data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles \n(not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{y^3}{x^2}",
"style": "rule"
} | {
"index": "37000",
"split": "train"
} |
math | [
{
"content": "A 1-liter carton of milk used to cost 80 rubles. Recently, in an effort to cut costs, the manufacturer reduced the carton size to 0.9 liters and increased the price to 99 rubles. By what percentage did the manufacturer's revenue increase? Please output the final answer within \\boxed{}.",
"rol... | {
"ground_truth": "37.5",
"style": "rule"
} | {
"index": "37001",
"split": "train"
} |
math | [
{
"content": "For how many bases between two and nine inclusive does the representation of $576_{10}$ have a final digit of 1? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "37002",
"split": "train"
} |
math | [
{
"content": "Solve for $c$: \n$$\\sqrt{9+\\sqrt{27+9c}} + \\sqrt{3+\\sqrt{3+c}} = 3+3\\sqrt{3}$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "37003",
"split": "train"
} |
math | [
{
"content": "Given that in triangle $\\triangle ABC$, $a=2$, $\\angle A=\\frac{π}{6}$, $b=2\\sqrt{3}$, find the measure of angle $\\angle C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "37004",
"split": "train"
} |
math | [
{
"content": "Let $f(x) = ax^6 + bx^4 - cx^2 + 3.$ If $f(91) = 1$, find $f(91) + f(-91)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "37005",
"split": "train"
} |
math | [
{
"content": "In a plane Cartesian coordinate system, a point whose x and y coordinates are both integers is called a \"lattice point.\" How many lattice points are there inside and on the boundaries of the triangle formed by the line $7x + 11y = 77$ and the coordinate axes? Please output the final answer withi... | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "37006",
"split": "train"
} |
math | [
{
"content": "For real numbers $x > 1,$ find the minimum value of\n\\[\\frac{x + 8}{\\sqrt{x - 1}}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "37007",
"split": "train"
} |
math | [
{
"content": "Let $[a]$ represent the greatest integer less than or equal to $a$. Determine the largest positive integer solution to the equation $\\left[\\frac{x}{7}\\right]=\\left[\\frac{x}{8}\\right]+1$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "104",
"style": "rule"
} | {
"index": "37008",
"split": "train"
} |
math | [
{
"content": "For letters \\( a \\sim z \\), encode them as $(\\mathbf{a}=1, \\mathbf{b}=2, \\ldots, \\mathbf{z}=26)$. Each English word (assuming all letters in the word are lowercase) can be assigned a product \\(\\mathbf{p}\\) calculated from these letter codes. For example, for the word \"good,\" its corres... | {
"ground_truth": "106",
"style": "rule"
} | {
"index": "37009",
"split": "train"
} |
math | [
{
"content": "Bethany has 11 pound coins and some 20 pence coins and some 50 pence coins in her purse. The mean value of the coins is 52 pence. Which could not be the number of coins in the purse? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "37010",
"split": "train"
} |
math | [
{
"content": "Calculate\n\n$$\n\\operatorname{tg} \\frac{\\pi}{47} \\cdot \\operatorname{tg} \\frac{2 \\pi}{47} + \\operatorname{tg} \\frac{2 \\pi}{47} \\cdot \\operatorname{tg} \\frac{3 \\pi}{47} + \\ldots + \\operatorname{tg} \\frac{k \\pi}{47} \\cdot \\operatorname{tg} \\frac{(k+1) \\pi}{47} + \\ldots + \\op... | {
"ground_truth": "-2021",
"style": "rule"
} | {
"index": "37011",
"split": "train"
} |
math | [
{
"content": "Add 53.463 to 12.9873 and round to the nearest thousandth. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "66.450",
"style": "rule"
} | {
"index": "37012",
"split": "train"
} |
math | [
{
"content": "To transmit a positive integer less than 1000, the Networked Number Node offers two options.\n\nOption 1. Pay $\\$$d to send each digit d. Therefore, 987 would cost $\\$$9 + $\\$$8 + $\\$$7 = $\\$$24 to transmit.\n\nOption 2. Encode integer into binary (base 2) first, and then pay $\\$$d to send e... | {
"ground_truth": "503",
"style": "rule"
} | {
"index": "37013",
"split": "train"
} |
math | [
{
"content": "Two runners started simultaneously in the same direction from the same point on a circular track. The first runner, moving ahead, caught up with the second runner at the moment when the second runner had only run half a lap. From that moment, the second runner doubled their speed. Will the first r... | {
"ground_truth": "2.5",
"style": "rule"
} | {
"index": "37014",
"split": "train"
} |
math | [
{
"content": "Find the smallest positive integer \\( n \\) such that:\n1. \\( n \\) has exactly 144 distinct positive divisors.\n2. There are ten consecutive integers among the positive divisors of \\( n \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "110880",
"style": "rule"
} | {
"index": "37015",
"split": "train"
} |
math | [
{
"content": "Four vertices of a rectangle include the points $(2, 3)$, $(2, 15)$, and $(13, 3)$. What is the area of the intersection of this rectangular region and the region inside the graph of the equation $(x - 13)^2 + (y - 3)^2 = 16$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "4\\pi",
"style": "rule"
} | {
"index": "37016",
"split": "train"
} |
math | [
{
"content": "For how many three-digit numbers can you subtract 297 and obtain a second three-digit number which is the original three-digit number reversed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "37017",
"split": "train"
} |
math | [
{
"content": "Let $n$ be the integer such that $0 \\le n < 29$ and $4n \\equiv 1 \\pmod{29}$. What is $\\left(3^n\\right)^4 - 3 \\pmod{29}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "37018",
"split": "train"
} |
math | [
{
"content": "Sarah and John run on a circular track. Sarah runs counterclockwise and completes a lap every 120 seconds, while John runs clockwise and completes a lap every 75 seconds. Both start from the same line at the same time. A photographer standing inside the track takes a picture at a random time betwe... | {
"ground_truth": "\\frac{1}{12}",
"style": "rule"
} | {
"index": "37019",
"split": "train"
} |
math | [
{
"content": "Evaluate $\\lfloor\\sqrt{63}\\rfloor$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "37020",
"split": "train"
} |
math | [
{
"content": "Compute the largest integer that can be expressed in the form $3^{x(3-x)}$ for some real number $x$ .\n\n[i]Proposed by James Lin Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37021",
"split": "train"
} |
math | [
{
"content": "In square $ABCD$, point $M$ is the midpoint of side $AB$ and point $N$ is the midpoint of side $BC$. What is the ratio of the area of triangle $AMN$ to the area of square $ABCD$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{8}",
"style": "rule"
} | {
"index": "37022",
"split": "train"
} |
math | [
{
"content": "How many arithmetic sequences, where the common difference is a natural number greater than 2, satisfy the conditions that the first term is 1783, the last term is 1993, and the number of terms is at least 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "37023",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = 2\\sqrt{3}\\sin^2(x) + 2\\sin(x)\\cos(x) - \\sqrt{3}$, where $x \\in \\left[ \\frac{\\pi}{3}, \\frac{11\\pi}{24} \\right]$:\n\n1. Determine the range of the function $f(x)$.\n2. Suppose an acute-angled triangle $ABC$ has two sides of lengths equal to the maximum and mini... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "37024",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), the three interior angles \\( \\angle A, \\angle B, \\angle C \\) satisfy \\( \\angle A = 3 \\angle B = 9 \\angle C \\). Find the value of \n\\[ \\cos A \\cdot \\cos B + \\cos B \\cdot \\cos C + \\cos C \\cdot \\cos A = \\quad . \\] Please output the final answe... | {
"ground_truth": "-1/4",
"style": "rule"
} | {
"index": "37025",
"split": "train"
} |
math | [
{
"content": "For the function $y=f(x)$, if there exists $x_{0} \\in D$ such that $f(-x_{0})+f(x_{0})=0$, then the function $f(x)$ is called a \"sub-odd function\" and $x_{0}$ is called a \"sub-odd point\" of the function. Consider the following propositions: \n$(1)$ Odd functions are necessarily \"sub-odd fun... | {
"ground_truth": "(1)(2)(4)(5)",
"style": "rule"
} | {
"index": "37026",
"split": "train"
} |
math | [
{
"content": "Let $C$ be a point on the parabola $y = x^2 - 4x + 7,$ and let $D$ be a point on the line $y = 3x - 5.$ Find the shortest distance $CD$ and also ensure that the projection of point $C$ over line $y = 3x - 5$ lands on point $D$. Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "\\frac{0.25}{\\sqrt{10}}",
"style": "rule"
} | {
"index": "37027",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\( Z \\) lies on \\( XY \\) and the three circles have diameters \\( XZ \\), \\( ZY \\), and \\( XY \\). If \\( XZ = 12 \\) and \\( ZY = 8 \\), then the ratio of the area of the shaded region to the area of the unshaded region is Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "12:13",
"style": "rule"
} | {
"index": "37028",
"split": "train"
} |
math | [
{
"content": "Defined on $\\mathbf{R}$, the function $f$ satisfies\n$$\nf(1+x)=f(9-x)=f(9+x).\n$$\n\nGiven $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2010",
"style": "rule"
} | {
"index": "37029",
"split": "train"
} |
math | [
{
"content": "Given the integers \\( 1, 2, 3, \\ldots, 40 \\), find the greatest possible sum of the positive differences between the integers in twenty pairs, where the positive difference is either 1 or 3. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "58",
"style": "rule"
} | {
"index": "37030",
"split": "train"
} |
math | [
{
"content": "I have three 30-sided dice that each have 6 purple sides, 8 green sides, 10 blue sides, and 6 silver sides. If I roll all three dice, what is the probability that they all show the same color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{2}{25}",
"style": "rule"
} | {
"index": "37031",
"split": "train"
} |
math | [
{
"content": "In a city with 10 parallel streets and 10 streets crossing them at right angles, what is the minimum number of turns that a closed bus route passing through all intersections can have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37032",
"split": "train"
} |
math | [
{
"content": "Khali has to shovel snow off the sidewalk in front of his house. The sidewalk is 20 feet long and 2 feet wide. If the snow is $\\frac{1}{2}$ foot deep, how many cubic feet of snow does Khali have to shovel off the sidewalk? Please output the final answer within \\boxed{}.",
"role": "user"
}
... | {
"ground_truth": "20",
"style": "rule"
} | {
"index": "37033",
"split": "train"
} |
math | [
{
"content": "Evaluate the infinite geometric series: $$\\frac{3}{2}-\\frac{2}{3}+\\frac{8}{27}-\\frac{32}{243}+\\dots$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{27}{26}",
"style": "rule"
} | {
"index": "37034",
"split": "train"
} |
math | [
{
"content": "What is the $1992^{\\text{nd}}$ letter in this sequence?\n\\[\\text{ABCDEDCBAABCDEDCBAABCDEDCBAABCDEDC}\\cdots\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "C",
"style": "rule"
} | {
"index": "37035",
"split": "train"
} |
math | [
{
"content": "Find all values of $x$ such that\n\\[3^x + 4^x + 5^x = 6^x.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "37036",
"split": "train"
} |
math | [
{
"content": "Prisha writes down one integer three times and another integer two times, with their sum being $105$, and one of the numbers is $15$. Calculate the other number. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37037",
"split": "train"
} |
math | [
{
"content": "Given a circle described by the equation $x^{2}-2x+y^{2}-2y+1=0$, find the cosine value of the angle between the two tangent lines drawn from an external point $P(3,2)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{5}",
"style": "rule"
} | {
"index": "37038",
"split": "train"
} |
math | [
{
"content": "What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "37039",
"split": "train"
} |
math | [
{
"content": "Triangle \\(A B C\\) has side lengths \\(A B = 65\\), \\(B C = 33\\), and \\(A C = 56\\). Find the radius of the circle tangent to sides \\(A C\\) and \\(B C\\) and to the circumcircle of triangle \\(A B C\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37040",
"split": "train"
} |
math | [
{
"content": "If \\(\\alpha, \\beta, \\gamma\\) are acute angles, and \\(\\sin ^{2} \\alpha+\\sin ^{2} \\beta+\\sin ^{2} \\gamma=1\\), what is the maximum value of \\(\\frac{\\sin \\alpha+\\sin \\beta+\\sin \\gamma}{\\cos \\alpha+\\cos \\beta+\\cos \\gamma}\\)? Please output the final answer within \\boxed{}.",... | {
"ground_truth": "\\frac{\\sqrt{2}}{2}",
"style": "rule"
} | {
"index": "37041",
"split": "train"
} |
math | [
{
"content": "Among the following numbers\n\n① $111111_{(2)}$ \n② $210_{(6)}$ \n③ $1000_{(4)}$ \n④ $81_{(8)}$\n\nThe largest number is \\_\\_\\_\\_\\_\\_\\_\\_, and the smallest number is \\_\\_\\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "111111_{(2)}",
"style": "rule"
} | {
"index": "37042",
"split": "train"
} |
math | [
{
"content": "What is the area, in square units, of triangle $ABC$ in the figure shown if points $A$, $B$, $C$ and $D$ are coplanar, angle $D$ is a right angle, $AC = 13$, $AB = 15$ and $DC = 5$? [asy]\npair A, B, C, D;\nA=(12,0);\nD=(0,0);\nC=(0,5);\nB=(0,9);\ndraw(A--B--C--A--D--C);\ndraw((0,.5)--(.5,.5)--(.5... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37043",
"split": "train"
} |
math | [
{
"content": "Given vectors $\\overrightarrow{m}=(\\cos \\alpha- \\frac{\\sqrt{2}}{3},-1)$ and $\\overrightarrow{n}=(\\sin \\alpha,1)$, vectors $\\overrightarrow{m}$ and $\\overrightarrow{n}$ are collinear, and $\\alpha \\in [-\\frac{\\pi}{2},0]$. \n$(1)$ Find the value of $\\sin \\alpha + \\cos \\alpha$; \n$... | {
"ground_truth": "\\frac{7}{12}",
"style": "rule"
} | {
"index": "37044",
"split": "train"
} |
math | [
{
"content": "In an arithmetic sequence \\(\\{a_{n}\\}\\), where \\(a_{1} > 0\\) and \\(5a_{8} = 8a_{13}\\), what is the value of \\(n\\) when the sum of the first \\(n\\) terms \\(S_{n}\\) is maximized? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "37045",
"split": "train"
} |
math | [
{
"content": "Wendy is playing darts with a circular dartboard of radius 20. Whenever she throws a dart, it lands uniformly at random on the dartboard. At the start of her game, there are 2020 darts placed randomly on the board. Every turn, she takes the dart farthest from the center, and throws it at the board... | {
"ground_truth": "6060",
"style": "rule"
} | {
"index": "37046",
"split": "train"
} |
math | [
{
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{335}{2011}",
"style": "rule"
} | {
"index": "37047",
"split": "train"
} |
math | [
{
"content": "A bin contains 10 black balls and 9 white balls. 4 balls are drawn at random. What is the probability of drawing 3 balls of one color and 1 ball of another color? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{160}{323}",
"style": "rule"
} | {
"index": "37048",
"split": "train"
} |
math | [
{
"content": "Given that $\\sin \\alpha - 2\\cos \\alpha = \\frac{\\sqrt{10}}{2}$, find $\\tan 2\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "37049",
"split": "train"
} |
math | [
{
"content": "Find $(-1)^{-10} + (-1)^{-9} + (-1)^{-8} + \\cdots + (-1)^9 + (-1)^{10}$.\n\n(The dots $\\cdots$ mean that there are 21 numbers being added, one for each integer from $-10$ to 10.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "37050",
"split": "train"
} |
math | [
{
"content": "Let $\\overline{AB}$ be a diameter in a circle with radius $6$. Let $\\overline{CD}$ be a chord in the circle that intersects $\\overline{AB}$ at point $E$ such that $BE = 3$ and $\\angle AEC = 60^{\\circ}$. Find the value of $CE^{2} + DE^{2}$. Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "108",
"style": "rule"
} | {
"index": "37051",
"split": "train"
} |
math | [
{
"content": "When $p(x) = Ax^5 + Bx^3 + Cx + 4$ is divided by $x - 3,$ the remainder is 11. Find the remainder when $p(x)$ is divided by $x + 3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-3",
"style": "rule"
} | {
"index": "37052",
"split": "train"
} |
math | [
{
"content": "Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the nex... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "37053",
"split": "train"
} |
math | [
{
"content": "From the numbers $1, 2, \\cdots, 10$, a number $a$ is randomly selected, and from the numbers $-1, -2, \\cdots, -10$, a number $b$ is randomly selected. What is the probability that $a^{2} + b$ is divisible by 3? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.3",
"style": "rule"
} | {
"index": "37054",
"split": "train"
} |
math | [
{
"content": "Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\\angle FAB = 120^\\circ,$ $\\overline{AB}\\parallel \\overline{DE},$ $\\overline{BC}\\parallel \\overline{EF,}$ $\\overline{CD}\\parallel \\overline{FA},$ and the y-coordinat... | {
"ground_truth": "51",
"style": "rule"
} | {
"index": "37055",
"split": "train"
} |
math | [
{
"content": "There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. How many cards were fl... | {
"ground_truth": "40",
"style": "rule"
} | {
"index": "37056",
"split": "train"
} |
math | [
{
"content": "How many terms are in the expansion of \\[(a+b+c)(d+e+f+g)?\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "12",
"style": "rule"
} | {
"index": "37057",
"split": "train"
} |
math | [
{
"content": "$15\\times\u0002 36$ -checkerboard is covered with square tiles. There are two kinds of tiles, with side $7$ or $5.$ \nTiles are supposed to cover whole squares of the board and be non-overlapping. \nWhat is the maximum number of squares to be covered? Please output the final answer within \\b... | {
"ground_truth": "540",
"style": "rule"
} | {
"index": "37058",
"split": "train"
} |
math | [
{
"content": "A pyramid has a triangular base with side lengths $20$, $20$, and $24$. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length $25$. The volume of the pyramid is $m\\sqrt{n}$, where $m$ and $n$ are positive integers, and $n$ is not div... | {
"ground_truth": "803",
"style": "rule"
} | {
"index": "37059",
"split": "train"
} |
math | [
{
"content": "Compute $\\arctan \\frac{2}{5} + \\arctan \\frac{5}{2}.$ Express your answer in radians. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{\\pi}{2}",
"style": "rule"
} | {
"index": "37060",
"split": "train"
} |
math | [
{
"content": "Find all complex numbers $z$ such that\n\\[z^2 = -77 - 36i.\\]Enter all complex numbers, separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2 - 9i, -2 + 9i",
"style": "rule"
} | {
"index": "37061",
"split": "train"
} |
math | [
{
"content": "In a certain sequence, the first term is $a_1 = 101$ and the second term is $a_2 = 102$. Furthermore, the values of the remaining terms are chosen so that $a_n + a_{n+1} + a_{n+2} = n + 2$ for all $n \\geq 1$. Determine $a_{50}$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "117",
"style": "rule"
} | {
"index": "37062",
"split": "train"
} |
math | [
{
"content": "There are integers $b,c$ for which both roots of the polynomial $x^2-x-1$ are also roots of the polynomial $x^5-bx-c$. Determine the product $bc$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "37063",
"split": "train"
} |
math | [
{
"content": "A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen? Please output the final answer within \\boxe... | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "37064",
"split": "train"
} |
math | [
{
"content": "For how many even integers $n$ between 1 and 200 is the greatest common divisor of 18 and $n$ equal to 4? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "34",
"style": "rule"
} | {
"index": "37065",
"split": "train"
} |
math | [
{
"content": "Calculate $(-2)^{23} + 2^{(2^4+5^2-7^2)}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-8388607.99609375",
"style": "rule"
} | {
"index": "37066",
"split": "train"
} |
math | [
{
"content": "The numbers from 1 to 1000 are written in a circle. Starting from the first one, every 15th number (i.e., the numbers 1, 16, 31, etc.) is crossed out, and during subsequent rounds, already crossed-out numbers are also taken into account. The crossing out continues until it turns out that all the n... | {
"ground_truth": "800",
"style": "rule"
} | {
"index": "37067",
"split": "train"
} |
math | [
{
"content": "The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n\\geq 2$. What is the smallest positive integer $k$ such that the product $a_1a_2\\cdots a_k$ is an integer? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "17",
"style": "rule"
} | {
"index": "37068",
"split": "train"
} |
math | [
{
"content": "Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex in the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies?... | {
"ground_truth": "81",
"style": "rule"
} | {
"index": "37069",
"split": "train"
} |
math | [
{
"content": "The median of the set of numbers $\\{$12, 38, 45, $x$, 14$\\}$ is five less than the mean. If $x$ is a negative integer, what is the value of $x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-14",
"style": "rule"
} | {
"index": "37070",
"split": "train"
} |
math | [
{
"content": "Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\pi",
"style": "rule"
} | {
"index": "37071",
"split": "train"
} |
math | [
{
"content": "Given the set of integers $\\{1, 2, 3, \\dots, 9\\}$, from which three distinct numbers are arbitrarily selected as the coefficients of the quadratic function $f_{(x)} = ax^2 + bx + c$, determine the total number of functions $f_{(x)}$ that satisfy $\\frac{f(1)}{2} \\in \\mathbb{Z}$. Please output... | {
"ground_truth": "264",
"style": "rule"
} | {
"index": "37072",
"split": "train"
} |
math | [
{
"content": "Horses $X, Y$ and $Z$ are entered in a three-horse race in which ties are not possible. The odds against $X$ winning are $3:1$ and the odds against $Y$ winning are $2:3$, what are the odds against $Z$ winning? (By \"odds against $H$ winning are $p:q$\" we mean the probability of $H$ winning the ra... | {
"ground_truth": "17:3",
"style": "rule"
} | {
"index": "37073",
"split": "train"
} |
math | [
{
"content": "Write $\\mathbf{2012}$ as the sum of $N$ distinct positive integers, where $N$ is at its maximum. What is the maximum value of $N$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "62",
"style": "rule"
} | {
"index": "37074",
"split": "train"
} |
math | [
{
"content": "Suppose lines $p$ and $q$ in the first quadrant are defined by the equations $y = -2x + 8$ and $y = -3x + 9$, respectively. What is the probability that a point randomly selected in the 1st quadrant and below $p$ will fall between $p$ and $q$? Express your answer as a decimal to the nearest hundre... | {
"ground_truth": "0.16",
"style": "rule"
} | {
"index": "37075",
"split": "train"
} |
math | [
{
"content": "In how many ways can the number 1500 be represented as a product of three natural numbers? Variations where the factors are the same but differ in order are considered identical. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "37076",
"split": "train"
} |
math | [
{
"content": "Given an arithmetic sequence $\\{a_n\\}$ with a common difference $d = -2$, and $a_1 + a_4 + a_7 + \\ldots + a_{97} = 50$, find the value of $a_3 + a_6 + a_9 + \\ldots + a_{99}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-82",
"style": "rule"
} | {
"index": "37077",
"split": "train"
} |
math | [
{
"content": "Given that point $P(-4,3)$ lies on the terminal side of angle $\\alpha$, find the value of $$\\frac{3\\sin^{2}\\frac{\\alpha}{2}+2\\sin\\frac{\\alpha}{2}\\cos\\frac{\\alpha}{2}+\\cos^{2}\\frac{\\alpha}{2}-2}{\\sin(\\frac{\\pi}{2}+\\alpha)\\tan(-3\\pi+\\alpha)+\\cos(6\\pi-\\alpha)}.$$ Please output... | {
"ground_truth": "-7",
"style": "rule"
} | {
"index": "37078",
"split": "train"
} |
math | [
{
"content": "Given an ellipse $C: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ ($a > b > 0$) with an eccentricity of $\\frac{\\sqrt{5}}{5}$, and the equation of the right directrix is $x=5$.\n\n(1) Find the equation of the ellipse; \n(2) A line $l$ with a slope of 1 passes through the right focus $F$ of the ellip... | {
"ground_truth": "\\frac{16\\sqrt{10}}{9}",
"style": "rule"
} | {
"index": "37079",
"split": "train"
} |
math | [
{
"content": "Given that $a, b \\in R^{+}$ and $a + b = 1$, find the supremum of $- \\frac{1}{2a} - \\frac{2}{b}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{9}{2}",
"style": "rule"
} | {
"index": "37080",
"split": "train"
} |
math | [
{
"content": "If the sequence \\(\\{a_n\\}\\) satisfies \\(a_1 = \\frac{2}{3}\\) and \\(a_{n+1} - a_n = \\sqrt{\\frac{2}{3} \\left(a_{n+1} + a_n\\right)}\\), then find the value of \\(a_{2015}\\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1354080",
"style": "rule"
} | {
"index": "37081",
"split": "train"
} |
math | [
{
"content": "Determine the area and the circumference of a circle with the center at the point \\( R(2, -1) \\) and passing through the point \\( S(7, 4) \\). Express your answer in terms of \\( \\pi \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10\\pi \\sqrt{2}",
"style": "rule"
} | {
"index": "37082",
"split": "train"
} |
math | [
{
"content": "A sequence is formed by arranging the following arrays into a row: $$( \\frac {1}{1}), ( \\frac {2}{1}, \\frac {1}{2}), ( \\frac {3}{1}, \\frac {2}{2}, \\frac {1}{3}), ( \\frac {4}{1}, \\frac {3}{2}, \\frac {2}{3}, \\frac {1}{4}), ( \\frac {5}{1}, \\frac {4}{2}, \\frac {3}{3}, \\frac {2}{4}, \\fra... | {
"ground_truth": "\\frac {5}{59}",
"style": "rule"
} | {
"index": "37083",
"split": "train"
} |
math | [
{
"content": "In the diagram, $ABCD$ is a square with side length $6,$ and $WXYZ$ is a rectangle with $ZY=10$ and $XY=6.$ Also, $AD$ and $WX$ are perpendicular. If the shaded area is equal to half of the area of $WXYZ,$ what is the length of $AP?$\n\n[asy]\ndraw((0,0)--(10,0)--(10,6)--(0,6)--cycle,black+linewid... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "37084",
"split": "train"
} |
math | [
{
"content": "The positive integers $a$ and $b$ are such that the numbers $15a+16b$ and $16a-15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "231361",
"style": "rule"
} | {
"index": "37085",
"split": "train"
} |
math | [
{
"content": "The three-tiered \"pyramid\" shown in the image is built from $1 \\mathrm{~cm}^{3}$ cubes and has a surface area of $42 \\mathrm{~cm}^{2}$. We made a larger \"pyramid\" based on this model, which has a surface area of $2352 \\mathrm{~cm}^{2}$. How many tiers does it have? Please output the final a... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "37086",
"split": "train"
} |
math | [
{
"content": "The polynomial $P(x) = x^3 + ax^2 + bx +c$ has the property that the mean of its zeros, the product of its zeros, and the sum of its coefficients are all equal. If the $y$-intercept of the graph of $y=\nP(x)$ is 2, what is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
... | {
"ground_truth": "-11",
"style": "rule"
} | {
"index": "37087",
"split": "train"
} |
math | [
{
"content": "Find the number by which the three numbers 480608, 508811, and 723217 when divided will give the same remainder. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "79",
"style": "rule"
} | {
"index": "37088",
"split": "train"
} |
math | [
{
"content": "The GDP of a certain city last year was 84,300,000 ten thousand yuan. Represent this number in scientific notation. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8.43 \\times 10^7",
"style": "rule"
} | {
"index": "37089",
"split": "train"
} |
math | [
{
"content": "Two right triangles \\( \\triangle AXY \\) and \\( \\triangle BXY \\) have a common hypotenuse \\( XY \\) and side lengths (in units) \\( AX=5 \\), \\( AY=10 \\), and \\( BY=2 \\). Sides \\( AY \\) and \\( BX \\) intersect at \\( P \\). Determine the area (in square units) of \\( \\triangle PXY \\... | {
"ground_truth": "25/3",
"style": "rule"
} | {
"index": "37090",
"split": "train"
} |
math | [
{
"content": "The faces of an octahedral die are labeled with digits $1$ through $8$. What is the probability, expressed as a common fraction, of rolling a sum of $15$ with a pair of such octahedral dice? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{32}",
"style": "rule"
} | {
"index": "37091",
"split": "train"
} |
math | [
{
"content": "Regular tetrahedron $A B C D$ is projected onto a plane sending $A, B, C$, and $D$ to $A^{\\prime}, B^{\\prime}, C^{\\prime}$, and $D^{\\prime}$ respectively. Suppose $A^{\\prime} B^{\\prime} C^{\\prime} D^{\\prime}$ is a convex quadrilateral with $A^{\\prime} B^{\\prime}=A^{\\prime} D^{\\prime}$ ... | {
"ground_truth": "2 \\sqrt[4]{6}",
"style": "rule"
} | {
"index": "37092",
"split": "train"
} |
math | [
{
"content": "In a certain class, with any distribution of 200 candies, there will be at least two students who receive the same number of candies (possibly none). What is the minimum number of students in such a class? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "37093",
"split": "train"
} |
math | [
{
"content": "In the plane Cartesian coordinate system \\( xOy \\), an ellipse \\( C \\) : \\( \\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1 \\) \\( (a>b>0) \\) has left and right foci \\( F_{1} \\) and \\( F_{2} \\) respectively. Chords \\( ST \\) and \\( UV \\) are parallel to the \\( x \\)-axis and \\( y \\)-a... | {
"ground_truth": "\\sqrt{15}",
"style": "rule"
} | {
"index": "37094",
"split": "train"
} |
math | [
{
"content": "A projectile is launched with an initial velocity $u$ at an angle $\\phi$ from the ground. The trajectory can be described by the parametric equations:\n\\[\nx = ut \\cos \\phi, \\quad y = ut \\sin \\phi - \\frac{1}{2} g t^2,\n\\]\nwhere $t$ is the time, $g$ is the acceleration due to gravity, and... | {
"ground_truth": "\\frac{\\pi}{8}",
"style": "rule"
} | {
"index": "37095",
"split": "train"
} |
math | [
{
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=5$ while $HQ=2$, then calculate $(BP)(PC)-(AQ)(QC)$. [asy]\nsize(150); defaultpen(linewidth(0.8));\npair B = (0,0), C = (3,0), A = (2,2), P = foot(A,B,C), Q = foot(B,A,C),H = inter... | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "37096",
"split": "train"
} |
math | [
{
"content": "What is the sum of all two-digit primes greater than 30 but less than 99, which are still prime when their two digits are interchanged? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "388",
"style": "rule"
} | {
"index": "37097",
"split": "train"
} |
math | [
{
"content": "Given the planar vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$, with $|\\overrightarrow {a}|=1$, $|\\overrightarrow {b}|=2$, and $\\overrightarrow {a} \\cdot \\overrightarrow {b} = 1$, find the angle between vectors $\\overrightarrow {a}$ and $\\overrightarrow {b}$. Please output the f... | {
"ground_truth": "\\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "37098",
"split": "train"
} |
math | [
{
"content": "Suppose $f(x) = x^2,$ and $g(x)$ is a polynomial such that $f(g(x)) = 4x^2 + 4x + 1$. Enter all possible polynomials $g(x),$ separated by commas. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2x-1",
"style": "rule"
} | {
"index": "37099",
"split": "train"
} |
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