data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Solve in $ \\mathbb{Z}^2 $ the equation: $ x^2\\left( 1+x^2 \\right) =-1+21^y. $ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(0, 0), (2, 1), (-2, 1)",
"style": "rule"
} | {
"index": "4400",
"split": "train"
} |
math | [
{
"content": "The area of polygon $ABCDEF$, in square units, is Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "46",
"style": "rule"
} | {
"index": "4401",
"split": "train"
} |
math | [
{
"content": "Let $P$ be a $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent... | {
"ground_truth": "2018",
"style": "rule"
} | {
"index": "4402",
"split": "train"
} |
math | [
{
"content": "The distance between two parallel lines $l_1: 3x + 4y - 2 = 0$ and $l_2: ax + 6y = 5$ is _______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{4}{15}",
"style": "rule"
} | {
"index": "4403",
"split": "train"
} |
math | [
{
"content": "Compute \\[\\sum_{k=2}^{31} \\log_3\\left(1 + \\frac{2}{k}\\right) \\log_k 3 \\log_{k+1} 3.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{0.5285}{3}",
"style": "rule"
} | {
"index": "4404",
"split": "train"
} |
math | [
{
"content": "The sequence $(x_n)$ is defined by $x_1 = 115$ and $x_k = x_{k - 1}^2 + x_{k - 1}$ for all $k \\ge 2.$ Compute\n\\[\\frac{1}{x_1 + 1} + \\frac{1}{x_2 + 1} + \\frac{1}{x_3 + 1} + \\dotsb.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{115}",
"style": "rule"
} | {
"index": "4405",
"split": "train"
} |
math | [
{
"content": "A reconnaissance team has 12 soldiers, including 3 radio operators. The 12 soldiers are randomly divided into three groups, with group sizes of 3, 4, and 5 soldiers, respectively. What is the probability that each group has exactly 1 radio operator? Please output the final answer within \\boxed{}.... | {
"ground_truth": "3/11",
"style": "rule"
} | {
"index": "4406",
"split": "train"
} |
math | [
{
"content": "A polynomial \\( P \\) is of the form \\( \\pm x^{6} \\pm x^{5} \\pm x^{4} \\pm x^{3} \\pm x^{2} \\pm x \\pm 1 \\). Given that \\( P(2)=27 \\), what is \\( P(3) \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "439",
"style": "rule"
} | {
"index": "4407",
"split": "train"
} |
math | [
{
"content": "In January 1859, an eight-year-old boy dropped a newly-hatched eel into a well in Sweden. The eel, named Ale, finally died in August 2014. How many years old was Åle when it died? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "155",
"style": "rule"
} | {
"index": "4408",
"split": "train"
} |
math | [
{
"content": "The sum of an infinite geometric series is $81$ times the series that results if the first five terms of the original series are removed. What is the value of the series' common ratio? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{3}",
"style": "rule"
} | {
"index": "4409",
"split": "train"
} |
math | [
{
"content": "What is the total volume in cubic feet of three boxes if each box is a cube with edge length 4 feet? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "192",
"style": "rule"
} | {
"index": "4410",
"split": "train"
} |
math | [
{
"content": "a) What is the maximum number of squares on an $8 \\times 8$ board that can be painted black such that in each corner of three squares, there is at least one unpainted square?\n\nb) What is the minimum number of squares on an $8 \\times 8$ board that need to be painted black such that in each corn... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "4411",
"split": "train"
} |
math | [
{
"content": "Given that $\\alpha$ and $\\beta$ are acute angles, $\\cos\\alpha=\\frac{{\\sqrt{5}}}{5}$, $\\cos({\\alpha-\\beta})=\\frac{3\\sqrt{10}}{10}$, find the value of $\\cos \\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\sqrt{2}}{10}",
"style": "rule"
} | {
"index": "4412",
"split": "train"
} |
math | [
{
"content": "Given the sequence $\\{a_n\\}$ where $a_1 = \\frac{1}{2}$ and $a_{n+1} = \\frac{1+a_n}{1-a_n}$ for $n \\in N^*$, find the smallest value of $n$ such that $a_1+a_2+a_3+…+a_n \\geqslant 72$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "238",
"style": "rule"
} | {
"index": "4413",
"split": "train"
} |
math | [
{
"content": "From Moscow to city \\( N \\), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \\( N \\) in 10 hours, including the waiting time. By how many times is the plane’s speed greater... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4414",
"split": "train"
} |
math | [
{
"content": "Evaluate the infinite geometric series: $$\\frac{1}{3}+\\frac{1}{6}+\\frac{1}{12}+\\frac{1}{24}+\\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{3}",
"style": "rule"
} | {
"index": "4415",
"split": "train"
} |
math | [
{
"content": "The radian measure of -150° is equal to what fraction of π. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "4416",
"split": "train"
} |
math | [
{
"content": "In acute triangle $\\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is given that $a-4\\cos C=c\\cos B$. \n$(1)$ Find the value of $b$; \n$(2)$ If $a^{2}+b^{2}+c^{2}=2\\sqrt{3}ab\\sin C$, find the area of $\\triangle ABC$. Please outp... | {
"ground_truth": "4\\sqrt{3}",
"style": "rule"
} | {
"index": "4417",
"split": "train"
} |
math | [
{
"content": "Given that the complex number \\( z \\) satisfies \\( |z|=1 \\), find the maximum value of \\( \\left| z^3 - 3z - 2 \\right| \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3\\sqrt{3}",
"style": "rule"
} | {
"index": "4418",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "676",
"style": "rule"
} | {
"index": "4419",
"split": "train"
} |
math | [
{
"content": "Masha talked a lot on the phone with her friends, and the charged battery discharged exactly after a day. It is known that the charge lasts for 5 hours of talk time or 150 hours of standby time. How long did Masha talk with her friends? Please output the final answer within \\boxed{}.",
"role"... | {
"ground_truth": "126/29",
"style": "rule"
} | {
"index": "4420",
"split": "train"
} |
math | [
{
"content": "If $x=11$, $y=-8$, and $2x-3z=5y$, what is the value of $z$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{62}{3}",
"style": "rule"
} | {
"index": "4421",
"split": "train"
} |
math | [
{
"content": "The number of intersection points between the graphs of the functions \\( y = \\sin x \\) and \\( y = \\log_{2021} |x| \\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1286",
"style": "rule"
} | {
"index": "4422",
"split": "train"
} |
math | [
{
"content": "Find all functions $ f: \\mathbb{Q}^{\\plus{}} \\mapsto \\mathbb{Q}^{\\plus{}}$ such that: \n\\[ f(x) \\plus{} f(y) \\plus{} 2xy f(xy) \\equal{} \\frac {f(xy)}{f(x\\plus{}y)}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{x^2}",
"style": "rule"
} | {
"index": "4423",
"split": "train"
} |
math | [
{
"content": "I am preparing for a French exam, which includes recalling vocabulary. There are 800 words in total on the exam, and the exam grade is based on the percentage of these words that I recall correctly. If I randomly guess the words I do not learn, my guessing has a 5% chance of being correct. What is... | {
"ground_truth": "716",
"style": "rule"
} | {
"index": "4424",
"split": "train"
} |
math | [
{
"content": "If the letters in the English word \"good\" are written in the wrong order, then the total number of possible mistakes is _____ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "4425",
"split": "train"
} |
math | [
{
"content": "Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \\in \\mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x... | {
"ground_truth": "{r=\\frac{a+1}a,a\\leq1010}",
"style": "rule"
} | {
"index": "4426",
"split": "train"
} |
math | [
{
"content": "In the Cartesian coordinate system, if the terminal side of angle α passes through point P(sin (5π/3), cos (5π/3)), evaluate sin(π+α). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "4427",
"split": "train"
} |
math | [
{
"content": "Each of the integers \\(1, 2, 3, \\ldots, 9\\) is assigned to a vertex of a regular 9-sided polygon (every vertex receives exactly one integer from \\(\\{1, 2, \\ldots, 9\\}\\), and no two vertices receive the same integer) so that the sum of the integers assigned to any three consecutive vertices... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "4428",
"split": "train"
} |
math | [
{
"content": "Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in 3, Anne and Carl in 5. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back? Express your answer as a fraction in lowest terms. Please output the final ... | {
"ground_truth": "\\frac{59}{20}",
"style": "rule"
} | {
"index": "4429",
"split": "train"
} |
math | [
{
"content": "Find $x$ if $\\log_x32 = \\dfrac{5}{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "4430",
"split": "train"
} |
math | [
{
"content": "Let \\(x_{1}, x_{2}, \\ldots, x_{200}\\) be natural numbers greater than 2 (not necessarily distinct). In a \\(200 \\times 200\\) table, the numbers are arranged as follows: at the intersection of the \\(i\\)-th row and the \\(k\\)-th column, the number \\(\\log _{x_{k}} \\frac{x_{i}}{9}\\) is wri... | {
"ground_truth": "-40000",
"style": "rule"
} | {
"index": "4431",
"split": "train"
} |
math | [
{
"content": "Let the sequence $\\{a_n\\}$ satisfy $a_1=2$, $a_2=6$, and $a_{n+2}-2a_{n+1}+a_n=2$. If $\\lfloor x \\rfloor$ represents the greatest integer not exceeding $x$, calculate $$\\left\\lfloor \\frac{2017}{a_1} + \\frac{2017}{a_2} + \\ldots + \\frac{2017}{a_{2017}} \\right\\rfloor.$$ Please output the ... | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "4432",
"split": "train"
} |
math | [
{
"content": "If each of the four numbers $3, 4, 6,$ and $7$ replaces a $\\square$, what is the largest possible sum of the fractions shown? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{23}{6}$",
"style": "rule"
} | {
"index": "4433",
"split": "train"
} |
math | [
{
"content": "Given that $A$, $B$, and $C$ are noncollinear points in the plane with integer coordinates\nsuch that the distances $AB$, $AC$, and $BC$ are integers, what is the smallest possible value of $AB$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "4434",
"split": "train"
} |
math | [
{
"content": "Mrs. Kučerová was on a seven-day vacation, and Káta walked her dog and fed her rabbits during this time. Káta received a large cake and 700 CZK as compensation. After another vacation, this time lasting four days, Káta received the same cake and 340 CZK for the same tasks.\n\nWhat was the cost of ... | {
"ground_truth": "140",
"style": "rule"
} | {
"index": "4435",
"split": "train"
} |
math | [
{
"content": "The positive integers are grouped as follows:\n\n\\( A_1 = \\{1\\}, A_2 = \\{2, 3, 4\\}, A_3 = \\{5, 6, 7, 8, 9\\} \\), and so on.\n\nIn which group does 2009 belong? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "45",
"style": "rule"
} | {
"index": "4436",
"split": "train"
} |
math | [
{
"content": "On the game show $\\text{\\emph{Wheel of Fortunes}}$, you encounter a spinner divided equally into 6 regions labeled as follows: \"Bankrupt,\" \"$600\", \"$100\", \"$2000\", \"$150\", \"$700\". What is the probability that you will accumulate exactly $1450$ in your first three spins without landin... | {
"ground_truth": "\\frac{6}{125}",
"style": "rule"
} | {
"index": "4437",
"split": "train"
} |
math | [
{
"content": "Circle $A$ has its center at $A(4, 4)$ and has a radius of 4 units. Circle $B$ has its center at $B(12, 4)$ and has a radius of 4 units. What is the area of the gray region bound by the circles and the $x$-axis? Express your answer in terms of $\\pi$. [asy]\nimport olympiad; size(150); defaultpen(... | {
"ground_truth": "32 - 8\\pi",
"style": "rule"
} | {
"index": "4438",
"split": "train"
} |
math | [
{
"content": "Let $f(x)$ and $g(x)$ be two monic cubic polynomials, and let $r$ be a real number. Two of the roots of $f(x)$ are $r + 1$ and $r + 7.$ Two of the roots of $g(x)$ are $r + 3$ and $r + 9,$ and\n\\[f(x) - g(x) = r\\]for all real numbers $x.$ Find $r.$ Please output the final answer within \\boxed... | {
"ground_truth": "32",
"style": "rule"
} | {
"index": "4439",
"split": "train"
} |
math | [
{
"content": "Given $$\\frac{\\pi}{2} < \\alpha < \\pi$$ and $$0 < \\beta < \\frac{\\pi}{2}$$, with $\\tan\\alpha = -\\frac{3}{4}$ and $\\cos(\\beta - \\alpha) = \\frac{5}{13}$, find the value of $\\sin\\beta$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{63}{65}",
"style": "rule"
} | {
"index": "4440",
"split": "train"
} |
math | [
{
"content": "Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6480/2401",
"style": "rule"
} | {
"index": "4441",
"split": "train"
} |
math | [
{
"content": "Let \\( x, y, z \\) be positive real numbers such that \\( xyz = 3 \\). Compute the minimum value of\n\\[ x^2 + 4xy + 12y^2 + 8yz + 3z^2. \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "162",
"style": "rule"
} | {
"index": "4442",
"split": "train"
} |
math | [
{
"content": "In the diagram, $D$ and $E$ are the midpoints of $\\overline{AB}$ and $\\overline{BC}$ respectively. Determine the area of $\\triangle DBC$.\n\n[asy]\nsize(180); defaultpen(linewidth(.7pt)+fontsize(10pt));\npair A, B, C, D, E, F;\nA=(0,6);\nB=(0,0);\nC=(8,0);\nD=(0,3);\nE=(4,0);\nF=(8/3,2);\ndraw... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4443",
"split": "train"
} |
math | [
{
"content": "How many even integers are there between \\( \\frac{12}{3} \\) and \\( \\frac{50}{2} \\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4444",
"split": "train"
} |
math | [
{
"content": "A sector with central angle $60^\\circ$ is cut from a circle of radius 10. Determine both the radius of the circle circumscribed about the sector and the inradius of the triangle formed by the sector. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "4445",
"split": "train"
} |
math | [
{
"content": "A standard die is rolled eight times. What is the probability that the product of all eight rolls is divisible by 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{247}{256}",
"style": "rule"
} | {
"index": "4446",
"split": "train"
} |
math | [
{
"content": "Find all real values of $x$ that satisfy $\\frac{x(x+1)}{(x-4)^2} \\ge 12.$ (Give your answer in interval notation.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "[3, 4) \\cup \\left(4, \\frac{64}{11}\\right]",
"style": "rule"
} | {
"index": "4447",
"split": "train"
} |
math | [
{
"content": "Suppose 5 different integers are randomly chosen from between 20 and 69, inclusive. What is the probability that they each have a different tens digit? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2500}{52969}",
"style": "rule"
} | {
"index": "4448",
"split": "train"
} |
math | [
{
"content": "What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "68",
"style": "rule"
} | {
"index": "4449",
"split": "train"
} |
math | [
{
"content": "Given $α∈(\\frac{\\pi}{2},π)$, $\\sin α =\\frac{\\sqrt{5}}{5}$.\n\n(Ⅰ) Find the value of $\\tan\\left( \\frac{\\pi}{4}+2α \\right)$;\n\n(Ⅱ) Find the value of $\\cos\\left( \\frac{5\\pi}{6}-2α \\right)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{3\\sqrt{3}+4}{10}",
"style": "rule"
} | {
"index": "4450",
"split": "train"
} |
math | [
{
"content": "Given that $a \\in \\mathbb{R}$, if the real part and the imaginary part of the complex number $\\frac{a + i}{1 + i}$ (where $i$ is the imaginary unit) are equal, then $\\_\\_\\_\\_\\_\\_$, $| \\overline{z}| = \\_\\_\\_\\_\\_\\_$. Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "4451",
"split": "train"
} |
math | [
{
"content": "Harry has 3 sisters and 5 brothers. His sister Harriet has $\\text{S}$ sisters and $\\text{B}$ brothers. What is the product of $\\text{S}$ and $\\text{B}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4452",
"split": "train"
} |
math | [
{
"content": "The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $4$. Another die's faces are marked with $2$, $2$, $5$, $6$, $7$, and $8$. Find the probability that the sum of the top two numbers will be $7$, $9$, or $11$.\nA) $\\frac{1}{18}$\nB) $\\frac{1}{9}$\nC) $\\frac{1}{3... | {
"ground_truth": "\\frac{4}{9}",
"style": "rule"
} | {
"index": "4453",
"split": "train"
} |
math | [
{
"content": "Solve for $z$ in the following equation: $2-iz = -1 + 3iz$.\n\nExpress your answer in standard form. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac34i",
"style": "rule"
} | {
"index": "4454",
"split": "train"
} |
math | [
{
"content": "Calculate the double integrals and change the order of integration.\n\n$$\n\\text{a) } \\int_{0}^{2} dx \\int_{0}^{3}\\left(x^{2} + 2xy\\right) dy\n$$\n\nb) $\\int_{-2}^{0} dy \\int_{0}^{y^{2}}(x+2y) dx$\n\nc) $\\int_{0}^{5} dx \\int_{0}^{5-x} \\sqrt{4 + x + y} dy$ Please output the final answer w... | {
"ground_truth": "\\frac{506}{3}",
"style": "rule"
} | {
"index": "4455",
"split": "train"
} |
math | [
{
"content": "A point $P$ is outside a circle and is $13$ inches from the center. A secant from $P$ cuts the circle at $Q$ and $R$ so that the external segment of the secant $PQ$ is $9$ inches and $QR$ is $7$ inches. The radius of the circle is: Please output the final answer within \\boxed{}.",
"role": "us... | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "4456",
"split": "train"
} |
math | [
{
"content": "For any positive real numbers \\(a\\) and \\(b\\), define \\(a \\circ b=a+b+2 \\sqrt{a b}\\). Find all positive real numbers \\(x\\) such that \\(x^{2} \\circ 9x=121\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{31-3\\sqrt{53}}{2}",
"style": "rule"
} | {
"index": "4457",
"split": "train"
} |
math | [
{
"content": "The asymptotes of a hyperbola are $y = x + 1$ and $y = 3 - x.$ Also, the hyperbola passes through $(3,3).$ Find the distance between the foci of the hyperbola. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 \\sqrt{6}",
"style": "rule"
} | {
"index": "4458",
"split": "train"
} |
math | [
{
"content": "Find the arithmetic square root of $4$, the square root of $5$, and the cube root of $-27$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "4459",
"split": "train"
} |
math | [
{
"content": "Determine the number of solutions to\n\\[2\\sin^3 x - 5 \\sin^2 x + 2 \\sin x = 0\\]in the range $0 \\le x \\le 2 \\pi.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "4460",
"split": "train"
} |
math | [
{
"content": "Complex numbers $p$, $q$, and $r$ form an equilateral triangle with side length 24 in the complex plane. If $|p + q + r| = 48$, find $|pq + pr + qr|$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "768",
"style": "rule"
} | {
"index": "4461",
"split": "train"
} |
math | [
{
"content": "Four friends rent a cottage for a total of £300 for the weekend. The first friend pays half of the sum of the amounts paid by the other three friends. The second friend pays one third of the sum of the amounts paid by the other three friends. The third friend pays one quarter of the sum of the amo... | {
"ground_truth": "65",
"style": "rule"
} | {
"index": "4462",
"split": "train"
} |
math | [
{
"content": "A line has equation $y=mx-50$ for some positive integer $m$. The line passes through the point $(a, 0)$ for some positive integer $a$. What is the sum of all possible values of $m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "93",
"style": "rule"
} | {
"index": "4463",
"split": "train"
} |
math | [
{
"content": "Consider a sequence $\\{a_n\\}$ where the sum of the first $n$ terms, $S_n$, satisfies $S_n = 2a_n - a_1$, and $a_1$, $a_2 + 1$, $a_3$ form an arithmetic sequence.\n\n(1) Find the general formula for the sequence $\\{a_n\\}$.\n(2) Let $b_n = \\log_2 a_n$ and $c_n = \\frac{3}{b_nb_{n+1}}$. Denote t... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "4464",
"split": "train"
} |
math | [
{
"content": "Two circles \\(C_{1}\\) and \\(C_{2}\\) touch each other externally and the line \\(l\\) is a common tangent. The line \\(m\\) is parallel to \\(l\\) and touches the two circles \\(C_{1}\\) and \\(C_{3}\\). The three circles are mutually tangent. If the radius of \\(C_{2}\\) is 9 and the radius of... | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4465",
"split": "train"
} |
math | [
{
"content": "A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "98",
"style": "rule"
} | {
"index": "4466",
"split": "train"
} |
math | [
{
"content": "Calculate the result of the expression \\((19 \\times 19 - 12 \\times 12) \\div \\left(\\frac{19}{12} - \\frac{12}{19}\\right)\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "228",
"style": "rule"
} | {
"index": "4467",
"split": "train"
} |
math | [
{
"content": "In $\\triangle{ABC}, AB=13, \\angle{A}=45^\\circ$, and $\\angle{C}=30^\\circ$. Let $H, D,$ and $M$ be points on the line $BC$ such that $AH\\perp{BC}$, $\\angle{BAD}=\\angle{CAD}$, and $BM=CM$. Point $N$ is the midpoint of the segment $HM$, and point $P$ is on ray $AD$ such that $PN\\perp{BC}$. Fi... | {
"ground_truth": "171",
"style": "rule"
} | {
"index": "4468",
"split": "train"
} |
math | [
{
"content": "In triangle \\( ABC \\), let \\( E \\) be the point where the side \\( AC \\) is divided into quarters closest to \\( C \\), and let \\( F \\) be the midpoint of side \\( BC \\). The line passing through points \\( E \\) and \\( F \\) intersects line \\( AB \\) at point \\( D \\). What percentage ... | {
"ground_truth": "112.5",
"style": "rule"
} | {
"index": "4469",
"split": "train"
} |
math | [
{
"content": "How many triangles are there with all sides being integers and the longest side being 11? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "4470",
"split": "train"
} |
math | [
{
"content": "To take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the probability that Margo is paired with her best friend, Irma? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{1}{29}",
"style": "rule"
} | {
"index": "4471",
"split": "train"
} |
math | [
{
"content": "Evaluate \\[(3a^3 - 7a^2 + a - 5)(4a - 6)\\] for $a = 2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "4472",
"split": "train"
} |
math | [
{
"content": "Given a regular quadrilateral pyramid $S-ABCD$, with a base side length of $2$ and a volume of $\\frac{{4\\sqrt{3}}}{3}$, the length of the lateral edge of this quadrilateral pyramid is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{5}",
"style": "rule"
} | {
"index": "4473",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c}$ be vectors such that $\\|\\mathbf{a}\\| = 1,$ $\\|\\mathbf{b}\\| = 5,$ $\\|\\mathbf{c}\\| = 3,$ and\n\\[\\mathbf{a} \\times (\\mathbf{a} \\times \\mathbf{b}) = \\mathbf{c}.\\]If $\\theta$ is the angle between $\\mathbf{a}$ and $\\mathbf{b},$ then fin... | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "4474",
"split": "train"
} |
math | [
{
"content": "Segments $AD=10$, $BE=6$, $CF=24$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn... | {
"ground_truth": "\\frac{40}{3}",
"style": "rule"
} | {
"index": "4475",
"split": "train"
} |
math | [
{
"content": "In the adjoining figure triangle $ABC$ is such that $AB = 4$ and $AC = 8$. IF $M$ is the midpoint of $BC$ and $AM = 3$, what is the length of $BC$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\sqrt{31}",
"style": "rule"
} | {
"index": "4476",
"split": "train"
} |
math | [
{
"content": "Let \\(\\mathbf{u}, \\mathbf{v},\\) and \\(\\mathbf{w}\\) be distinct non-zero vectors such that no two are parallel. The vectors are related via:\n\\[(\\mathbf{u} \\times \\mathbf{v}) \\times \\mathbf{w} = \\frac{1}{2} \\|\\mathbf{v}\\| \\|\\mathbf{w}\\| \\mathbf{u}.\\]\nLet \\(\\phi\\) be the an... | {
"ground_truth": "\\frac{\\sqrt{3}}{2}",
"style": "rule"
} | {
"index": "4477",
"split": "train"
} |
math | [
{
"content": "$\\frac{1000^2}{252^2-248^2}$ equals Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "500",
"style": "rule"
} | {
"index": "4478",
"split": "train"
} |
math | [
{
"content": "How many times does the digit 9 appear in the list of all integers from 1 to 1000? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "300",
"style": "rule"
} | {
"index": "4479",
"split": "train"
} |
math | [
{
"content": "Find all real numbers $x$ satisfying the equation $x^{3}-8=16 \\sqrt[3]{x+1}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2,1 \\pm \\sqrt{5}",
"style": "rule"
} | {
"index": "4480",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a positive common difference $d > 0$, and $a_2$, $a_5 - 1$, $a_{10}$ form a geometric sequence. If $a_1 = 5$, and $S_n$ is the sum of the first $n$ terms of the sequence $\\{a_n\\}$, determine the minimum value of $$\\frac{2S_{n} + n + 32}{a_{n} + 1}$$... | {
"ground_truth": "\\frac{20}{3}",
"style": "rule"
} | {
"index": "4481",
"split": "train"
} |
math | [
{
"content": "Given angles $\\alpha$ and $\\beta$ have their vertices at the origin of coordinates, and their initial sides coincide with the positive half-axis of the x-axis, $\\alpha, \\beta \\in (0, \\pi)$. The terminal side of angle $\\beta$ intersects the unit circle at a point whose x-coordinate is $-\\fr... | {
"ground_truth": "\\frac{56}{65}",
"style": "rule"
} | {
"index": "4482",
"split": "train"
} |
math | [
{
"content": "Find the sum of all real numbers $x$ such that $5 x^{4}-10 x^{3}+10 x^{2}-5 x-11=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "4483",
"split": "train"
} |
math | [
{
"content": "The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The te... | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "4484",
"split": "train"
} |
math | [
{
"content": "Find the smallest natural number ending in the digit 4 that becomes 4 times larger when its last digit is moved to the beginning of the number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "102564",
"style": "rule"
} | {
"index": "4485",
"split": "train"
} |
math | [
{
"content": "In the sequence $1,2,1,2,2,1,2,2,2,1,2,2,2,2,1,2, \\cdots$ where the number of 2s between consecutive 1s increases by 1 each time, what is the sum of the first 1234 terms? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2419",
"style": "rule"
} | {
"index": "4486",
"split": "train"
} |
math | [
{
"content": "A rectangular prism has edges $a=b=8$ units and $c=27$ units. Divide the prism into four parts from which a cube can be assembled. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "4487",
"split": "train"
} |
math | [
{
"content": "Given a polynomial \\( P(x) \\) with integer coefficients. It is known that \\( P(1) = 2013 \\), \\( P(2013) = 1 \\), and \\( P(k) = k \\), where \\( k \\) is some integer. Find \\( k \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1007",
"style": "rule"
} | {
"index": "4488",
"split": "train"
} |
math | [
{
"content": "Simplify and write the result as a common fraction: $$\\sqrt[4]{\\sqrt[3]{\\sqrt{\\frac{1}{65536}}}}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{2^{\\frac{2}{3}}}",
"style": "rule"
} | {
"index": "4489",
"split": "train"
} |
math | [
{
"content": "Two arithmetic sequences $C$ and $D$ both begin with 50. Sequence $C$ has a common difference of 12 and is increasing, while sequence $D$ has a common difference of 8 and is decreasing. What is the absolute value of the difference between the 31st term of sequence $C$ and the 31st term of sequence... | {
"ground_truth": "600",
"style": "rule"
} | {
"index": "4490",
"split": "train"
} |
math | [
{
"content": "A ray of light passing through the point $A = (-3,9,11),$ reflects off the plane $x + y + z = 12$ at $B,$ and then passes through the point $C = (3,5,9).$ Find the point $B.$\n\n[asy]\nimport three;\n\nsize(180);\ncurrentprojection = perspective(6,3,2);\n\ntriple A, B, C;\n\nA = (0,-0.5,0.5*1.5);... | {
"ground_truth": "\\left( -\\frac{5}{3}, \\frac{16}{3}, \\frac{25}{3} \\right)",
"style": "rule"
} | {
"index": "4491",
"split": "train"
} |
math | [
{
"content": "What is the ratio of $x$ to $y$ if: $\\frac{10x-3y}{13x-2y} = \\frac{3}{5}$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{9}{11}",
"style": "rule"
} | {
"index": "4492",
"split": "train"
} |
math | [
{
"content": "Let $(F_n)$ be the sequence defined recursively by $F_1=F_2=1$ and $F_{n+1}=F_n+F_{n-1}$ for $n\\geq 2$. Find all pairs of positive integers $(x,y)$ such that\n$$5F_x-3F_y=1.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(2,3);(5,8);(8,13)",
"style": "rule"
} | {
"index": "4493",
"split": "train"
} |
math | [
{
"content": "The transformation $T,$ taking vectors to vectors, has the following properties:\n\n(i) $T(a \\mathbf{v} + b \\mathbf{w}) = a T(\\mathbf{v}) + b T(\\mathbf{w})$ for all vectors $\\mathbf{v}$ and $\\mathbf{w},$ and for all scalars $a$ and $b.$\n(ii) $T(\\mathbf{v} \\times \\mathbf{w}) = T(\\mathbf{... | {
"ground_truth": "\\begin{pmatrix} 7 \\\\ 8 \\\\ 11 \\end{pmatrix}",
"style": "rule"
} | {
"index": "4494",
"split": "train"
} |
math | [
{
"content": "Find the number of six-digit palindromes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9000",
"style": "rule"
} | {
"index": "4495",
"split": "train"
} |
math | [
{
"content": "Nathan and his two younger twin sisters' ages multiply to 72. Find the sum of their three ages. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "4496",
"split": "train"
} |
math | [
{
"content": "Given $x < 0$, the maximum value of $3x +\\dfrac{4}{x}$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-4\\sqrt{3}",
"style": "rule"
} | {
"index": "4497",
"split": "train"
} |
math | [
{
"content": "Calculate the sum of the $2023$ roots of $(x-1)^{2023} + 2(x-2)^{2022} + 3(x-3)^{2021} + \\cdots + 2022(x-2022)^2 + 2023(x-2023)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2021",
"style": "rule"
} | {
"index": "4498",
"split": "train"
} |
math | [
{
"content": "Simplify first, then evaluate: $(\\frac{{x-1}}{{x-3}}-\\frac{{x+1}}{x})÷\\frac{{{x^2}+3x}}{{{x^2}-6x+9}}$, where $x$ satisfies $x^{2}+2x-6=0$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{1}{2}",
"style": "rule"
} | {
"index": "4499",
"split": "train"
} |
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