data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "A rectangular prism has a volume of \\(8 \\mathrm{~cm}^{3}\\), total surface area of \\(32 \\mathrm{~cm}^{2}\\), and its length, width, and height are in geometric progression. Find the sum of all its edge lengths (in cm). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "34200",
"split": "train"
} |
math | [
{
"content": "Square $ABCD$ is constructed along diameter $AB$ of a semicircle, where both the square and semicircle are coplanar. Line segment $AB$ has a length of 8 centimeters. If point $M$ is the midpoint of arc $AB$, what is the length of segment $MD$? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "4\\sqrt{10}",
"style": "rule"
} | {
"index": "34201",
"split": "train"
} |
math | [
{
"content": "Abe holds 1 green and 1 red jelly bean in his hand. Bob holds 1 green, 1 yellow, and 2 red jelly beans in his hand. Each randomly picks a jelly bean to show the other. What is the probability that the colors match? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{8}",
"style": "rule"
} | {
"index": "34202",
"split": "train"
} |
math | [
{
"content": "In a lottery game, the host randomly selects one of the four identical empty boxes numbered $1$, $2$, $3$, $4$, puts a prize inside, and then closes all four boxes. The host knows which box contains the prize. When a participant chooses a box, before opening the chosen box, the host randomly opens... | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "34203",
"split": "train"
} |
math | [
{
"content": "What is the minimum value of $z$ if $z=x^2+2y^2+6x-4y+22?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "34204",
"split": "train"
} |
math | [
{
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 9x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "34205",
"split": "train"
} |
math | [
{
"content": "A three-digit number has a remainder of 2 when divided by 4, 5, and 6. If three digits are appended to this three-digit number to make it a six-digit number divisible by 4, 5, and 6, what is the smallest six-digit number that meets this condition? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "122040",
"style": "rule"
} | {
"index": "34206",
"split": "train"
} |
math | [
{
"content": "Let $A M O L$ be a quadrilateral with $A M=10, M O=11$, and $O L=12$. Given that the perpendicular bisectors of sides $A M$ and $O L$ intersect at the midpoint of segment $A O$, find the length of side LA. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\sqrt{77}$",
"style": "rule"
} | {
"index": "34207",
"split": "train"
} |
math | [
{
"content": "Convert the binary number $101101110_{(2)}$ to a decimal number and then to an octal number $({\\ }\\_{(8)})$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "556_{(8)}",
"style": "rule"
} | {
"index": "34208",
"split": "train"
} |
math | [
{
"content": "The digits of a certain three-digit number form a geometric progression. If the digits of the hundreds and units places are swapped, the new three-digit number will be 594 less than the original number. If, in the original number, the hundreds digit is removed and the remaining two-digit number ha... | {
"ground_truth": "842",
"style": "rule"
} | {
"index": "34209",
"split": "train"
} |
math | [
{
"content": "Find the value of $c$ such that all the roots of the polynomial $x^3 - 5x^2 + 2bx - c$ are real and positive, given that one root is twice another and four times the third. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1000}{343}",
"style": "rule"
} | {
"index": "34210",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has $BC=20.$ The incircle of the triangle evenly trisects the median $AD.$ If the area of the triangle is $m \\sqrt{n}$ where $m$ and $n$ are integers and $n$ is not divisible by the square of a prime, find $m+n.$ Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "34211",
"split": "train"
} |
math | [
{
"content": "Points $A$, $B$, $C$, $D$, and $E$ are located in 3-dimensional space with $AB= BC= CD= DE= EA= 2$ and $\\angle ABC = \\angle CDE = \\angle\nDEA = 90^\\circ$. The plane of triangle $ABC$ is parallel to $\\overline{DE}$. What is the area of triangle $BDE$? Please output the final answer within \\b... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "34212",
"split": "train"
} |
math | [
{
"content": "Suppose that $x$ and $y$ are real numbers with $-4 \\leq x \\leq -2$ and $2 \\leq y \\leq 4$. What is the greatest possible value of $\\frac{x+y}{x}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "34213",
"split": "train"
} |
math | [
{
"content": "What is the least common multiple of 6, 8, and 10? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "34214",
"split": "train"
} |
math | [
{
"content": "When rolling a fair 6-sided die, what is the probability of a 2 or 4 being rolled? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "34215",
"split": "train"
} |
math | [
{
"content": "How many distinct sequences of five letters can be made from the letters in FREQUENCY if each sequence must begin with F, end with Y, and no letter can appear in a sequence more than once? Further, the second letter must be a vowel. Please output the final answer within \\boxed{}.",
"role": "u... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "34216",
"split": "train"
} |
math | [
{
"content": "If $2x - y = 5$ and $x + 2y = 5$, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "34217",
"split": "train"
} |
math | [
{
"content": "In a $3 \\times 3$ grid (each cell is a $1 \\times 1$ square), two identical pieces are placed, with at most one piece per cell. There are ___ distinct ways to arrange the pieces. (If two arrangements can be made to coincide by rotation, they are considered the same arrangement). Please output the... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34218",
"split": "train"
} |
math | [
{
"content": "Chelsea goes to La Verde's at MIT and buys 100 coconuts, each weighing 4 pounds, and 100 honeydews, each weighing 5 pounds. She wants to distribute them among \\( n \\) bags, so that each bag contains at most 13 pounds of fruit. What is the minimum \\( n \\) for which this is possible? Please outp... | {
"ground_truth": "75",
"style": "rule"
} | {
"index": "34219",
"split": "train"
} |
math | [
{
"content": "Using systematic sampling method to select 32 people from 960 for a questionnaire survey, they are randomly numbered from 1 to 960. After grouping, the number drawn by simple random sampling in the first group is 9. Among the 32 people drawn, those with numbers in the interval [1,450] will fill ou... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34220",
"split": "train"
} |
math | [
{
"content": "Given the function $y=\\sin (\\omega x+\\frac{\\pi }{3})+2$, its graph shifts to the right by $\\frac{4\\pi }{3}$ units and coincides with the original graph. Find the minimum value of $|\\omega|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {3}{2}",
"style": "rule"
} | {
"index": "34221",
"split": "train"
} |
math | [
{
"content": "Let $u_0 = \\frac{1}{3}$, and for $k \\ge 0$, let $u_{k+1} = \\frac{3}{2}u_k - \\frac{3}{2}u_k^2$. This sequence tends to a limit; call it $M$. Determine the least value of $k$ such that $|u_k - M| \\le \\frac{1}{2^{1000}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "34222",
"split": "train"
} |
math | [
{
"content": "Find the interval of all $x$ such that both $2x$ and $3x$ are in the interval $(1,2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(\\frac{1}{2},\\frac{2}{3}\\right)",
"style": "rule"
} | {
"index": "34223",
"split": "train"
} |
math | [
{
"content": "Today, Ivan the Confessor prefers continuous functions $f:[0,1]\\to\\mathbb{R}$ satisfying $f(x)+f(y)\\geq |x-y|$ for all pairs $x,y\\in [0,1]$. Find the minimum of $\\int_0^1 f$ over all preferred functions.\n\n( Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "34224",
"split": "train"
} |
math | [
{
"content": "What is the largest integer $n$ that satisfies $(100^2-99^2)(99^2-98^2)\\dots(3^2-2^2)(2^2-1^2)$ is divisible by $3^n$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "34225",
"split": "train"
} |
math | [
{
"content": "Given that $(2-x)^{5}=a\\_{0}+a\\_{1}x+a\\_{2}x^{2}+…+a\\_{5}x^{5}$, find the value of $\\frac{a\\_0+a\\_2+a\\_4}{a\\_1+a\\_3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{61}{60}",
"style": "rule"
} | {
"index": "34226",
"split": "train"
} |
math | [
{
"content": "A circle passing through the vertex \\( P \\) of triangle \\( PQR \\) touches side \\( QR \\) at point \\( F \\) and intersects sides \\( PQ \\) and \\( PR \\) at points \\( M \\) and \\( N \\), respectively, different from vertex \\( P \\). Find the ratio \\( QF : FR \\) if it is known that the l... | {
"ground_truth": "1/2",
"style": "rule"
} | {
"index": "34227",
"split": "train"
} |
math | [
{
"content": "How many sets of two or more consecutive positive integers have a sum of $15$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "34228",
"split": "train"
} |
math | [
{
"content": "If a school bus leaves school with 48 students on board, and one-half of the students get off the bus at each of the first three stops, how many students remain on the bus after the third stop? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "34229",
"split": "train"
} |
math | [
{
"content": "In an extended game, each of 6 players, including Hugo, rolls a standard 8-sided die. The winner is the one who rolls the highest number. In the case of a tie for the highest roll, the tied players will re-roll until a single winner emerges. What is the probability that Hugo's first roll was a 7, ... | {
"ground_truth": "\\frac{8856}{32768}",
"style": "rule"
} | {
"index": "34230",
"split": "train"
} |
math | [
{
"content": "Welcome to the USAYNO, where each question has a yes/no answer. Choose any subset of the following six problems to answer. If you answer $n$ problems and get them all correct, you will receive $\\max (0,(n-1)(n-2))$ points. If any of them are wrong, you will receive 0 points. Your answer should be... | {
"ground_truth": "NNNYYY",
"style": "rule"
} | {
"index": "34231",
"split": "train"
} |
math | [
{
"content": "Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and heads towards point B... | {
"ground_truth": "7.2",
"style": "rule"
} | {
"index": "34232",
"split": "train"
} |
math | [
{
"content": "Let \\( N \\) be the total number of students in the school before the New Year, among which \\( M \\) are boys, making up \\( k \\) percent of the total. This means \\( M = \\frac{k}{100} N \\), or \\( 100M = kN \\).\n\nAfter the New Year, the number of boys became \\( M+1 \\), and the total numb... | {
"ground_truth": "197",
"style": "rule"
} | {
"index": "34233",
"split": "train"
} |
math | [
{
"content": "In right triangle $ABC$, $\\sin A = \\frac{3}{5}$ and $\\sin B = 1$. Find $\\sin C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "34234",
"split": "train"
} |
math | [
{
"content": "An isosceles trapezoid $A B C D$ with bases $A B$ and $C D$ has $A B=13, C D=17$, and height 3. Let $E$ be the intersection of $A C$ and $B D$. Circles $\\Omega$ and $\\omega$ are circumscribed about triangles $A B E$ and $C D E$. Compute the sum of the radii of $\\Omega$ and $\\omega$. Please out... | {
"ground_truth": "39",
"style": "rule"
} | {
"index": "34235",
"split": "train"
} |
math | [
{
"content": "As shown in the diagram, in the square \\(ABCD\\), \\(AB = 2\\). Draw an arc with center \\(C\\) and radius equal to \\(CD\\), and another arc with center \\(B\\) and radius equal to \\(BA\\). The two arcs intersect at \\(E\\). What is the area of the sector \\(BAE\\)? Please output the final answ... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "34236",
"split": "train"
} |
math | [
{
"content": "What is the correct ordering of the three numbers $\\frac{5}{19}$, $\\frac{7}{21}$, and $\\frac{9}{23}$, in increasing order? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5}{19} < \\frac{7}{21} < \\frac{9}{23}",
"style": "rule"
} | {
"index": "34237",
"split": "train"
} |
math | [
{
"content": "Consider the case when all numbers are equal. $\\frac{5}{4} n + \\frac{5}{4} = n$. If the first number is -5, then all numbers will be equal to -5. The same applies to all cases where the first number is equal to $-5 + 1024n$, $n \\in \\mathbb{Z}$. Please output the final answer within \\boxed{}."... | {
"ground_truth": "-5",
"style": "rule"
} | {
"index": "34238",
"split": "train"
} |
math | [
{
"content": "Evaluate\n\\[i^{14762} + i^{14763} + i^{14764} + i^{14765}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0",
"style": "rule"
} | {
"index": "34239",
"split": "train"
} |
math | [
{
"content": "Let \\( x, y, z \\) be complex numbers such that:\n\\[\nxy + 3y = -9,\nyz + 3z = -9,\nzx + 3x = -9.\n\\]\nDetermine all possible values of \\( xyz \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "27",
"style": "rule"
} | {
"index": "34240",
"split": "train"
} |
math | [
{
"content": "What is $100(100-3)-(100 \\cdot 100-3)$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-297",
"style": "rule"
} | {
"index": "34241",
"split": "train"
} |
math | [
{
"content": "Given a positive geometric sequence $\\{a_{n}\\}$, if ${a_m}{a_n}=a_3^2$, find the minimum value of $\\frac{2}{m}+\\frac{1}{{2n}}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "34242",
"split": "train"
} |
math | [
{
"content": "A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 3-primable positive integers are there that are less than 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "28",
"style": "rule"
} | {
"index": "34243",
"split": "train"
} |
math | [
{
"content": "Three boys \\( B_{1}, B_{2}, B_{3} \\) and three girls \\( G_{1}, G_{2}, G_{3} \\) are to be seated in a row according to the following rules:\n1) A boy will not sit next to another boy and a girl will not sit next to another girl,\n2) Boy \\( B_{1} \\) must sit next to girl \\( G_{1} \\).\nIf \\(... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "34244",
"split": "train"
} |
math | [
{
"content": "A large supermarket purchased a popular disinfectant laundry detergent. Due to the rise in raw material prices, the cost price per bottle of detergent this year increased by $4$ compared to last year. The quantity of detergent purchased for $1440$ yuan this year is the same as the quantity purchas... | {
"ground_truth": "8100",
"style": "rule"
} | {
"index": "34245",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive integer $n$ for which $11n-8$ and $5n + 9$ share a common factor greater than $1$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "34246",
"split": "train"
} |
math | [
{
"content": "Two lines with slopes $\\frac{1}{2}$ and $2$ intersect at $(2,2)$. What is the area of the triangle enclosed by these two lines and the line $x+y=10$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "34247",
"split": "train"
} |
math | [
{
"content": "Given a sequence ${{a_{n}}}$ where all terms are non-zero, the sum of the first $n$ terms is ${{S_{n}}}$, and it satisfies ${{a_{1}}=a,}$ $2{{S_{n}}={{a_{n}}{{a_{n+1}}}}}$.\n\n(I) Find the value of ${{a_{2}}}$;\n(II) Find the general formula for the $n^{th}$ term of the sequence;\n(III) If $a=-9$,... | {
"ground_truth": "-15",
"style": "rule"
} | {
"index": "34248",
"split": "train"
} |
math | [
{
"content": "You have a whole pizza in the refrigerator. On your first trip to the refrigerator, you eat half the pizza. On each successive trip, you eat half of the remaining pizza. After five trips to the refrigerator, what fractional part of the pizza have you eaten? Please output the final answer within \\... | {
"ground_truth": "\\frac{31}{32}",
"style": "rule"
} | {
"index": "34249",
"split": "train"
} |
math | [
{
"content": "How many one-thirds are in one-sixth? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "34250",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\\\{a_{n}\\\\}$, let $S_{n}$ denote the sum of its first $n$ terms. If $a_{4}=-12$ and $a_{8}=-4$:\n\n$(1)$ Find the general term formula for the sequence;\n$(2)$ Find the minimum value of $S_{n}$ and the corresponding value of $n$. Please output the final answer wit... | {
"ground_truth": "-90",
"style": "rule"
} | {
"index": "34251",
"split": "train"
} |
math | [
{
"content": "If $x + x^2 + x^3 + \\ldots + x^9 + x^{10} = a_0 + a_1(1 + x) + a_2(1 + x)^2 + \\ldots + a_9(1 + x)^9 + a_{10}(1 + x)^{10}$, then $a_9 = \\_\\_\\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-9",
"style": "rule"
} | {
"index": "34252",
"split": "train"
} |
math | [
{
"content": "Five packages are delivered to five houses, one to each house. If these packages are randomly delivered, what is the probability that exactly three of them are delivered to the correct houses? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": ... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "34253",
"split": "train"
} |
math | [
{
"content": "The graph of $y = f(x)$ is shown below.\n\n[asy]\nunitsize(0.5 cm);\n\nreal func(real x) {\n real y;\n if (x >= -3 && x <= 0) {y = -2 - x;}\n if (x >= 0 && x <= 2) {y = sqrt(4 - (x - 2)^2) - 2;}\n if (x >= 2 && x <= 3) {y = 2*(x - 2);}\n return(y);\n}\n\nint i, n;\n\nfor (i = -5; i <= 5; ++i)... | {
"ground_truth": "\\text{E}",
"style": "rule"
} | {
"index": "34254",
"split": "train"
} |
math | [
{
"content": "Consider an infinite grid of unit squares. An $n$-omino is a subset of $n$ squares that is connected. Below are depicted examples of 8 -ominoes. Two $n$-ominoes are considered equivalent if one can be obtained from the other by translations and rotations. What is the number of distinct 15 -ominoes... | {
"ground_truth": "3426576",
"style": "rule"
} | {
"index": "34255",
"split": "train"
} |
math | [
{
"content": "How many rational terms are in the expansion of\n\na) $(\\sqrt{2}+\\sqrt[4]{3})^{100}$\n\nb) $(\\sqrt{2}+\\sqrt[3]{3})^{300}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "34256",
"split": "train"
} |
math | [
{
"content": "There is a solid iron cone with a base radius of $3cm$ and a slant height of $5cm$. After melting it at high temperature and casting it into a solid iron sphere (without considering any loss), the radius of this iron sphere is _______ $cm$. Please output the final answer within \\boxed{}.",
"r... | {
"ground_truth": "\\sqrt[3]{9}",
"style": "rule"
} | {
"index": "34257",
"split": "train"
} |
math | [
{
"content": "Given $x, y \\in (-1, 1)$, find the minimum value of the expression $$\\sqrt {(x+1)^{2}+(y-1)^{2}}+\\sqrt {(x+1)^{2}+(y+1)^{2}}+\\sqrt {(x-1)^{2}+(y+1)^{2}}+\\sqrt {(x-1)^{2}+(y-1)^{2}}.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4\\sqrt{2}",
"style": "rule"
} | {
"index": "34258",
"split": "train"
} |
math | [
{
"content": "Let $P(x)=x^{3}+a x^{2}+b x+2015$ be a polynomial all of whose roots are integers. Given that $P(x) \\geq 0$ for all $x \\geq 0$, find the sum of all possible values of $P(-1)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9496",
"style": "rule"
} | {
"index": "34259",
"split": "train"
} |
math | [
{
"content": "Find the number of pairs of natural numbers \\((x, y)\\) such that \\(1 \\leq x, y \\leq 1000\\) and \\(x^2 + y^2\\) is divisible by 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200000",
"style": "rule"
} | {
"index": "34260",
"split": "train"
} |
math | [
{
"content": "Let $\\triangle PQR$ be a triangle in the plane, and let $S$ be a point outside the plane of $\\triangle PQR$, so that $SPQR$ is a pyramid whose faces are all triangles.\n\nSuppose that every edge of $SPQR$ has length $18$ or $41$, but no face of $SPQR$ is equilateral. Then what is the surface are... | {
"ground_truth": "1440",
"style": "rule"
} | {
"index": "34261",
"split": "train"
} |
math | [
{
"content": "The sum of the three largest natural divisors of a natural number \\( N \\) is 10 times the sum of its three smallest natural divisors. Find all possible values of \\( N \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "34262",
"split": "train"
} |
math | [
{
"content": "Consider finding the result when we compute the series $$1^3 + 2^3 + 3^3 + \\dots + 49^3 + 50^3$$ and the series $$(-1)^3 + (-2)^3 + (-3)^3 + \\dots + (-49)^3 + (-50)^3,$$ then subtract the second series' result from the first series' result. What is the sum? Please output the final answer within ... | {
"ground_truth": "3251250",
"style": "rule"
} | {
"index": "34263",
"split": "train"
} |
math | [
{
"content": "Given that $\\tan α$ and $\\tan β$ are the roots of the equation $x^{2}+3 \\sqrt {3}x+4=0$, and $\\(- \\frac {π}{2} < α < \\frac {π}{2}\\)$, $\\(- \\frac {π}{2} < β < \\frac {π}{2}\\)$, find $α+β$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac {2\\pi}{3}",
"style": "rule"
} | {
"index": "34264",
"split": "train"
} |
math | [
{
"content": "What is the smallest integer larger than $(\\sqrt{5}+\\sqrt{3})^4$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "248",
"style": "rule"
} | {
"index": "34265",
"split": "train"
} |
math | [
{
"content": "A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{6}\\pi",
"style": "rule"
} | {
"index": "34266",
"split": "train"
} |
math | [
{
"content": "If the set $\\{1, a, \\frac{b}{a}\\}$ equals $\\{0, a^2, a + b\\}$, find the value of $a - b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34267",
"split": "train"
} |
math | [
{
"content": "If $A=2+i$, $O=-4$, $P=-i$, and $S=2+4i$, find $A-O+P+S$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8+4i",
"style": "rule"
} | {
"index": "34268",
"split": "train"
} |
math | [
{
"content": "The positive integers from 1 to 576 are written in a 24 by 24 grid so that the first row contains the numbers 1 to 24, the second row contains the numbers 25 to 48, and so on. An 8 by 8 square is drawn around 64 of these numbers. The sum of the numbers in the four corners of the 8 by 8 square is 1... | {
"ground_truth": "499",
"style": "rule"
} | {
"index": "34269",
"split": "train"
} |
math | [
{
"content": "A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "22\\%",
"style": "rule"
} | {
"index": "34270",
"split": "train"
} |
math | [
{
"content": "The expression $\\log_{y^6}{x}\\cdot\\log_{x^5}{y^2}\\cdot\\log_{y^4}{x^3}\\cdot\\log_{x^3}{y^4}\\cdot\\log_{y^2}{x^5}$ can be written as $a\\log_y{x}$ for what constant $a$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac16",
"style": "rule"
} | {
"index": "34271",
"split": "train"
} |
math | [
{
"content": "Find all integers $n \\geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \\hdots, a_n$ with $\\text{max}(a_1,a_2,\\hdots,a_n) \\leq n \\cdot \\text{min}(a_1,a_2,\\hdots,a_n)$, there exist three that are the side lengths of an acute triangle. Please output the final answer within \\bo... | {
"ground_truth": "n \\geq 13",
"style": "rule"
} | {
"index": "34272",
"split": "train"
} |
math | [
{
"content": "Let $s$ be the limiting sum of the geometric series $4 - \\frac{8}{3} + \\frac{16}{9} - \\dots$, as the number of terms increases without bound. Then $s$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.4",
"style": "rule"
} | {
"index": "34273",
"split": "train"
} |
math | [
{
"content": "The sales tax rate in Rubenenkoville is 6%. During a sale at the Bergville Coat Closet, the price of a coat is discounted 20% from its $90.00 price. Two clerks, Jack and Jill, calculate the bill independently. Jack rings up $90.00 and adds 6% sales tax, then subtracts 20% from this total. Jill rin... | {
"ground_truth": "$0",
"style": "rule"
} | {
"index": "34274",
"split": "train"
} |
math | [
{
"content": "For any real number \\( x \\), let \\( f(x) \\) be the minimum of the values \\( 4x + 1 \\), \\( x + 2 \\), and \\( -2x + 4 \\). What is the maximum value of \\( f(x) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{8}{3}",
"style": "rule"
} | {
"index": "34275",
"split": "train"
} |
math | [
{
"content": "For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only?\n\t\n\tAlexey Glebov Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1023",
"style": "rule"
} | {
"index": "34276",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\(PQRS\\) is a rectangle with \\(SR = 15\\). Point \\(T\\) is above \\(PS\\) and point \\(U\\) is on \\(PS\\) so that \\(TU\\) is perpendicular to \\(PS\\). If \\(PT = 10\\) and \\(US = 4\\) and the area of \\(PQRS\\) is 180, what is the area of \\(\\triangle PTS\\)? Please output... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "34277",
"split": "train"
} |
math | [
{
"content": "If $|x-\\log y|=x+\\log y$ where $x$ and $\\log y$ are real, then Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x(y-1)=0",
"style": "rule"
} | {
"index": "34278",
"split": "train"
} |
math | [
{
"content": "Given a sequence $\\{a_{n}\\}$ that satisfies ${a}_{n+1}=\\frac{1}{3}{a}_{n}$, if $a_{4}+a_{5}=4$, calculate $a_{2}+a_{3}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "34279",
"split": "train"
} |
math | [
{
"content": "Given a geometric sequence $\\{a_n\\}$ satisfies $a_2a_5=2a_3$, and $a_4$, $\\frac{5}{4}$, $2a_7$ form an arithmetic sequence, the maximum value of $a_1a_2a_3…a_n$ is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1024",
"style": "rule"
} | {
"index": "34280",
"split": "train"
} |
math | [
{
"content": "Three triangles. Inside triangle $ABC$, a random point $M$ is chosen. What is the probability that the area of one of the triangles $ABM$, $BCM$, or $CAM$ will be greater than the sum of the areas of the other two? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.75",
"style": "rule"
} | {
"index": "34281",
"split": "train"
} |
math | [
{
"content": "(1) Simplify: $f(α)= \\dfrac {\\sin (α+ \\dfrac {3}{2}π)\\sin (-α+π)\\cos (α+ \\dfrac {π}{2})}{\\cos (-α -π )\\cos (α - \\dfrac {π}{2})\\tan (α +π )}$\n(2) Evaluate: $\\tan 675^{\\circ}+\\sin (-330^{\\circ})+\\cos 960^{\\circ}$ Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "34282",
"split": "train"
} |
math | [
{
"content": "If $y = -x^2 + 5$ and $x$ is a real number, then what is the maximum value possible for $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "34283",
"split": "train"
} |
math | [
{
"content": "Observe:\n$$\n\\begin{array}{l}\n1 \\times 2 \\times 3 \\times 4 + 1 = 5^{2} \\\\\n2 \\times 3 \\times 4 \\times 5 + 1 = 11^{2} \\\\\n3 \\times 4 \\times 5 \\times 6 + 1 = 19^{2} \\\\\n\\ldots \\ldots\n\\end{array}\n$$\n\nCalculate $\\sqrt{2020 \\times 2021 \\times 2022 \\times 2023 + 1}=$ Please ... | {
"ground_truth": "4086461",
"style": "rule"
} | {
"index": "34284",
"split": "train"
} |
math | [
{
"content": "The noon temperatures for seven consecutive days were $80^{\\circ}$, $79^{\\circ}$, $81^{\\circ}$, $85^{\\circ}$, $87^{\\circ}$, $89^{\\circ}$, and $87^{\\circ}$ Fahrenheit. What is the mean noon temperature, in degrees Fahrenheit, for the week? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "84",
"style": "rule"
} | {
"index": "34285",
"split": "train"
} |
math | [
{
"content": "In the final stage of a professional bowling competition, the top five players compete as follows: \n\n- The fifth place player competes against the fourth place player.\n- The loser of the match receives the 5th place award.\n- The winner then competes against the third place player.\n- The loser... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "34286",
"split": "train"
} |
math | [
{
"content": "$A B C$ is a triangle with points $E, F$ on sides $A C, A B$, respectively. Suppose that $B E, C F$ intersect at $X$. It is given that $A F / F B=(A E / E C)^{2}$ and that $X$ is the midpoint of $B E$. Find the ratio $C X / X F$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "34287",
"split": "train"
} |
math | [
{
"content": "Given $|x|=4$, $|y|=2$, and $x<y$, then the value of $x\\div y$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "34288",
"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 \\([1, 3]\\). Find \\( \\underbrace{f(f(\\ldots f}_{2017}\\left(\\frac{3+\\sqrt{7}}{2}\\right)) \\ldots) \\). If necessary, round your answer to two decimal places. Pleas... | {
"ground_truth": "0.18",
"style": "rule"
} | {
"index": "34289",
"split": "train"
} |
math | [
{
"content": "A quadrilateral is inscribed in a circle of radius $200\\sqrt{2}$. Three of the sides of this quadrilateral have length $200$. What is the length of the fourth side? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "34290",
"split": "train"
} |
math | [
{
"content": "Lydia likes a five-digit number if none of its digits are divisible by 3. Find the total sum of the digits of all five-digit numbers that Lydia likes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "174960",
"style": "rule"
} | {
"index": "34291",
"split": "train"
} |
math | [
{
"content": "What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 360 = 0$ has integral solutions with one root being a multiple of the other? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "34292",
"split": "train"
} |
math | [
{
"content": "A cryptographer designed the following method to encode natural numbers: first, represent the natural number in base 5, then map the digits in the base 5 representation to the elements of the set $\\{V, W, X, Y, Z\\}$ in a one-to-one correspondence. Using this correspondence, he found that three c... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "34293",
"split": "train"
} |
math | [
{
"content": "Triangle $\\triangle DEF$ has a right angle at $F$, $\\angle D = 60^\\circ$, and $DF=12$. Find the radius of the incircle of $\\triangle DEF$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6(\\sqrt{3}-1)",
"style": "rule"
} | {
"index": "34294",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xOy$, the equation of line $C_1$ is $y=-\\sqrt{3}x$, and the parametric equations of curve $C_2$ are given by $\\begin{cases}x=-\\sqrt{3}+\\cos\\varphi\\\\y=-2+\\sin\\varphi\\end{cases}$. Establish a polar coordinate system with the coordinate origin as the po... | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "34295",
"split": "train"
} |
math | [
{
"content": "A basketball player scored 18, 22, 15, and 20 points respectively in her first four games of a season. Her points-per-game average was higher after eight games than it was after these four games. If her average after nine games was greater than 19, determine the least number of points she could ha... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "34296",
"split": "train"
} |
math | [
{
"content": "When one ounce of water is added to a mixture of acid and water, the new mixture is $20\\%$ acid. When one ounce of acid is added to the new mixture, the result is $33\\frac13\\%$ acid. The percentage of acid in the original mixture is Please output the final answer within \\boxed{}.",
"role":... | {
"ground_truth": "25\\%",
"style": "rule"
} | {
"index": "34297",
"split": "train"
} |
math | [
{
"content": "How can 13 rectangles of sizes $1 \\times 1, 2 \\times 1, 3 \\times 1, \\ldots, 13 \\times 1$ be combined to form a rectangle, where all sides are greater than 1? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13 \\times 7",
"style": "rule"
} | {
"index": "34298",
"split": "train"
} |
math | [
{
"content": "Define a $\\it{good\\ word}$ 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 or D, B is never immediately followed by C, C is never immediately followed by A, and D is neve... | {
"ground_truth": "512",
"style": "rule"
} | {
"index": "34299",
"split": "train"
} |
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