data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "The maximum value of the function $y=4^x+2^{x+1}+5$, where $x\\in[1,2]$, is to be found. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "7800",
"split": "train"
} |
math | [
{
"content": "Which of the following numbers is closest to 1: $\frac{11}{10}$, $\frac{111}{100}$, 1.101, $\frac{1111}{1000}$, 1.011? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1.011",
"style": "rule"
} | {
"index": "7801",
"split": "train"
} |
math | [
{
"content": "If the square roots of a positive number are $a+2$ and $2a-11$, find the positive number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "7802",
"split": "train"
} |
math | [
{
"content": "If $2 x^{2}=9 x-4$ and $x \neq 4$, what is the value of $2 x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7803",
"split": "train"
} |
math | [
{
"content": "In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which have at least 3 roads... | {
"ground_truth": "1004",
"style": "rule"
} | {
"index": "7804",
"split": "train"
} |
math | [
{
"content": "As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\\omega$ and bisect its perimeter. Let $(P, P')$ an... | {
"ground_truth": "60^\\circ",
"style": "rule"
} | {
"index": "7805",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer $n$ such that $\\sigma(n) = 28$ , where $\\sigma(n)$ is the sum of the divisors of $n$ , including $n$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7806",
"split": "train"
} |
math | [
{
"content": "Find $\\begin{pmatrix} 3 \\\\ -7 \\end{pmatrix} + \\begin{pmatrix} -6 \\\\ 11 \\end{pmatrix}.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -3 \\\\ 4 \\end{pmatrix}",
"style": "rule"
} | {
"index": "7807",
"split": "train"
} |
math | [
{
"content": "Calculate the result of the expression:\n\n\\[ 2013 \\times \\frac{5.7 \\times 4.2 + \\frac{21}{5} \\times 4.3}{\\frac{14}{73} \\times 15 + \\frac{5}{73} \\times 177 + 656} \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "126",
"style": "rule"
} | {
"index": "7808",
"split": "train"
} |
math | [
{
"content": "In the middle of the school year, $40\\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does t... | {
"ground_truth": "170",
"style": "rule"
} | {
"index": "7809",
"split": "train"
} |
math | [
{
"content": "In a bag, there are 70 balls which differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.\n\nWhat is the minimum number of balls that must be drawn from the bag, without seeing them, to ensure that there are at least 10 balls of one color among them? Please output the ... | {
"ground_truth": "38",
"style": "rule"
} | {
"index": "7810",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin(\\theta + 3\\pi) = -\\frac{2}{3}$, find the value of $\\frac{\\tan(-5\\pi - \\theta) \\cdot \\cos(\\theta - 2\\pi) \\cdot \\sin(-3\\pi - \\theta)}{\\tan(\\frac{7\\pi}{2} + \\theta) \\cdot \\sin(-4\\pi + \\theta) \\cdot \\cot(-\\theta - \\frac{\\pi}{2})} + 2 \\tan(6\\pi - \\theta)... | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "7811",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)= \\frac {1}{2}x^{2}-2\\ln x+a(a\\in\\mathbb{R})$, $g(x)=-x^{2}+3x-4$. \n$(1)$ Find the intervals of monotonicity for $f(x)$; \n$(2)$ Let $a=0$, the line $x=t$ intersects the graphs of $f(x)$ and $g(x)$ at points $M$ and $N$ respectively. When $|MN|$ reaches its minimum v... | {
"ground_truth": "\\frac {3+ \\sqrt {33}}{6}",
"style": "rule"
} | {
"index": "7812",
"split": "train"
} |
math | [
{
"content": "Three merchants - Sosipatra Titovna, Olympiada Karpovna, and Poliksena Uvarovna - sat down to drink tea. Olympiada Karpovna and Sosipatra Titovna together drank 11 cups, Poliksena Uvarovna and Olympiada Karpovna drank 15 cups, and Sosipatra Titovna and Poliksena Uvarovna drank 14 cups. How many cu... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "7813",
"split": "train"
} |
math | [
{
"content": "A dot is marked at each vertex of a triangle $A B C$. Then, 2,3 , and 7 more dots are marked on the sides $A B, B C$, and $C A$, respectively. How many triangles have their vertices at these dots? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "357",
"style": "rule"
} | {
"index": "7814",
"split": "train"
} |
math | [
{
"content": "Rectangle $EFGH$ has area $4032$. An ellipse with area $4032\\pi$ passes through points $E$ and $G$ and has foci at $F$ and $H$. Determine the perimeter of the rectangle $EFGH$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8\\sqrt{2016}",
"style": "rule"
} | {
"index": "7815",
"split": "train"
} |
math | [
{
"content": "Let $P$ be an interior point of triangle $ABC$ and extend lines from the vertices through $P$ to the opposite sides. Let $a$, $b$, $c$, and $d$ denote the lengths of the segments indicated in the figure. Find the product $abc$ if $a + b + c = 43$ and $d = 3$. Please output the final answer within ... | {
"ground_truth": "441",
"style": "rule"
} | {
"index": "7816",
"split": "train"
} |
math | [
{
"content": "Into how many regions do the x-axis and the graphs of \\( y = 2 - x^2 \\) and \\( y = x^2 - 1 \\) split the plane? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7817",
"split": "train"
} |
math | [
{
"content": "Let $a_{1}=3$, and for $n>1$, let $a_{n}$ be the largest real number such that $$4\\left(a_{n-1}^{2}+a_{n}^{2}\\right)=10 a_{n-1} a_{n}-9$$ What is the largest positive integer less than $a_{8}$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "335",
"style": "rule"
} | {
"index": "7818",
"split": "train"
} |
math | [
{
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "7819",
"split": "train"
} |
math | [
{
"content": "Three distinct vertices of a regular 2020-gon are chosen uniformly at random. The probability that the triangle they form is isosceles can be expressed as $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a+b$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "773",
"style": "rule"
} | {
"index": "7820",
"split": "train"
} |
math | [
{
"content": "Given that the random variable $\\xi$ follows a normal distribution $N(4, 6^2)$, and $P(\\xi \\leq 5) = 0.89$, find the probability $P(\\xi \\leq 3)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.11",
"style": "rule"
} | {
"index": "7821",
"split": "train"
} |
math | [
{
"content": "Find the sum of all prime numbers whose representation in base 14 has the form $101010...101$ (alternating ones and zeros). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "197",
"style": "rule"
} | {
"index": "7822",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, $c$ are the sides opposite to $\\angle A$, $\\angle B$, $\\angle C$ respectively. If $\\cos 2B + \\cos B + \\cos (A-C) = 1$ and $b = \\sqrt{7}$, find the minimum value of $a^2 + c^2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "7823",
"split": "train"
} |
math | [
{
"content": "Two people, A and B, are working together to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they have completed half of the document, A's typing speed triples, while B takes a 5-minute break and then continues typing at his original speed... | {
"ground_truth": "18000",
"style": "rule"
} | {
"index": "7824",
"split": "train"
} |
math | [
{
"content": "Find the largest positive integer $m$ such that an $m \\times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$. Please output the final answer within \\boxe... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "7825",
"split": "train"
} |
math | [
{
"content": "A square has a side length of $4$. Within this square, two equilateral triangles are placed such that one has its base along the bottom side of the square, and the other is rotated such that its vertex touches the midpoint of the top side of the square, and its base is parallel to the bottom of th... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "7826",
"split": "train"
} |
math | [
{
"content": "Given an isosceles trapezoid \\(ABCD\\), where \\(AD \\parallel BC\\), \\(BC = 2AD = 4\\), \\(\\angle ABC = 60^\\circ\\), and \\(\\overrightarrow{CE} = \\frac{1}{3} \\overrightarrow{CD}\\), find \\(\\overrightarrow{CA} \\cdot \\overrightarrow{BE}\\). Please output the final answer within \\boxed{}... | {
"ground_truth": "-10",
"style": "rule"
} | {
"index": "7827",
"split": "train"
} |
math | [
{
"content": "Determine the residue of $-998\\pmod{28}$. Your answer should be an integer in the range $0,1,2,\\ldots,25,26,27$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "7828",
"split": "train"
} |
math | [
{
"content": "A right isosceles triangle is inscribed in a triangle with a base of 30 and a height of 10 such that its hypotenuse is parallel to the base of the given triangle, and the vertex of the right angle lies on this base. Find the hypotenuse. Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7829",
"split": "train"
} |
math | [
{
"content": "Calculate the values of:\n(1) $8^{\\frac{2}{3}} - (0.5)^{-3} + \\left(\\frac{1}{\\sqrt{3}}\\right)^{-2} \\times \\left(\\frac{81}{16}\\right)^{-\\frac{1}{4}}$;\n(2) $\\log 5 \\cdot \\log 8000 + (\\log 2^{\\sqrt{3}})^2 + e^{\\ln 1} + \\ln(e \\sqrt{e})$. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{11}{2}",
"style": "rule"
} | {
"index": "7830",
"split": "train"
} |
math | [
{
"content": "Given that $x$ is a multiple of $12600$, what is the greatest common divisor of $g(x) = (5x + 7)(11x + 3)(17x + 8)(4x + 5)$ and $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "840",
"style": "rule"
} | {
"index": "7831",
"split": "train"
} |
math | [
{
"content": "The numbers from 1 to 150, inclusive, are placed in a bag and a number is randomly selected from the bag. What is the probability it is not a perfect power (integers that can be expressed as $x^{y}$ where $x$ is an integer and $y$ is an integer greater than 1. For example, $2^{4}=16$ is a perfect ... | {
"ground_truth": "\\frac{133}{150}",
"style": "rule"
} | {
"index": "7832",
"split": "train"
} |
math | [
{
"content": "Given that the perimeter of triangle \\( \\triangle ABC \\) is 20, the radius of the inscribed circle is \\( \\sqrt{3} \\), and \\( BC = 7 \\). Find the value of \\( \\tan A \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{3}",
"style": "rule"
} | {
"index": "7833",
"split": "train"
} |
math | [
{
"content": "Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:\n\nFirst method:\n\na) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:\n\n\\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 3... | {
"ground_truth": "-12.6",
"style": "rule"
} | {
"index": "7834",
"split": "train"
} |
math | [
{
"content": "If $x, y$ and $2x + \\frac{y}{2}$ are not zero, then\n$\\left( 2x + \\frac{y}{2} \\right)^{-1} \\left[(2x)^{-1} + \\left( \\frac{y}{2} \\right)^{-1} \\right]$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{xy}",
"style": "rule"
} | {
"index": "7835",
"split": "train"
} |
math | [
{
"content": "From the numbers 0, 1, 2, 3, and 4, select three different digits to form a three-digit number and calculate the total number of such numbers that are odd. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7836",
"split": "train"
} |
math | [
{
"content": "Given $\\overrightarrow{a}=(-3,4)$, $\\overrightarrow{b}=(5,2)$, find $|\\overrightarrow{a}|$, $|\\overrightarrow{b}|$, and $\\overrightarrow{a}\\cdot \\overrightarrow{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "7837",
"split": "train"
} |
math | [
{
"content": "Let $N = 34 \\cdot 34 \\cdot 63 \\cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1 : 14",
"style": "rule"
} | {
"index": "7838",
"split": "train"
} |
math | [
{
"content": "Points $A,B,C$ and $D$ lie on a line in that order, with $AB = CD$ and $BC = 16$. Point $E$ is not on the line, and $BE = CE = 13$. The perimeter of $\\triangle AED$ is three times the perimeter of $\\triangle BEC$. Find $AB$.\nA) $\\frac{32}{3}$\nB) $\\frac{34}{3}$\nC) $\\frac{36}{3}$\nD) $\\frac... | {
"ground_truth": "\\frac{34}{3}",
"style": "rule"
} | {
"index": "7839",
"split": "train"
} |
math | [
{
"content": "Given that the parabola $y=x^2-5x+2$ is symmetric about the point $(3,2)$ with $y=ax^2+bx+c$, find the value of $3a+3c+b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8",
"style": "rule"
} | {
"index": "7840",
"split": "train"
} |
math | [
{
"content": "In a right triangle \\(ABC\\) with a right angle at \\(B\\) and \\(\\angle A = 30^\\circ\\), a height \\(BD\\) is drawn. Then, in triangle \\(BDC\\), a median \\(DE\\) is drawn, and in triangle \\(DEC\\), an angle bisector \\(EF\\) is drawn. Find the ratio \\( \\frac{FC}{AC} \\). Please output the... | {
"ground_truth": "1/8",
"style": "rule"
} | {
"index": "7841",
"split": "train"
} |
math | [
{
"content": "If the sum of all the angles except one of a convex polygon is $2190^{\\circ}$, then the number of sides of the polygon must be Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "7842",
"split": "train"
} |
math | [
{
"content": "Calculate the value of the expression given by: $$2 + \\cfrac{3}{4 + \\cfrac{5}{6}}.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{76}{29}",
"style": "rule"
} | {
"index": "7843",
"split": "train"
} |
math | [
{
"content": "Side $AB$ of triangle $ABC$ was divided into $n$ equal parts (dividing points $B_0 = A, B_1, B_2, ..., B_n = B$ ), and side $AC$ of this triangle was divided into $(n + 1)$ equal parts (dividing points $C_0 = A, C_1, C_2, ..., C_{n+1} = C$ ). Colored are the triangles $C_iB_iC_{i+1}$ ... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7844",
"split": "train"
} |
math | [
{
"content": "Simplify the product \\[\\frac{8}{4}\\cdot\\frac{12}{8}\\cdot\\frac{16}{12} \\dotsm \\frac{4n+4}{4n} \\dotsm \\frac{2008}{2004}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "502",
"style": "rule"
} | {
"index": "7845",
"split": "train"
} |
math | [
{
"content": "Twelve chess players played a round-robin tournament. Each player then wrote 12 lists. In the first list, only the player himself was included, and in the $(k+1)$-th list, the players included those who were in the $k$-th list as well as those whom they defeated. It turned out that each player's 1... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "7846",
"split": "train"
} |
math | [
{
"content": "The quadratic \\( x^2 + 1800x + 1800 \\) can be written in the form \\( (x+b)^2 + c \\), where \\( b \\) and \\( c \\) are constants. What is \\( \\frac{c}{b} \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-898",
"style": "rule"
} | {
"index": "7847",
"split": "train"
} |
math | [
{
"content": "Find the number of triples $(x,y,z)$ of real numbers that satisfy\n\\begin{align*}\nx &= 2018 - 2019 \\operatorname{sign}(y + z), \\\\\ny &= 2018 - 2019 \\operatorname{sign}(x + z), \\\\\nz &= 2018 - 2019 \\operatorname{sign}(x + y).\n\\end{align*}Note: For a real number $a,$\n\\[\\operatorname{si... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "7848",
"split": "train"
} |
math | [
{
"content": "A thousand integer divisions are made: $2018$ is divided by each of the integers from $ 1$ to $1000$ . Thus, a thousand integer quotients are obtained with their respective remainders. Which of these thousand remainders is the bigger? Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "672",
"style": "rule"
} | {
"index": "7849",
"split": "train"
} |
math | [
{
"content": "At Beaumont High School, there are 20 players on the basketball team. All 20 players are taking at least one of biology or chemistry. (Biology and chemistry are two different science courses at the school.) If there are 8 players taking biology and 4 players are taking both sciences, how many p... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "7850",
"split": "train"
} |
math | [
{
"content": "What is half of the absolute value of the difference of the squares of 21 and 15 added to the absolute value of the difference of their cubes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3051",
"style": "rule"
} | {
"index": "7851",
"split": "train"
} |
math | [
{
"content": "Arrange 6 volunteers for 3 different tasks, each task requires 2 people. Due to the work requirements, A and B must work on the same task, and C and D cannot work on the same task. How many different arrangements are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7852",
"split": "train"
} |
math | [
{
"content": "A pentagon is inscribed around a circle, with the lengths of its sides being whole numbers, and the lengths of the first and third sides equal to 1. Into what segments does the point of tangency divide the second side? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7853",
"split": "train"
} |
math | [
{
"content": "Solve\n\\[(x^3 + 3x^2 \\sqrt{2} + 6x + 2 \\sqrt{2}) + (x + \\sqrt{2}) = 0.\\]Enter all the solutions, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\sqrt{2}, -\\sqrt{2} + i, -\\sqrt{2} - i",
"style": "rule"
} | {
"index": "7854",
"split": "train"
} |
math | [
{
"content": "Determine the number of all positive integers which cannot be written in the form $80k + 3m$ , where $k,m \\in N = \\{0,1,2,...,\\}$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "7855",
"split": "train"
} |
math | [
{
"content": "There are several balls of the same shape and size in a bag, including $a+1$ red balls, $a$ yellow balls, and $1$ blue ball. Now, randomly draw a ball from the bag, with the rule that drawing a red ball earns $1$ point, a yellow ball earns $2$ points, and a blue ball earns $3$ points. If the expec... | {
"ground_truth": "\\frac{3}{10}",
"style": "rule"
} | {
"index": "7856",
"split": "train"
} |
math | [
{
"content": "The cube shown is divided into 64 small cubes. Exactly one of the cubes is grey, as shown in the diagram. Two cubes are said to be 'neighbours' if they have a common face.\n\nOn the first day, the white neighbours of the grey cube are changed to grey. On the second day, the white neighbours of all... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7857",
"split": "train"
} |
math | [
{
"content": "\\( A, B, C \\) are positive integers. It is known that \\( A \\) has 7 divisors, \\( B \\) has 6 divisors, \\( C \\) has 3 divisors, \\( A \\times B \\) has 24 divisors, and \\( B \\times C \\) has 10 divisors. What is the minimum value of \\( A + B + C \\)? Please output the final answer within ... | {
"ground_truth": "91",
"style": "rule"
} | {
"index": "7858",
"split": "train"
} |
math | [
{
"content": "Given that $\\binom{24}{3}=2024$, $\\binom{24}{4}=10626$, and $\\binom{24}{5}=42504$, find $\\binom{26}{6}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "230230",
"style": "rule"
} | {
"index": "7859",
"split": "train"
} |
math | [
{
"content": "In quadrilateral $ABCD$, sides $\\overline{AB}$ and $\\overline{BC}$ both have length 10, sides $\\overline{CD}$ and $\\overline{DA}$ both have length 17, and the measure of angle $ADC$ is $60^\\circ$. What is the length of diagonal $\\overline{AC}$? [asy]\ndraw((0,0)--(17,0));\ndraw(rotate(301, (... | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "7860",
"split": "train"
} |
math | [
{
"content": "Compute $i^{600} + i^{599} + \\cdots + i + 1$, where $i^2=-1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "7861",
"split": "train"
} |
math | [
{
"content": "Given that $\\frac{\\cos \\alpha + \\sin \\alpha}{\\cos \\alpha - \\sin \\alpha} = 2$, find the value of $\\frac{1 + \\sin 4\\alpha - \\cos 4\\alpha}{1 + \\sin 4\\alpha + \\cos 4\\alpha}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "7862",
"split": "train"
} |
math | [
{
"content": "In the quadrilateral $MARE$ inscribed in a unit circle $\\omega,$ $AM$ is a diameter of $\\omega,$ and $E$ lies on the angle bisector of $\\angle RAM.$ Given that triangles $RAM$ and $REM$ have the same area, find the area of quadrilateral $MARE.$ Please output the final answer wi... | {
"ground_truth": "\\frac{8\\sqrt{2}}{9}",
"style": "rule"
} | {
"index": "7863",
"split": "train"
} |
math | [
{
"content": "Jenny wants to create all the six-letter words where the first two letters are the same as the last two letters. How many combinations of letters satisfy this property? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17576",
"style": "rule"
} | {
"index": "7864",
"split": "train"
} |
math | [
{
"content": "The sum of the first $n$ terms of an arithmetic sequence $\\{a_n\\}$ is $S_n$. Given that $S_{10}=0$ and $S_{15}=25$, find the minimum value of $nS_n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-49",
"style": "rule"
} | {
"index": "7865",
"split": "train"
} |
math | [
{
"content": "Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\\frac{x^{2}}{16}{+}\\frac{y^{2}}{b^{2}}{=}1$, and the line $l$ passing through $F_1$ intersects the ellipse at points $A$ and $B$. If the maximum value of $|AF_2|+|BF_2|$ is $10$, find the eccentricity of the ellipse. Please o... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7866",
"split": "train"
} |
math | [
{
"content": "What are the last three digits of \\(2003^N\\), where \\(N = 2002^{2001}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "241",
"style": "rule"
} | {
"index": "7867",
"split": "train"
} |
math | [
{
"content": "Two cards are dealt at random from a standard deck of 52 cards. What is the probability that the first card is a King and the second card is a $\\heartsuit$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\dfrac{1}{52}",
"style": "rule"
} | {
"index": "7868",
"split": "train"
} |
math | [
{
"content": "An integer between $1000$ and $9999$, inclusive, is chosen at random. What is the probability that it is an odd integer whose digits are all distinct? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{56}{225}",
"style": "rule"
} | {
"index": "7869",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively, and they satisfy the equation $$\\frac {2c-b}{a} = \\frac {\\cos{B}}{\\cos{A}}$$. If $a = 2\\sqrt {5}$, find the maximum value of $b + c$. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "4\\sqrt{5}",
"style": "rule"
} | {
"index": "7870",
"split": "train"
} |
math | [
{
"content": "Let $ABCDEF$ be a regular hexagon, and let $G$ , $H$ , $I$ , $J$ , $K$ , and $L$ be the midpoints of sides $AB$ , $BC$ , $CD$ , $DE$ , $EF$ , and $FA$ , respectively. The intersection of lines $\\overline{AH}$ , $\\overline{BI}$ , $\\overline{CJ}$ , $\\overline{DK}$ , $\\overli... | {
"ground_truth": "4/7",
"style": "rule"
} | {
"index": "7871",
"split": "train"
} |
math | [
{
"content": "Given that the positive numbers $x$ and $y$ satisfy the equation $$3x+y+ \\frac {1}{x}+ \\frac {2}{y}= \\frac {13}{2}$$, find the minimum value of $$x- \\frac {1}{y}$$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "- \\frac {1}{2}",
"style": "rule"
} | {
"index": "7872",
"split": "train"
} |
math | [
{
"content": "Consider a large square divided into a grid of \\(5 \\times 5\\) smaller squares, each with side length \\(1\\) unit. A shaded region within the large square is formed by connecting the centers of four smaller squares, creating a smaller square inside. Calculate the ratio of the area of the shaded... | {
"ground_truth": "\\frac{2}{25}",
"style": "rule"
} | {
"index": "7873",
"split": "train"
} |
math | [
{
"content": "If the complex number \\( z \\) satisfies \\( |z| = 2 \\), then the maximum value of \\( \\frac{\\left|z^{2}-z+1\\right|}{|2z-1-\\sqrt{3}i|} \\) is _______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{2}",
"style": "rule"
} | {
"index": "7874",
"split": "train"
} |
math | [
{
"content": "Xiaoming started adding the page numbers of a book from page 1, sequentially until the end, and obtained a sum of 4979. Later, he discovered that one sheet (which includes two consecutive pages) was missing from the book. How many pages did the book originally have? Please output the final answer ... | {
"ground_truth": "100",
"style": "rule"
} | {
"index": "7875",
"split": "train"
} |
math | [
{
"content": "Find the integer $n,$ $-180 \\le n \\le 180,$ such that $\\cos n^\\circ = \\cos 430^\\circ.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-70",
"style": "rule"
} | {
"index": "7876",
"split": "train"
} |
math | [
{
"content": "Given that x > 0, y > 0, and x + 2y = 4, find the minimum value of $$\\frac {(x+1)(2y+1)}{xy}$$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {9}{2}",
"style": "rule"
} | {
"index": "7877",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $C$: $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), with the circle centered at the right focus $F$ of $C$ ($c$, $0$) and with radius $a$ intersects one of the asymptotes of $C$ at points $A$ and $B$. If $|AB| = \\frac{2}{3}c$, determine the eccentricity of the h... | {
"ground_truth": "\\frac{3\\sqrt{5}}{5}",
"style": "rule"
} | {
"index": "7878",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)=\\frac{1}{2}x^{2}-a\\ln x+b$ where $a\\in R$.\n(I) If the equation of the tangent line to the curve $y=f(x)$ at $x=1$ is $3x-y-3=0$, find the values of the real numbers $a$ and $b$.\n(II) If $x=1$ is the extreme point of the function $f(x)$, find the value of the real numb... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7879",
"split": "train"
} |
math | [
{
"content": "Define the determinant operation $\\begin{vmatrix} a_{1} & a_{2} \\\\ b_{1} & b_{2}\\end{vmatrix} =a_{1}b_{2}-a_{2}b_{1}$, and consider the function $f(x)= \\begin{vmatrix} \\sqrt {3} & \\sin x \\\\ 1 & \\cos x\\end{vmatrix}$. If the graph of this function is translated to the left by $t(t > 0)$ u... | {
"ground_truth": "\\dfrac{5\\pi}{6}",
"style": "rule"
} | {
"index": "7880",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, given $A(1,4)$, $B(4,1)$, $C(0,-4)$, find the minimum value of $\\overrightarrow{PA} \\cdot \\overrightarrow{PB} + \\overrightarrow{PB} \\cdot \\overrightarrow{PC} + \\overrightarrow{PC} \\cdot \\overrightarrow{PA}$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "- \\dfrac {62}{3}",
"style": "rule"
} | {
"index": "7881",
"split": "train"
} |
math | [
{
"content": "In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \\cdot B \\cdot C = 2023$. What is the largest possible value of the sum $A+B+C$? Please output the final answer within \\boxed{}.",
"role": "user"
}... | {
"ground_truth": "297",
"style": "rule"
} | {
"index": "7882",
"split": "train"
} |
math | [
{
"content": "A point has rectangular coordinates $(-5,-7,4)$ and spherical coordinates $(\\rho, \\theta, \\phi).$ Find the rectangular coordinates of the point with spherical coordinates $(\\rho, \\theta, -\\phi).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(5,7,4)",
"style": "rule"
} | {
"index": "7883",
"split": "train"
} |
math | [
{
"content": "The expression $\\dfrac{\\sqrt[3]{5}}{\\sqrt[5]{5}}$ equals 5 raised to what power? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2/15",
"style": "rule"
} | {
"index": "7884",
"split": "train"
} |
math | [
{
"content": "The shortest distance from a point on the curve $y=\\ln x$ to the line $y=x+2$ is what value? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "7885",
"split": "train"
} |
math | [
{
"content": "The diagram shows a shape made from ten squares of side-length \\(1 \\mathrm{~cm}\\), joined edge to edge. What is the length of its perimeter, in centimetres?\nA) 14\nB) 18\nC) 30\nD) 32\nE) 40 Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "7886",
"split": "train"
} |
math | [
{
"content": "Given $f(x) = \\frac {\\log_{2}x-1}{2\\log_{2}x+1}$ (where $x > 2$), and $f(x_1) + f(2x_2) = \\frac {1}{2}$, find the minimum value of $f(x_1x_2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {1}{3}",
"style": "rule"
} | {
"index": "7887",
"split": "train"
} |
math | [
{
"content": "Let \\( f : \\mathbb{C} \\to \\mathbb{C} \\) be defined by \\( f(z) = z^2 - 2iz + 2 \\). Determine how many complex numbers \\( z \\) exist such that \\( \\text{Im}(z) > 0 \\) and both the real and the imaginary parts of \\( f(z) \\) are integers within \\( |a|, |b| \\leq 5 \\). Please output the ... | {
"ground_truth": "110",
"style": "rule"
} | {
"index": "7888",
"split": "train"
} |
math | [
{
"content": "Given that \\( F_1 \\) and \\( F_2 \\) are the left and right foci of the ellipse \\( C: \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) where \\( a > b > 0 \\), and \\( P \\) is a point on the ellipse \\( C \\). The incenter of triangle \\( \\triangle F_{1}PF_{2} \\) is \\( I \\). If there exists... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "7889",
"split": "train"
} |
math | [
{
"content": "Let $a_{10} = 10$, and for each positive integer $n >10$ let $a_n = 100a_{n - 1} + n$. Find the least positive $n > 10$ such that $a_n$ is a multiple of $99$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "7890",
"split": "train"
} |
math | [
{
"content": "What is the sum of the reciprocals of the natural-number factors of 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "7891",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\\frac{MO}{MF}$ is _______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "7892",
"split": "train"
} |
math | [
{
"content": "For a positive number $x$, define $f(x)=\\frac{x}{x+1}$. For example, $f(1)=\\frac{1}{1+1}=\\frac{1}{2}$, $f(2)=\\frac{2}{2+1}=\\frac{2}{3}$, $f(\\frac{1}{2})=\\frac{\\frac{1}{2}}{\\frac{1}{2}+1}=\\frac{1}{3}$. \n$(1)$ Find the value of: $f(3)+f(\\frac{1}{3})=$______; $f(4)+f(\\frac{1}{4})=$_____... | {
"ground_truth": "2022.5",
"style": "rule"
} | {
"index": "7893",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x)$ and its derivative $f''(x)$ on $\\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \\leqslant f(-a)+2a+1$. Please output the final answer within \\b... | {
"ground_truth": "-\\dfrac {1}{2}",
"style": "rule"
} | {
"index": "7894",
"split": "train"
} |
math | [
{
"content": "Find all pairs of distinct rational numbers $(a,b)$ such that $a^a=b^b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left(\\left(\\frac{u}{v}\\right)^{\\frac{u}{v-u}}, \\left(\\frac{u}{v}\\right)^{\\frac{v}{v-u}}\\right)",
"style": "rule"
} | {
"index": "7895",
"split": "train"
} |
math | [
{
"content": "Onkon wants to cover his room's floor with his favourite red carpet. How many square yards of red carpet are required to cover a rectangular floor that is $12$ feet long and $9$ feet wide? (There are 3 feet in a yard.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "7896",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC$, $a=3$, $\\angle C = \\frac{2\\pi}{3}$, and the area of $ABC$ is $\\frac{3\\sqrt{3}}{4}$. Find the lengths of sides $b$ and $c$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{13}",
"style": "rule"
} | {
"index": "7897",
"split": "train"
} |
math | [
{
"content": "\"The Nine Chapters on the Mathematical Art\" is an ancient Chinese mathematical text, which records: \"If it can be halved, then halve it; if not, juxtapose the numerator and denominator, subtract the lesser from the greater, continue to subtract in turn, seeking their equality. Use the equal num... | {
"ground_truth": "273",
"style": "rule"
} | {
"index": "7898",
"split": "train"
} |
math | [
{
"content": "Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{500}{1001}",
"style": "rule"
} | {
"index": "7899",
"split": "train"
} |
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