,1), \\quad(0,0,-1)\\). Given two vectors \\(v\\) and \\(w\\) on the board, a move consists of erasing \\(v\\) and \\(w\\) and replacing them with \\(\\frac{1}{\\sqrt{2}}(v+w)\\) ... | {
"ground_truth": "2 \\sqrt{3}",
"style": "rule"
} | {
"index": "32016",
"split": "train"
} |
math | [
{
"content": "A line connects points $(2,1)$ and $(7,3)$ on a square that has vertices at $(2,1)$, $(7,1)$, $(7,6)$, and $(2,6)$. What fraction of the area of the square is above this line? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "32017",
"split": "train"
} |
math | [
{
"content": "The function $g$ is defined on the set of integers and satisfies \\[g(n)= \\begin{cases} n-4 & \\mbox{if }n\\ge 1010 \\\\ g(g(n+7)) & \\mbox{if }n<1010. \\end{cases}\\] Find $g(77)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1011",
"style": "rule"
} | {
"index": "32018",
"split": "train"
} |
math | [
{
"content": "If point P lies on the graph of the function $y=e^x$ and point Q lies on the graph of the function $y=\\ln x$, then the minimum distance between points P and Q is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "32019",
"split": "train"
} |
math | [
{
"content": "A truck delivered 4 bags of cement. They are stacked in the truck. A worker can carry one bag at a time either from the truck to the gate or from the gate to the shed. The worker can carry the bags in any order, each time taking the top bag, carrying it to the respective destination, and placing i... | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "32020",
"split": "train"
} |
math | [
{
"content": "The vertices of a $3 \\times 1 \\times 1$ rectangular prism are $A, B, C, D, E, F, G$, and $H$ so that $A E, B F$, $C G$, and $D H$ are edges of length 3. Point $I$ and point $J$ are on $A E$ so that $A I=I J=J E=1$. Similarly, points $K$ and $L$ are on $B F$ so that $B K=K L=L F=1$, points $M$ an... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "32021",
"split": "train"
} |
math | [
{
"content": "Two students, A and B, each select two courses from four elective courses. The probability that they share exactly one course in common is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "32022",
"split": "train"
} |
math | [
{
"content": "The teacher and two boys and two girls stand in a row for a photo, with the requirement that the two girls must stand together and the teacher cannot stand at either end. Calculate the number of different arrangements. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32023",
"split": "train"
} |
math | [
{
"content": "A fair six-sided die has faces numbered $1, 2, 3, 4, 5, 6$. The die is rolled four times, and the results are $a, b, c, d$. What is the probability that one of the numbers in the set $\\{a, a+b, a+b+c, a+b+c+d\\}$ is equal to 4? Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "$\\frac{343}{1296}$",
"style": "rule"
} | {
"index": "32024",
"split": "train"
} |
math | [
{
"content": "Find the remainder when $x^{2015} + 1$ is divided by $x^8 - x^6 + x^4 - x^2 + 1.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-x^5 + 1",
"style": "rule"
} | {
"index": "32025",
"split": "train"
} |
math | [
{
"content": "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "32026",
"split": "train"
} |
math | [
{
"content": "A wishing well is located at the point $(11,11)$ in the $xy$ -plane. Rachelle randomly selects an integer $y$ from the set $\\left\\{ 0, 1, \\dots, 10 \\right\\}$ . Then she randomly selects, with replacement, two integers $a,b$ from the set $\\left\\{ 1,2,\\dots,10 \\right\\}$ . The pr... | {
"ground_truth": "111",
"style": "rule"
} | {
"index": "32027",
"split": "train"
} |
math | [
{
"content": "If $a,b,c>0$, find the smallest possible value of\n\\[\\left\\lfloor{\\frac{a+b}{c}}\\right\\rfloor+\\left\\lfloor{\\frac{b+c}{a}}\\right\\rfloor+\\left\\lfloor{\\frac{c+a}{b}}\\right\\rfloor.\\](Note that $\\lfloor{x}\\rfloor$ denotes the greatest integer less than or equal to $x$.) Please output... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "32028",
"split": "train"
} |
math | [
{
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(6,0)$, $(6,2)$, and $(0,2)$. What is the probability that $x + y < 3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "32029",
"split": "train"
} |
math | [
{
"content": "In unit square $A B C D$, points $E, F, G$ are chosen on side $B C, C D, D A$ respectively such that $A E$ is perpendicular to $E F$ and $E F$ is perpendicular to $F G$. Given that $G A=\\frac{404}{1331}$, find all possible values of the length of $B E$. Please output the final answer within \\box... | {
"ground_truth": "\\frac{9}{11}",
"style": "rule"
} | {
"index": "32030",
"split": "train"
} |
math | [
{
"content": "Maximum difference in weights of two bags is achieved by taking the largest and smallest possible values of the two different brands. Given the weights of three brands of flour are $\\left(25\\pm 0.1\\right)kg$, $\\left(25\\pm 0.2\\right)kg$, and $\\left(25\\pm 0.3\\right)kg$, calculate the maximu... | {
"ground_truth": "0.6",
"style": "rule"
} | {
"index": "32031",
"split": "train"
} |
math | [
{
"content": "In the equation, $\\frac15+\\frac{5}{x}=\\frac{12}{x}+\\frac{1}{12}$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "32032",
"split": "train"
} |
math | [
{
"content": "Two people, A and B, take turns to draw candies from a bag. A starts by taking 1 candy, then B takes 2 candies, A takes 4 candies next, then B takes 8 candies, and so on. This continues. When the number of candies remaining in the bag is less than the number they are supposed to take, they take al... | {
"ground_truth": "260",
"style": "rule"
} | {
"index": "32033",
"split": "train"
} |
math | [
{
"content": "What is the smallest palindrome that is larger than 2015? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2112",
"style": "rule"
} | {
"index": "32034",
"split": "train"
} |
math | [
{
"content": "Richard is building a rectangular backyard from 360 feet of fencing. The fencing must cover three sides of the backyard (the fourth side is bordered by Richard's house). What is the maximum area of this backyard? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16200",
"style": "rule"
} | {
"index": "32035",
"split": "train"
} |
math | [
{
"content": "The symbol $\\lfloor x \\rfloor$ denotes the largest integer not exceeding $x$. For example, $\\lfloor 3 \\rfloor = 3,$ and $\\lfloor 9/2 \\rfloor = 4.$ Compute \\[\\lfloor \\sqrt{1} \\rfloor + \\lfloor \\sqrt{2} \\rfloor + \\lfloor \\sqrt{3} \\rfloor + \\cdots + \\lfloor \\sqrt{16} \\rfloor.\\] P... | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "32036",
"split": "train"
} |
math | [
{
"content": "How many strictly positive integers less than or equal to 120 are not divisible by 3, 5, or 7? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "32037",
"split": "train"
} |
math | [
{
"content": "A square has an area of $25.$ A rectangle has the same width as the square. The length of the rectangle is double its width. What is the area of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "32038",
"split": "train"
} |
math | [
{
"content": "Four teams play in a tournament in which each team plays exactly one game against each of the other three teams. At the end of each game, either the two teams tie or one team wins and the other team loses. A team is awarded 3 points for a win, 0 points for a loss, and 1 point for a tie. If $S$ is ... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "32039",
"split": "train"
} |
math | [
{
"content": "How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "48",
"style": "rule"
} | {
"index": "32040",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=2\\sin x\\cos x-2\\cos ^{2}(x+\\frac{π}{4})$. \n$(1)$ Find the intervals where $f(x)$ is monotonically increasing and its center of symmetry. \n$(2)$ Given $x\\in (0,\\frac{π}{2})$, if $f(x+\\frac{π}{6})=\\frac{3}{5}$, find the value of $\\cos 2x$. Please output the final answer within... | {
"ground_truth": "\\frac{4\\sqrt{3}-3}{10}",
"style": "rule"
} | {
"index": "32041",
"split": "train"
} |
math | [
{
"content": "Given an integer coefficient polynomial\n$$\nf(x)=x^{5}+a_{1} x^{4}+a_{2} x^{3}+a_{3} x^{2}+a_{4} x+a_{5}\n$$\n\nIf $f(\\sqrt{3}+\\sqrt{2})=0$ and $f(1)+f(3)=0$, then $f(-1)=$ $\\qquad$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "32042",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$ satisfies $a_1=\\frac{1}{2}$, $a_{n+1}=1-\\frac{1}{a_n} (n\\in N^*)$, find the maximum positive integer $k$ such that $a_1+a_2+\\cdots +a_k < 100$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "199",
"style": "rule"
} | {
"index": "32043",
"split": "train"
} |
math | [
{
"content": "The base of a pyramid is a parallelogram with sides measuring 10 cm and 18 cm, and an area of 90 cm². The height of the pyramid passes through the intersection point of the diagonals of the base and is 6 cm. Determine the lateral surface area of the pyramid. Please output the final answer within \... | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "32044",
"split": "train"
} |
math | [
{
"content": "Let $\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\\omega+c\\omega^2,$ where $0\\leq a \\leq 1,0\\leq b\\leq 1,$ and $0\\leq c\\leq 1.$ What is the area of $S$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{3}{2}\\sqrt3",
"style": "rule"
} | {
"index": "32045",
"split": "train"
} |
math | [
{
"content": "Given that $sinα + cosα = \\frac{\\sqrt{2}}{3}, α ∈ (0, π)$, find the value of $sin(α + \\frac{π}{12})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{2} + \\sqrt{3}}{6}",
"style": "rule"
} | {
"index": "32046",
"split": "train"
} |
math | [
{
"content": "Given the polar equation of curve $C$ is $\\rho\\sin^2\\theta-8\\cos\\theta=0$, with the pole as the origin of the Cartesian coordinate system $xOy$, and the polar axis as the positive half-axis of $x$. In the Cartesian coordinate system, a line $l$ with an inclination angle $\\alpha$ passes throu... | {
"ground_truth": "16\\sqrt{2}",
"style": "rule"
} | {
"index": "32047",
"split": "train"
} |
math | [
{
"content": "Find the sum of $202_4 + 330_4 + 1000_4$. Express your answer first in base 4, then convert that sum to base 10. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "158",
"style": "rule"
} | {
"index": "32048",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $AB=2 \\cdot AC$. Let $D$ and $E$ be on $\\overline{AB}$ and $\\overline{BC}$, respectively, such that $\\angle BAE = \\angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose that $\\triangle CFE$ is equilateral. What is $\\angle ACB$? Please output the f... | {
"ground_truth": "90^{\\circ}",
"style": "rule"
} | {
"index": "32049",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the product of the first 10 primes, and let $$S=\\sum_{x y \\mid n} \\varphi(x) \\cdot y$$ where $\\varphi(x)$ denotes the number of positive integers less than or equal to $x$ that are relatively prime to $x$, and the sum is taken over ordered pairs $(x, y)$ of positive integers for wh... | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "32050",
"split": "train"
} |
math | [
{
"content": "Given that \\( a_{k} \\) is the number of integer terms in \\( \\log_{2} k, \\log_{3} k, \\cdots, \\log_{2018} k \\). Calculate \\( \\sum_{k=1}^{2018} a_{k} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4102",
"style": "rule"
} | {
"index": "32051",
"split": "train"
} |
math | [
{
"content": "Three couples dine at the same restaurant every Saturday at the same table. The table is round and the couples agreed that:\n(a) under no circumstances should husband and wife sit next to each other; and\n(b) the seating arrangement of the six people at the table must be different each Saturday.\n... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "32052",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle XYZ$ be a right triangle with $\\angle Y$ as the right angle. A circle with a diameter of $YZ$ intersects side $XZ$ at point $W$. Given that $XW = 2$ and $YW = 3$, find the length of $ZW$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "32053",
"split": "train"
} |
math | [
{
"content": "Let \\(ABC\\) be a triangle with circumradius \\(R = 17\\) and inradius \\(r = 7\\). Find the maximum possible value of \\(\\sin \\frac{A}{2}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{17 + \\sqrt{51}}{34}",
"style": "rule"
} | {
"index": "32054",
"split": "train"
} |
math | [
{
"content": "ABC is a triangle. D is the midpoint of AB, E is a point on the side BC such that BE = 2 EC and ∠ADC = ∠BAE. Find ∠BAC. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32055",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=13, A C=14$, and $B C=15$. Let $G$ be the point on $A C$ such that the reflection of $B G$ over the angle bisector of $\\angle B$ passes through the midpoint of $A C$. Let $Y$ be the midpoint of $G C$ and $X$ be a point on segment $A G$ such that $\\frac{A X}{X ... | {
"ground_truth": "\\frac{1170 \\sqrt{37}}{1379}",
"style": "rule"
} | {
"index": "32056",
"split": "train"
} |
math | [
{
"content": "Compute the multiplicative inverse of $201$ modulo $299$. Express your answer as an integer from $0$ to $298$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "180",
"style": "rule"
} | {
"index": "32057",
"split": "train"
} |
math | [
{
"content": "Given $a$, $b$, and $c$ are the three sides of $\\triangle ABC$, and $3a^2+3b^2-3c^2+2ab=0$, then $\\tan C= \\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2 \\sqrt {2}",
"style": "rule"
} | {
"index": "32058",
"split": "train"
} |
math | [
{
"content": "Larry now only likes numbers that end with two digits making the whole number divisible by 4. He has some favorite numbers like 120, 1156, and 504. How many different pairs of the last two digits are possible in numbers that Larry likes? Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "25",
"style": "rule"
} | {
"index": "32059",
"split": "train"
} |
math | [
{
"content": "Real numbers \\(a, b, c\\) and a positive number \\(\\lambda\\) satisfy \\(f(x) = x^3 + a x^2 + b x + c\\), which has 3 real roots \\(x_1, x_2, x_3\\), such that:\n(1) \\(x_2 - x_1 = \\lambda\\);\n(2) \\(x_3 > \\frac{1}{2}(x_1 + x_2)\\).\nFind the maximum value of \\(\\frac{2a^3 + 27c - 9ab}{\\lam... | {
"ground_truth": "\\frac{3 \\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "32060",
"split": "train"
} |
math | [
{
"content": "Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$ . Segments $AC$ and $BD$ both have length $5$ . Given that the length of $CD$ can be expressed as $\\frac{a}{b}$ where $a$ and $b$ are relativ... | {
"ground_truth": "132",
"style": "rule"
} | {
"index": "32061",
"split": "train"
} |
math | [
{
"content": "Find the last two digits (in order) of \\( 7^{30105} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "32062",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a point outside a circle $\\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\\Gamma$ . Let segment $PO$ intersect circle $\\Gamma$ at $C$ . A tangent to circle $\\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$... | {
"ground_truth": "12 \\sqrt{3}",
"style": "rule"
} | {
"index": "32063",
"split": "train"
} |
math | [
{
"content": "One day the Beverage Barn sold $252$ cans of soda to $100$ customers, and every customer bought at least one can of soda. What is the maximum possible median number of cans of soda bought per customer on that day? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.5",
"style": "rule"
} | {
"index": "32064",
"split": "train"
} |
math | [
{
"content": "My three-digit code is 023. Reckha can't choose a code that is the same as mine in two or more of the three digit-positions, nor that is the same as mine except for switching the positions of two digits (so 320 and 203, for example, are forbidden, but 302 is fine). Reckha can otherwise choose any ... | {
"ground_truth": "969",
"style": "rule"
} | {
"index": "32065",
"split": "train"
} |
math | [
{
"content": "In the diagram, each of the five boxes is to contain a number. Each number in a shaded box must be the average of the number in the box to the left of it and the number in the box to the right of it. Given the numbers:\n\n| 8 | | | 26 | $x$ |\n\nWhat is the value of $x$ ? Please output the final... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "32066",
"split": "train"
} |
math | [
{
"content": "Let $A$ and $B$ be two subsets of the set $\\{1,2, \\cdots, 20\\}$ such that $A \\cap B = \\varnothing$, and if $n \\in A$, then $2n + 2 \\in B$. Let $M(A)$ be the sum of the elements in $A$. Find the maximum value of $M(A)$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "32067",
"split": "train"
} |
math | [
{
"content": "In the spring round of the 2000 City Tournament, high school students in country $N$ were presented with six problems. Each problem was solved by exactly 1000 students, and no two students solved all six problems together. What is the smallest possible number of high school students in country $N$... | {
"ground_truth": "2000",
"style": "rule"
} | {
"index": "32068",
"split": "train"
} |
math | [
{
"content": "A triangle has vertices \\( P=(-7,4) \\), \\( Q=(-14,-20) \\), and \\( R=(2,-8) \\). The equation of the bisector of \\( \\angle P \\) can be written in the form \\( ax+2y+c=0 \\). Find \\( a+c \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "32069",
"split": "train"
} |
math | [
{
"content": "A cube with an edge length of 2 decimeters is first cut 4 times horizontally, then 5 times vertically. The total surface area of all the small pieces after cutting is ____ square decimeters. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "96",
"style": "rule"
} | {
"index": "32070",
"split": "train"
} |
math | [
{
"content": "Given that the total number of units produced by the workshops A, B, C, and D is 2800, and workshops A and C together contributed 60 units to the sample, determine the total number of units produced by workshops B and D. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1600",
"style": "rule"
} | {
"index": "32071",
"split": "train"
} |
math | [
{
"content": "Remove all positive integers that are divisible by 7, and arrange the remaining numbers in ascending order to form a sequence $\\{a_n\\}$. Calculate the value of $a_{100}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "116",
"style": "rule"
} | {
"index": "32072",
"split": "train"
} |
math | [
{
"content": "Given that the polar coordinate equation of circle $C$ is $ρ=2\\cos θ$, the parametric equation of line $l$ is $\\begin{cases}x= \\frac{1}{2}+ \\frac{ \\sqrt{3}}{2}t \\\\ y= \\frac{1}{2}+ \\frac{1}{2}t\\end{cases} (t\\text{ is a parameter})$, and the polar coordinates of point $A$ are $(\\frac{ \\... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "32073",
"split": "train"
} |
math | [
{
"content": "Compute: $$\\left( \\frac {8}{27}\\right)\\,^{- \\frac {2}{3}}-\\log {\\sqrt {2}}-\\log {\\sqrt {5}}=$$ \\_\\_\\_\\_\\_\\_ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{7}{4}",
"style": "rule"
} | {
"index": "32074",
"split": "train"
} |
math | [
{
"content": "Suppose that $X_1, X_2, \\dots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S = \\sum_{i=1}^k X_i/2^i$, where $k$ is the least positive integer such that $X_k < X_{k+1}$, or $k = \\infty$ if there is no such integer. Find the expected value of $S$. ... | {
"ground_truth": "2e^{1/2}-3",
"style": "rule"
} | {
"index": "32075",
"split": "train"
} |
math | [
{
"content": "The equation\n\\[(x - \\sqrt[3]{17})(x - \\sqrt[3]{67})(x - \\sqrt[3]{97}) = \\frac{1}{2}\\]\nhas three distinct solutions $u,$ $v,$ and $w.$ Calculate the value of $u^3 + v^3 + w^3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "181.5",
"style": "rule"
} | {
"index": "32076",
"split": "train"
} |
math | [
{
"content": "Given $\\angle1+\\angle2=180^\\circ$ and $\\angle3=\\angle4,$ find $\\angle4.$ Express your answer in degrees. [asy]\n/* AMC8 1997 #12 Problem */\npair A=(0,0), B=(24,0), C=(48,0), D=(18,24), E=(12,48);\npen p=1mm+black;\ndraw(A--C);\ndraw(A--E);\ndraw(B--E);\ndraw(D--C);\nlabel(\"70\", A, NE);\nl... | {
"ground_truth": "35^\\circ",
"style": "rule"
} | {
"index": "32077",
"split": "train"
} |
math | [
{
"content": "Given that the batch of rice contains 1512 bushels and a sample of 216 grains contains 27 grains of wheat, calculate the approximate amount of wheat mixed in this batch of rice. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "189",
"style": "rule"
} | {
"index": "32078",
"split": "train"
} |
math | [
{
"content": "Find the largest value of $n$ such that $5x^2 + nx + 90$ can be factored as the product of two linear factors with integer coefficients. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "451",
"style": "rule"
} | {
"index": "32079",
"split": "train"
} |
math | [
{
"content": "If $A:B:C = 2:1:4$, what is the value of $(3A + 2B) \\div (4C - A)$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{7}",
"style": "rule"
} | {
"index": "32080",
"split": "train"
} |
math | [
{
"content": "In a pot, there are 6 sesame-filled dumplings, 5 peanut-filled dumplings, and 4 red bean paste-filled dumplings. These three types of dumplings look exactly the same from the outside. If 4 dumplings are randomly selected from the pot, the probability that at least one dumpling of each type is sele... | {
"ground_truth": "\\dfrac{48}{91}",
"style": "rule"
} | {
"index": "32081",
"split": "train"
} |
math | [
{
"content": "Given rectangle ABCD, AB=4, BC=8, points K and L are midpoints of BC and AD, respectively, and point M is the midpoint of KL. What is the area of the quadrilateral formed by the rectangle diagonals and segments KM and LM. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "32082",
"split": "train"
} |
math | [
{
"content": "There are $2012$ backgammon checkers (stones, pieces) with one side is black and the other side is white.\nThese checkers are arranged into a line such that no two consequtive checkers are in same color. At each move, we are chosing two checkers. And we are turning upside down of the two checker... | {
"ground_truth": "1006",
"style": "rule"
} | {
"index": "32083",
"split": "train"
} |
math | [
{
"content": "Suppose $a$, $b$, $c$, and $d$ are integers satisfying the equations: $a-b+c=7$, $b-c+d=8$, $c-d+a=4$, and $d-a+b=1$. Determine the value of $2a+2b+2c+2d$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "32084",
"split": "train"
} |
math | [
{
"content": "$1.$ A bottle in the shape of a cone lies on its base. Water is poured into the bottle until its level reaches a distance of 8 centimeters from the vertex of the cone (measured vertically). We now turn the bottle upside down without changing the amount of water it contains; This leaves an empty sp... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "32085",
"split": "train"
} |
math | [
{
"content": "Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve *Daily Challenge* problems $D_1, D_2, D_3$ . However, he insists that carrot $C_i$ must be eaten only after solving *Daily Challenge* problem $D_i$ . In how many satisfactory orders can he complete all six ... | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "32086",
"split": "train"
} |
math | [
{
"content": "There are 2 teachers, 3 male students, and 4 female students taking a photo together. How many different standing arrangements are there under the following conditions? (Show the process, and represent the final result with numbers) \n(1) The male students must stand together; \n(2) The female s... | {
"ground_truth": "1728",
"style": "rule"
} | {
"index": "32087",
"split": "train"
} |
math | [
{
"content": "In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a + b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "667",
"style": "rule"
} | {
"index": "32088",
"split": "train"
} |
math | [
{
"content": "The fictional country of Isoland uses a 6-letter license plate system using the same 12-letter alphabet as the Rotokas of Papua New Guinea (A, E, G, I, K, O, P, R, T, U, V). Design a license plate that starts with a vowel (A, E, I, O, U), ends with a consonant (G, K, P, R, T, V), contains no repea... | {
"ground_truth": "151200",
"style": "rule"
} | {
"index": "32089",
"split": "train"
} |
math | [
{
"content": "Let \\( a, b, c \\) be non-negative real numbers satisfying\n\\[ a + b + c = 8 \\]\nand\n\\[ ab + bc + ca = 16.\\]\nLet \\( m = \\min \\{ ab, bc, ca \\} \\). Find the maximum possible value of \\( m \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "16/9",
"style": "rule"
} | {
"index": "32090",
"split": "train"
} |
math | [
{
"content": "On the lateral side \\( C D \\) of the trapezoid \\( A B C D \\) (\\( A D \\parallel B C \\)), a point \\( M \\) is marked. From the vertex \\( A \\), a perpendicular \\( A H \\) is dropped onto the segment \\( B M \\). It turns out that \\( A D = H D \\). Find the length of the segment \\( A D \\... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "32091",
"split": "train"
} |
math | [
{
"content": "Find one fourth of 12.8, expressed as a simplified improper fraction and also as a mixed number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 \\frac{1}{5}",
"style": "rule"
} | {
"index": "32092",
"split": "train"
} |
math | [
{
"content": "Evaluate the series $$\\frac{2^1}{8^1 - 1} + \\frac{2^2}{8^2 - 1} + \\frac{2^3}{8^3 - 1} + \\frac{2^4}{8^4 - 1} + \\cdots.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "32093",
"split": "train"
} |
math | [
{
"content": "When $\\{a,0,-1\\} = \\{4,b,0\\}$, find the values of $a$ and $b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "32094",
"split": "train"
} |
math | [
{
"content": "Let $P(x) = x^2 + ax + b$ be a quadratic polynomial. For how many pairs $(a, b)$ of positive integers where $a, b < 1000$ do the quadratics $P(x+1)$ and $P(x) + 1$ have at least one root in common? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "32095",
"split": "train"
} |
math | [
{
"content": "Mientka Publishing Company prices its bestseller Where's Walter? as follows:\n$C(n) = \\begin{cases} 12n, & \\text{if } 1 \\le n \\le 24 \\\\ 11n, & \\text{if } 25 \\le n \\le 48 \\\\ 10n, & \\text{if } 49 \\le n \\end{cases}$\nwhere $n$ is the number of books ordered, and $C(n)$ is the cost in do... | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "32096",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid with a perimeter of 8 and an area of 2, a circle can be inscribed. Find the distance from the point of intersection of the diagonals of the trapezoid to its shorter base. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2 - \\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "32097",
"split": "train"
} |
math | [
{
"content": "Xiao Ming has multiple 1 yuan, 2 yuan, and 5 yuan banknotes. He wants to buy a kite priced at 18 yuan using no more than 10 of these banknotes and must use at least two different denominations. How many different ways can he pay? Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "32098",
"split": "train"
} |
math | [
{
"content": "Let $r_{1}, r_{2}, r_{3}, r_{4}$ be the four roots of the polynomial $x^{4}-4 x^{3}+8 x^{2}-7 x+3$. Find the value of $\\frac{r_{1}^{2}}{r_{2}^{2}+r_{3}^{2}+r_{4}^{2}}+\\frac{r_{2}^{2}}{r_{1}^{2}+r_{3}^{2}+r_{4}^{2}}+\\frac{r_{3}^{2}}{r_{1}^{2}+r_{2}^{2}+r_{4}^{2}}+\\frac{r_{4}^{2}}{r_{1}^{2}+r_{2... | {
"ground_truth": "-4",
"style": "rule"
} | {
"index": "32099",
"split": "train"
} |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.