data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "To popularize knowledge of fire safety, a certain school organized a competition on related knowledge. The competition is divided into two rounds, and each participant must participate in both rounds. If a participant wins in both rounds, they are considered to have won the competition. It is know... | {
"ground_truth": "\\frac{223}{300}",
"style": "rule"
} | {
"index": "37400",
"split": "train"
} |
math | [
{
"content": "What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "37401",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$, its sum of the first $n$ terms is $S_n$. It is known that $a_2=2$, $S_5=15$, and $b_n=\\frac{1}{a_{n+1}^2-1}$. Find the sum of the first 10 terms of the sequence $\\{b_n\\}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {175}{264}",
"style": "rule"
} | {
"index": "37402",
"split": "train"
} |
math | [
{
"content": "Let $A_{1} A_{2} \\ldots A_{19}$ be a regular nonadecagon. Lines $A_{1} A_{5}$ and $A_{3} A_{4}$ meet at $X$. Compute $\\angle A_{7} X A_{5}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1170^{\\circ}}{19}",
"style": "rule"
} | {
"index": "37403",
"split": "train"
} |
math | [
{
"content": "Consider five-dimensional Cartesian space $\\mathbb{R}^{5}=\\left\\{\\left(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\\right) \\mid x_{i} \\in \\mathbb{R}\\right\\}$ and consider the hyperplanes with the following equations: - $x_{i}=x_{j}$ for every $1 \\leq i<j \\leq 5$; - $x_{1}+x_{2}+x_{3}+x_{4}+x_{5}=... | {
"ground_truth": "480",
"style": "rule"
} | {
"index": "37404",
"split": "train"
} |
math | [
{
"content": "Calculate:<br/>$(1)2(\\sqrt{3}-\\sqrt{5})+3(\\sqrt{3}+\\sqrt{5})$;<br/>$(2)-{1}^{2}-|1-\\sqrt{3}|+\\sqrt[3]{8}-(-3)×\\sqrt{9}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11 - \\sqrt{3}",
"style": "rule"
} | {
"index": "37405",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \\frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).\n(1) Find the length of side c.\n(2) Find the value of sin(B - $\\frac{\\pi}{3}$). Please output the final answer wi... | {
"ground_truth": "\\frac{2\\sqrt{14} - 5\\sqrt{3}}{18}",
"style": "rule"
} | {
"index": "37406",
"split": "train"
} |
math | [
{
"content": "Given $\\cos \\left(\\alpha- \\frac {\\beta}{2}\\right)=- \\frac {1}{9}$ and $\\sin \\left( \\frac {\\alpha}{2}-\\beta\\right)= \\frac {2}{3}$, with $0 < \\beta < \\frac {\\pi}{2} < \\alpha < \\pi$, find $\\sin \\frac {\\alpha+\\beta}{2}=$ ______. Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac {22}{27}",
"style": "rule"
} | {
"index": "37407",
"split": "train"
} |
math | [
{
"content": "The sum of the digits in the product of $\\overline{A A A A A A A A A} \\times \\overline{B B B B B B B B B}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "37408",
"split": "train"
} |
math | [
{
"content": "Compute $\\arctan ( \\tan 65^\\circ - 2 \\tan 40^\\circ )$. (Express your answer in degrees as an angle between $0^\\circ$ and $180^\\circ$.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "25^\\circ",
"style": "rule"
} | {
"index": "37409",
"split": "train"
} |
math | [
{
"content": "(1) Given the hyperbola $C$: $\\dfrac{x^{2}}{a^{2}} - \\dfrac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$, its right vertex is $A$, and a circle $A$ with center $A$ and radius $b$ intersects one of the asymptotes of the hyperbola $C$ at points $M$ and $N$. If $\\angle MAN = 60^{\\circ}$, then the eccentri... | {
"ground_truth": "\\dfrac{4}{5}",
"style": "rule"
} | {
"index": "37410",
"split": "train"
} |
math | [
{
"content": "A bicycle trip is 30 km long. Ari rides at an average speed of 20 km/h. Bri rides at an average speed of 15 km/h. If Ari and Bri begin at the same time, how many minutes after Ari finishes the trip will Bri finish? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37411",
"split": "train"
} |
math | [
{
"content": "Find $x+y+z$ when $$ a_1x+a_2y+a_3z= a $$ $$ b_1x+b_2y+b_3z=b $$ $$ c_1x+c_2y+c_3z=c $$ Given that $$ a_1\\left(b_2c_3-b_3c_2\\right)-a_2\\left(b_1c_3-b_3c_1\\right)+a_3\\left(b_1c_2-b_2c_1\\right)=9 $$ $$ a\\left(b_2c_3-b_3c_2\\right)-a_2\\left(bc_3-b_3c\\right)+a_3\\left(bc_2-b_... | {
"ground_truth": "16/9",
"style": "rule"
} | {
"index": "37412",
"split": "train"
} |
math | [
{
"content": "For how many ordered pairs of positive integers $(a, b)$ such that $a \\le 50$ is it true that $x^2 - ax + b$ has integer roots? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "625",
"style": "rule"
} | {
"index": "37413",
"split": "train"
} |
math | [
{
"content": "The perimeter of a square with side length $x$ units is equal to the circumference of a circle with radius 2 units. What is the value of $x$? Express your answer as a decimal to the nearest hundredth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3.14",
"style": "rule"
} | {
"index": "37414",
"split": "train"
} |
math | [
{
"content": "A $7 \\times 7$ board is either empty or contains an invisible $2 \\times 2$ ship placed \"by the cells.\" You are allowed to place detectors in some cells of the board and then activate them all at once. An activated detector signals if its cell is occupied by the ship. What is the minimum number... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "37415",
"split": "train"
} |
math | [
{
"content": "Travel along the alley clockwise.\n\nIn 1 hour of walking, the pedestrian walked 6 kilometers and did not reach point $B$ (a whole $2 \\pi - 6$ km!), so the third option is clearly longer than the first and can be excluded.\n\nIn the first case, when moving along the alley, they would need to cove... | {
"ground_truth": "0.21",
"style": "rule"
} | {
"index": "37416",
"split": "train"
} |
math | [
{
"content": "Two ferries cross a river with constant speeds, turning at the shores without losing time. They start simultaneously from opposite shores and meet for the first time 700 feet from one shore. They continue to the shores, return, and meet for the second time 400 feet from the opposite shore. Determi... | {
"ground_truth": "1400",
"style": "rule"
} | {
"index": "37417",
"split": "train"
} |
math | [
{
"content": "A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10000",
"style": "rule"
} | {
"index": "37418",
"split": "train"
} |
math | [
{
"content": "Let \\( A, B, C \\) be points on the same plane with \\( \\angle ACB = 120^\\circ \\). There is a sequence of circles \\( \\omega_0, \\omega_1, \\omega_2, \\ldots \\) on the same plane (with corresponding radii \\( r_0, r_1, r_2, \\ldots \\) where \\( r_0 > r_1 > r_2 > \\cdots \\)) such that each ... | {
"ground_truth": "\\frac{3}{2} + \\sqrt{3}",
"style": "rule"
} | {
"index": "37419",
"split": "train"
} |
math | [
{
"content": "Place each of the digits 6, 7, 8 and 9 in exactly one square to make the smallest possible product. What is this product? [asy]draw((0,.5)--(10,.5),linewidth(1));\ndraw((4,1)--(6,1)--(6,3)--(4,3)--(4,1),linewidth(1));\ndraw((7,1)--(9,1)--(9,3)--(7,3)--(7,1),linewidth(1));\ndraw((7,4)--(9,4)--(9,6)... | {
"ground_truth": "5372",
"style": "rule"
} | {
"index": "37420",
"split": "train"
} |
math | [
{
"content": "A six place number is formed by repeating a three place number; for example, $256256$ or $678678$, etc. Any number of this form is always exactly divisible by: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1001",
"style": "rule"
} | {
"index": "37421",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[\\sin^2 4^\\circ + \\sin^2 8^\\circ + \\sin^2 12^\\circ + \\dots + \\sin^2 176^\\circ.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{45}{2}",
"style": "rule"
} | {
"index": "37422",
"split": "train"
} |
math | [
{
"content": "The least common multiple of $x$ and $y$ is $18$, and the least common multiple of $y$ and $z$ is $20$. Determine the least possible value of the least common multiple of $x$ and $z$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90",
"style": "rule"
} | {
"index": "37423",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$ . (Note: $n$ is written in the usual base ten notation.) Please output the final answer w... | {
"ground_truth": "9999",
"style": "rule"
} | {
"index": "37424",
"split": "train"
} |
math | [
{
"content": "Given the line $y=-x+1$ and the ellipse $\\frac{x^{2}}{a^{2}}+ \\frac{y^{2}}{b^{2}}=1(a > b > 0)$, they intersect at points $A$ and $B$. $OA \\perp OB$, where $O$ is the origin. If the eccentricity of the ellipse $e \\in [\\frac{1}{2}, \\frac{\\sqrt{3}}{2}]$, find the maximum value of $a$. Please ... | {
"ground_truth": "\\frac{\\sqrt{10}}{2}",
"style": "rule"
} | {
"index": "37425",
"split": "train"
} |
math | [
{
"content": "Let $\\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\\clubsuit(8)=8$ and $\\clubsuit(123)=1+2+3=6$. For how many two-digit values of $x$ is $\\clubsuit(\\clubsuit(x))=3$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "37426",
"split": "train"
} |
math | [
{
"content": "On the board we write a series of $n$ numbers, where $n \\geq 40$ , and each one of them is equal to either $1$ or $-1$ , such that the following conditions both hold:\n\n(i) The sum of every $40$ consecutive numbers is equal to $0$ .\n(ii) The sum of every $42$ consecutive numbers is n... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37427",
"split": "train"
} |
math | [
{
"content": "Alice wants to write down a list of prime numbers less than 100, using each of the digits 1, 2, 3, 4, and 5 once and no other digits. Which prime number must be in her list? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "41",
"style": "rule"
} | {
"index": "37428",
"split": "train"
} |
math | [
{
"content": "In a division problem, the dividend is 12, and the divisor is a natural number less than 12. What is the sum of all possible different remainders? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37429",
"split": "train"
} |
math | [
{
"content": "At the \"Economics and Law\" congress, a \"Best of the Best\" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each particip... | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "37430",
"split": "train"
} |
math | [
{
"content": "Given four one-inch squares are placed with their bases on a line. The second square from the left is lifted out and rotated 30 degrees before reinserting it such that it just touches the adjacent square on its right. Determine the distance in inches from point B, the highest point of the rotated ... | {
"ground_truth": "\\frac{2 + \\sqrt{3}}{4}",
"style": "rule"
} | {
"index": "37431",
"split": "train"
} |
math | [
{
"content": "Let $P(x) = b_0 + b_1x + \\dots + b_nx^n$ be a polynomial with integer coefficients, and $0 \\le b_i < 5$ for all $0 \\le i \\le n$.\n\nGiven that $P(\\sqrt{5}) = 40 + 31\\sqrt{5}$, compute $P(3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "381",
"style": "rule"
} | {
"index": "37432",
"split": "train"
} |
math | [
{
"content": "The length of a rectangle is three times its width. The perimeter is 160 cm. What is the number of square centimeters in the area of the rectangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1200",
"style": "rule"
} | {
"index": "37433",
"split": "train"
} |
math | [
{
"content": "Given a convex quadrilateral \\(ABCD\\) with \\(\\angle C = 57^{\\circ}\\), \\(\\sin \\angle A + \\sin \\angle B = \\sqrt{2}\\), and \\(\\cos \\angle A + \\cos \\angle B = 2 - \\sqrt{2}\\), find the measure of angle \\(D\\) in degrees. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "168",
"style": "rule"
} | {
"index": "37434",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, side $a = \\sqrt{3}$, side $b = \\sqrt{3}$, and side $c > 3$. Let $x$ be the largest number such that the magnitude, in degrees, of the angle opposite side $c$ exceeds $x$. Then $x$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120^{\\circ}",
"style": "rule"
} | {
"index": "37435",
"split": "train"
} |
math | [
{
"content": "From 50 products, 10 are selected for inspection. The total number of items is \\_\\_\\_\\_\\_\\_\\_, and the sample size is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "37436",
"split": "train"
} |
math | [
{
"content": "The measures of the three interior angles of a triangle are $50^\\circ$, $55^\\circ$ and $x^\\circ$. What is the degree measure of the largest interior angle of this triangle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "75^\\circ",
"style": "rule"
} | {
"index": "37437",
"split": "train"
} |
math | [
{
"content": "There is a unique two-digit positive integer $t$ for which the last two digits of $13 \\cdot t$ are $26$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "37438",
"split": "train"
} |
math | [
{
"content": "There are two red, two black, two white, and a positive but unknown number of blue socks in a drawer. It is empirically determined that if two socks are taken from the drawer without replacement, the probability they are of the same color is $\\frac{1}{5}$. How many blue socks are there in the dra... | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "37439",
"split": "train"
} |
math | [
{
"content": "Compute $\\arccos (\\sin 3)$. All functions are in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3 - \\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "37440",
"split": "train"
} |
math | [
{
"content": "Square \\(ABCD\\) has sides of length 14. A circle is drawn through \\(A\\) and \\(D\\) so that it is tangent to \\(BC\\). What is the radius of the circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8.75",
"style": "rule"
} | {
"index": "37441",
"split": "train"
} |
math | [
{
"content": "Four brothers have together forty-eight Kwanzas. If the first brother's money were increased by three Kwanzas, if the second brother's money were decreased by three Kwanzas, if the third brother's money were triplicated and if the last brother's money were reduced by a third, then all brothers wou... | {
"ground_truth": "6, 12, 3, 27",
"style": "rule"
} | {
"index": "37442",
"split": "train"
} |
math | [
{
"content": "For natural numbers \\\\(m\\\\) greater than or equal to \\\\(2\\\\) and their powers of \\\\(n\\\\), the following decomposition formula is given: \n\\\\(2^{2}=1+3\\\\) \\\\(3^{2}=1+3+5\\\\) \\\\(4^{2}=1+3+5+7\\\\) \\\\(…\\\\) \n\\\\(2^{3}=3+5\\\\) \\\\(3^{3}=7+9+11\\\\) \\\\(…\\\\) ... | {
"ground_truth": "125",
"style": "rule"
} | {
"index": "37443",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$, with sides of length $5$, $6$, and $7$, has one vertex on the positive $x$-axis, one on the positive $y$-axis, and one on the positive $z$-axis. Let $O$ be the origin. What is the volume of tetrahedron $OABC$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\sqrt{95}",
"style": "rule"
} | {
"index": "37444",
"split": "train"
} |
math | [
{
"content": "Let $S$ be the set of $10$-tuples of non-negative integers that have sum $2019$. For any tuple in $S$, if one of the numbers in the tuple is $\\geq 9$, then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\\in S$ we can get from $... | {
"ground_truth": "10^8",
"style": "rule"
} | {
"index": "37445",
"split": "train"
} |
math | [
{
"content": "Find all possible positive integers represented in decimal as $13 x y 45 z$, which are divisible by 792, where $x, y, z$ are unknown digits. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1380456",
"style": "rule"
} | {
"index": "37446",
"split": "train"
} |
math | [
{
"content": "A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a $2 / 3$ chance of catching each individual error still in the ... | {
"ground_truth": "\\frac{416}{729}",
"style": "rule"
} | {
"index": "37447",
"split": "train"
} |
math | [
{
"content": "Let \\( f(x) = x^2 + px + q \\). It is known that the inequality \\( |f(x)| > \\frac{1}{2} \\) has no solutions on the interval \\([3, 5]\\). Find \\(\\underbrace{f(f(\\ldots f}_{2017}\\left(\\frac{7+\\sqrt{15}}{2}\\right)) \\ldots)\\). Round the answer to hundredths if necessary. Please output th... | {
"ground_truth": "1.56",
"style": "rule"
} | {
"index": "37448",
"split": "train"
} |
math | [
{
"content": "Consider a cube PQRSTUVW with a side length s. Let M and N be the midpoints of edges PU and RW, and let K be the midpoint of QT. Find the ratio of the area of triangle MNK to the area of one of the faces of the cube. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "37449",
"split": "train"
} |
math | [
{
"content": "The shape of a bridge arch is a parabola. It is known that the width of the parabolic arch is 8 meters, and the area of the parabolic arch is 160 square meters. Then, the height of the parabolic arch is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37450",
"split": "train"
} |
math | [
{
"content": "What is the largest power of 2 by which the number \\(10^{10} - 2^{10}\\) is divisible? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "37451",
"split": "train"
} |
math | [
{
"content": "How many subsets $S$ of the set $\\{1,2, \\ldots, 10\\}$ satisfy the property that, for all $i \\in[1,9]$, either $i$ or $i+1$ (or both) is in $S$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "144",
"style": "rule"
} | {
"index": "37452",
"split": "train"
} |
math | [
{
"content": "Given the lines $l_{1}$: $x+ay-a+2=0$ and $l_{2}$: $2ax+(a+3)y+a-5=0$. \n$(1)$ When $a=1$, find the coordinates of the intersection point of lines $l_{1}$ and $l_{2}$. \n$(2)$ If $l_{1}$ is parallel to $l_{2}$, find the value of $a$. Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "a = \\frac{3}{2}",
"style": "rule"
} | {
"index": "37453",
"split": "train"
} |
math | [
{
"content": "Given that $-1 - 4\\sqrt{2}$ is a root of the equation \\[x^3 + ax^2 + bx + 31 = 0\\]and that $a$ and $b$ are rational numbers, compute $a.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "37454",
"split": "train"
} |
math | [
{
"content": "There are 3 rods with several golden disks of different sizes placed on them. Initially, 5 disks are arranged on the leftmost rod (A) in descending order of size. According to the rule that only one disk can be moved at a time and a larger disk can never be placed on top of a smaller one, the goal... | {
"ground_truth": "31",
"style": "rule"
} | {
"index": "37455",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a}$ and $\\mathbf{b}$ be two vectors such that\n\\[\\|\\mathbf{a} + \\mathbf{b}\\| = \\|\\mathbf{b}\\|.\\]Find the angle between the vectors $\\mathbf{a} + 2 \\mathbf{b}$ and $\\mathbf{a},$ in degrees Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "90^\\circ",
"style": "rule"
} | {
"index": "37456",
"split": "train"
} |
math | [
{
"content": "The digits $1,2,3,4,5,6$ are randomly chosen (without replacement) to form the three-digit numbers $M=\\overline{A B C}$ and $N=\\overline{D E F}$. For example, we could have $M=413$ and $N=256$. Find the expected value of $M \\cdot N$. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "143745",
"style": "rule"
} | {
"index": "37457",
"split": "train"
} |
math | [
{
"content": "If 700 were expressed as a sum of at least three distinct powers of 2, what would be the least possible sum of the exponents of these powers? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37458",
"split": "train"
} |
math | [
{
"content": "1-2-3+4+5-6-7+8+9-10-11+\\cdots + 1992+1993-1994-1995+1996= Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "37459",
"split": "train"
} |
math | [
{
"content": "We call a positive integer $t$ good if there is a sequence $a_{0}, a_{1}, \\ldots$ of positive integers satisfying $a_{0}=15, a_{1}=t$, and $a_{n-1} a_{n+1}=\\left(a_{n}-1\\right)\\left(a_{n}+1\\right)$ for all positive integers $n$. Find the sum of all good numbers. Please output the final answer... | {
"ground_truth": "296",
"style": "rule"
} | {
"index": "37460",
"split": "train"
} |
math | [
{
"content": "In a square, 20 points were marked and connected by non-intersecting segments with each other and with the vertices of the square, dividing the square into triangles. How many triangles were formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "42",
"style": "rule"
} | {
"index": "37461",
"split": "train"
} |
math | [
{
"content": "In an isosceles triangle \\(ABC\\), the base \\(AC\\) is equal to \\(x\\), and the lateral side is equal to 12. On the ray \\(AC\\), point \\(D\\) is marked such that \\(AD = 24\\). From point \\(D\\), a perpendicular \\(DE\\) is dropped to the line \\(AB\\). Find \\(x\\) given that \\(BE = 6\\). ... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "37462",
"split": "train"
} |
math | [
{
"content": "In a certain city, vehicle license plates are numbered consecutively from \"10000\" to \"99999\". How many license plates out of these 90,000 have the digit 9 appearing at least once and where the sum of the digits is a multiple of 9? Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "4168",
"style": "rule"
} | {
"index": "37463",
"split": "train"
} |
math | [
{
"content": "Eastbound traffic flows at 80 miles per hour and westbound traffic flows at 60 miles per hour. An eastbound driver observes 30 westbound vehicles in a 10-minute period. Calculate the number of westbound vehicles in a 150-mile section of the highway. Please output the final answer within \\boxed{}.... | {
"ground_truth": "193",
"style": "rule"
} | {
"index": "37464",
"split": "train"
} |
math | [
{
"content": "The ratio of the sums of the first \\( n \\) terms of two arithmetic sequences is \\(\\frac{9n+2}{n+7}\\). Find the ratio of their 5th terms. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{83}{16}",
"style": "rule"
} | {
"index": "37465",
"split": "train"
} |
math | [
{
"content": "For each ordered pair of real numbers $(x,y)$ satisfying \\[\\log_2(2x+y) = \\log_4(x^2+xy+7y^2)\\]there is a real number $K$ such that \\[\\log_3(3x+y) = \\log_9(3x^2+4xy+Ky^2).\\]Find the product of all possible values of $K$. Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "189",
"style": "rule"
} | {
"index": "37466",
"split": "train"
} |
math | [
{
"content": "When the height of a cylinder is doubled and its radius is increased by $200\\%$, the cylinder's volume is multiplied by a factor of $X$. What is the value of $X$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "37467",
"split": "train"
} |
math | [
{
"content": "Given the sequence \\(\\left\\{a_{n}\\right\\}\\) with the general term\n\\[ a_{n} = n^{4} + 6n^{3} + 11n^{2} + 6n, \\]\nfind the sum of the first 12 terms \\( S_{12} \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104832",
"style": "rule"
} | {
"index": "37468",
"split": "train"
} |
math | [
{
"content": "An urn contains $k$ balls labeled with $k$, for all $k = 1, 2, \\ldots, 2016$. What is the minimum number of balls we must draw, without replacement and without looking at the balls, to ensure that we have 12 balls with the same number? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "22122",
"style": "rule"
} | {
"index": "37469",
"split": "train"
} |
math | [
{
"content": "Given that $\\operatorname{tg} \\theta$ and $\\operatorname{ctg} \\theta$ are the real roots of the equation $2x^{2} - 2kx = 3 - k^{2}$, and $\\alpha < \\theta < \\frac{5 \\pi}{4}$, find the value of $\\cos \\theta - \\sin \\theta$. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "-\\sqrt{\\frac{5 - 2\\sqrt{5}}{5}}",
"style": "rule"
} | {
"index": "37470",
"split": "train"
} |
math | [
{
"content": "A long thin strip of paper is $1024$ units in length, $1$ unit in width, and is divided into $1024$ unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a $512$ by $1$ stri... | {
"ground_truth": "593",
"style": "rule"
} | {
"index": "37471",
"split": "train"
} |
math | [
{
"content": "What is the integer formed by the rightmost two digits of the integer equal to \\(4^{127} + 5^{129} + 7^{131}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "37472",
"split": "train"
} |
math | [
{
"content": "If $p, q,$ and $r$ are three non-zero integers such that $p + q + r = 26$ and\\[\\frac{1}{p} + \\frac{1}{q} + \\frac{1}{r} + \\frac{360}{pqr} = 1,\\] compute $pqr$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "576",
"style": "rule"
} | {
"index": "37473",
"split": "train"
} |
math | [
{
"content": "A square with a side length of 100 cm is drawn on a board. Alexei intersected it with two lines parallel to one pair of its sides. After that, Danil intersected the square with two lines parallel to the other pair of sides. As a result, the square was divided into 9 rectangles, with the dimensions... | {
"ground_truth": "2400",
"style": "rule"
} | {
"index": "37474",
"split": "train"
} |
math | [
{
"content": "Determine the product of the solutions of the equation $-21 = -x^2 + 4x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-21",
"style": "rule"
} | {
"index": "37475",
"split": "train"
} |
math | [
{
"content": "How many of the numbers in Grace's sequence, starting from 43 and each number being 4 less than the previous one, are positive? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37476",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\\overline{BE}=\\overline{ED}$. Point $F$ is on $AB$. Then, if $\\overline{BF}=5$, $\\overline{BA}$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37477",
"split": "train"
} |
math | [
{
"content": "For how many integers $n$ with $1 \\le n \\le 2023$ is the product\n\\[\n \\prod_{k=0}^{n-1} \\left( \\left( 1 + e^{2 \\pi i k / n} \\right)^n + 1 \\right) \n\\]equal to zero, where $n$ needs to be an even multiple of $5$? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "202",
"style": "rule"
} | {
"index": "37478",
"split": "train"
} |
math | [
{
"content": "Find the area of rhombus $EFGH$ given that the radii of the circles circumscribed around triangles $EFG$ and $EGH$ are $15$ and $30$, respectively. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "37479",
"split": "train"
} |
math | [
{
"content": "There are knights, liars, and followers living on an island; each knows who is who among them. All 2018 islanders were arranged in a row and asked to answer \"Yes\" or \"No\" to the question: \"Are there more knights on the island than liars?\". They answered in turn such that everyone else could ... | {
"ground_truth": "1009",
"style": "rule"
} | {
"index": "37480",
"split": "train"
} |
math | [
{
"content": "Alice places a coin, heads up, on a table then turns off the light and leaves the room. Bill enters the room with 2 coins and flips them onto the table and leaves. Carl enters the room, in the dark, and removes a coin at random. Alice reenters the room, turns on the light and notices that both coi... | {
"ground_truth": "3/5",
"style": "rule"
} | {
"index": "37481",
"split": "train"
} |
math | [
{
"content": "$908 \\times 501 - [731 \\times 1389 - (547 \\times 236 + 842 \\times 731 - 495 \\times 361)] =$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5448",
"style": "rule"
} | {
"index": "37482",
"split": "train"
} |
math | [
{
"content": "There are 8 sprinters in the Olympic 100-meter finals. Three of the sprinters are Americans. The gold medal goes to first place, silver to second, and bronze to third. In how many ways can the medals be awarded if at most one American gets a medal? Please output the final answer within \\boxed{... | {
"ground_truth": "240",
"style": "rule"
} | {
"index": "37483",
"split": "train"
} |
math | [
{
"content": "Let $m$ and $n$ satisfy $mn = 6$ and $m+n = 7$. Additionally, suppose $m^2 - n^2 = 13$. Find the value of $|m-n|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{7}",
"style": "rule"
} | {
"index": "37484",
"split": "train"
} |
math | [
{
"content": "Define a sequence of convex polygons \\( P_n \\) as follows. \\( P_0 \\) is an equilateral triangle with side length 1. \\( P_{n+1} \\) is obtained from \\( P_n \\) by cutting off the corners one-third of the way along each side (for example, \\( P_1 \\) is a regular hexagon with side length \\(\\... | {
"ground_truth": "\\frac{\\sqrt{3}}{7}",
"style": "rule"
} | {
"index": "37485",
"split": "train"
} |
math | [
{
"content": "A regular decagon is formed by connecting three sequentially adjacent vertices of the decagon. Find the probability that all three sides of the triangle are also sides of the decagon. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "37486",
"split": "train"
} |
math | [
{
"content": "Given that the diagonals of a rhombus are always perpendicular bisectors of each other, what is the area of a rhombus with side length $\\sqrt{113}$ units and diagonals that differ by 10 units? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "72",
"style": "rule"
} | {
"index": "37487",
"split": "train"
} |
math | [
{
"content": "The function $y=f(x)$ is an even function with the smallest positive period of 4, and when $x \\in [-2, 0]$, $f(x) = 2x + 1$. If there exist $x_1, x_2, \\ldots, x_n$ satisfying $0 \\leq x_1 < x_2 < \\ldots < x_n$, and $|f(x_1) - f(x_2)| + |f(x_2) - f(x_3)| + \\ldots + |f(x_{n-1}) - f(x_n)| = 2016$... | {
"ground_truth": "1513",
"style": "rule"
} | {
"index": "37488",
"split": "train"
} |
math | [
{
"content": "Let \\(x\\) and \\(y\\) be positive real numbers such that\n\\[\n\\frac{1}{x + 1} + \\frac{1}{y + 1} = \\frac{1}{2}.\n\\]\nFind the minimum value of \\(x + 3y.\\) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4 + 4 \\sqrt{3}",
"style": "rule"
} | {
"index": "37489",
"split": "train"
} |
math | [
{
"content": "Real numbers $r$ and $s$ are roots of $p(x)=x^3+ax+b$, and $r+4$ and $s-3$ are roots of $q(x)=x^3+ax+b+240$. Find the sum of all possible values of $|b|$.\nHint\n\\[\\color{red}\\boxed{\\boxed{\\color{blue}\\textbf{Use Vieta's Formulae!}}}\\] Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "420",
"style": "rule"
} | {
"index": "37490",
"split": "train"
} |
math | [
{
"content": "(1) If the terminal side of angle $\\theta$ passes through $P(-4t, 3t)$ ($t>0$), find the value of $2\\sin\\theta + \\cos\\theta$. \n(2) Given that a point $P$ on the terminal side of angle $\\alpha$ has coordinates $(x, -\\sqrt{3})$ ($x\\neq 0$), and $\\cos\\alpha = \\frac{\\sqrt{2}}{4}x$, find ... | {
"ground_truth": "\\frac{2}{5}",
"style": "rule"
} | {
"index": "37491",
"split": "train"
} |
math | [
{
"content": "How many five-digit numbers are there that are divisible by 5 and do not contain repeating digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5712",
"style": "rule"
} | {
"index": "37492",
"split": "train"
} |
math | [
{
"content": "Two distinct natural numbers end with 8 zeros and have exactly 90 divisors. Find their sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "700000000",
"style": "rule"
} | {
"index": "37493",
"split": "train"
} |
math | [
{
"content": "When each edge of a cube is increased by $50\\%$, by what percent is the surface area of the cube increased? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "125\\%",
"style": "rule"
} | {
"index": "37494",
"split": "train"
} |
math | [
{
"content": "How many distinct four-digit numbers are divisible by 3 and have 23 as their last two digits? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37495",
"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": "37496",
"split": "train"
} |
math | [
{
"content": "Zan has created this iterative rule for generating sequences of whole numbers:\n\n1) If a number is 25 or less, double the number.\n\n2) If a number is greater than 25, subtract 12 from it.\n\nLet $F$ be the first number in a sequence generated by the rule above. $F$ is a \"sweet number\" if 16 is... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "37497",
"split": "train"
} |
math | [
{
"content": "What is the smallest possible sum of two consecutive integers whose product is greater than 420? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "43",
"style": "rule"
} | {
"index": "37498",
"split": "train"
} |
math | [
{
"content": "Petya approaches the entrance door with a combination lock, which has buttons numbered from 0 to 9. To open the door, three correct buttons need to be pressed simultaneously. Petya does not remember the code and tries combinations one by one. Each attempt takes Petya 2 seconds.\n\na) How much time... | {
"ground_truth": "\\frac{29}{120}",
"style": "rule"
} | {
"index": "37499",
"split": "train"
} |
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