data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "For positive integers $n$ and $k$, let $\\mho(n, k)$ be the number of distinct prime divisors of $n$ that are at least $k$. Find the closest integer to $$\\sum_{n=1}^{\\infty} \\sum_{k=1}^{\\infty} \\frac{\\mho(n, k)}{3^{n+k-7}}$$ Please output the final answer within \\boxed{}.",
"role": "use... | {
"ground_truth": "167",
"style": "rule"
} | {
"index": "5500",
"split": "train"
} |
math | [
{
"content": "If $x = 151$ and $x^3y - 3x^2y + 3xy = 3423000$, what is the value of $y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3423000}{3375001}",
"style": "rule"
} | {
"index": "5501",
"split": "train"
} |
math | [
{
"content": "Compute $\\begin{pmatrix} 1 & -1 \\\\ 1 & 0 \\end{pmatrix}^3.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\begin{pmatrix} -1 & 0 \\\\ 0 & -1 \\end{pmatrix}.",
"style": "rule"
} | {
"index": "5502",
"split": "train"
} |
math | [
{
"content": "The complete graph of $y=f(x)$, which consists of five line segments, is shown in red below. (On this graph, the distance between grid lines is $1$.)\n\nWhat is the sum of the $x$-coordinates of all points where $f(x) = 1.8$?\n\n[asy]\n\nsize(150);\n\nreal ticklen=3;\n\nreal tickspace=2;\n\nreal t... | {
"ground_truth": "4.5",
"style": "rule"
} | {
"index": "5503",
"split": "train"
} |
math | [
{
"content": "Three workers can complete a certain task. The second and third worker together can complete it twice as fast as the first worker; the first and third worker together can complete it three times faster than the second worker. How many times faster can the first and second worker together complete ... | {
"ground_truth": "7/5",
"style": "rule"
} | {
"index": "5504",
"split": "train"
} |
math | [
{
"content": "A line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 1 \\\\ 1 \\end{pmatrix} + t \\begin{pmatrix} 2 \\\\ -3 \\end{pmatrix}.\\]A second line is parameterized by\n\\[\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} 5 \\\\ -9 \\end{pmatrix} + u \\begin... | {
"ground_truth": "\\begin{pmatrix} 7 \\\\ -8 \\end{pmatrix}",
"style": "rule"
} | {
"index": "5505",
"split": "train"
} |
math | [
{
"content": "Find the largest integer less than 80 that leaves a remainder of 3 when divided by 5. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "78",
"style": "rule"
} | {
"index": "5506",
"split": "train"
} |
math | [
{
"content": "Half the people in a room left. One third of those remaining started to dance. There were then $12$ people who were not dancing. The original number of people in the room was Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "5507",
"split": "train"
} |
math | [
{
"content": "Yann writes down the first $n$ consecutive positive integers, $1,2,3,4, \\ldots, n-1, n$. He removes four different integers $p, q, r, s$ from the list. At least three of $p, q, r, s$ are consecutive and $100<p<q<r<s$. The average of the integers remaining in the list is 89.5625. What is the numbe... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "5508",
"split": "train"
} |
math | [
{
"content": "Let \\( p(x) \\) be a polynomial with integer coefficients such that \\( p(m) - p(n) \\) divides \\( m^2 - n^2 \\) for all integers \\( m \\) and \\( n \\). If \\( p(0) = 1 \\) and \\( p(1) = 2 \\), find the largest possible value of \\( p(100) \\). Please output the final answer within \\boxed{}.... | {
"ground_truth": "10001",
"style": "rule"
} | {
"index": "5509",
"split": "train"
} |
math | [
{
"content": "A person rolled a fair six-sided die $100$ times and obtained a $6$ $19$ times. What is the approximate probability of rolling a $6$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.19",
"style": "rule"
} | {
"index": "5510",
"split": "train"
} |
math | [
{
"content": "Some marbles in a bag are red and the rest are blue. If one red marble is removed, then one-seventh of the remaining marbles are red. If two blue marbles are removed instead of one red, then one-fifth of the remaining marbles are red. How many marbles were in the bag originally? Please output the ... | {
"ground_truth": "22",
"style": "rule"
} | {
"index": "5511",
"split": "train"
} |
math | [
{
"content": "We define a number as an ultimate mountain number if it is a 4-digit number and the third digit is larger than the second and fourth digit but not necessarily the first digit. For example, 3516 is an ultimate mountain number. How many 4-digit ultimate mountain numbers are there? Please output the ... | {
"ground_truth": "204",
"style": "rule"
} | {
"index": "5512",
"split": "train"
} |
math | [
{
"content": "A right triangle $\\triangle ABC$ has sides $AC=3$, $BC=4$, and $AB=5$. When this triangle is rotated around the right-angle side $BC$, the surface area of the resulting solid is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24\\pi",
"style": "rule"
} | {
"index": "5513",
"split": "train"
} |
math | [
{
"content": "Express $0.\\overline{1}+0.\\overline{01}+0.\\overline{0001}$ as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1213}{9999}",
"style": "rule"
} | {
"index": "5514",
"split": "train"
} |
math | [
{
"content": "Suppose $x-3$ and $y+3$ are multiples of $7$.\n\nWhat is the smallest positive integer, $n,$ for which $x^2+xy+y^2+n$ is a multiple of $7$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "5515",
"split": "train"
} |
math | [
{
"content": "When the base-10 integers 200 and 1200 are expressed in base 2, how many more digits does 1200 have than 200 (after being converted)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "5516",
"split": "train"
} |
math | [
{
"content": "Let $a$ and $b$ be positive real numbers such that $a + 2b = 1.$ Find the minimum value of\n\\[\\frac{1}{a} + \\frac{2}{b}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "5517",
"split": "train"
} |
math | [
{
"content": "**\nHow many non-similar regular 500-pointed stars are there?\n\n** Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "99",
"style": "rule"
} | {
"index": "5518",
"split": "train"
} |
math | [
{
"content": "Two circles of radius 2 are centered at $(2,0)$ and at $(0,2)$. What is the area of the intersection of the interiors of the two circles? Express your answer in fully expanded form in terms of $\\pi$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2\\pi-4",
"style": "rule"
} | {
"index": "5519",
"split": "train"
} |
math | [
{
"content": "Let $n = 2^4 \\cdot 3^5 \\cdot 4^6\\cdot 6^7$. How many natural-number factors does $n$ have? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "312",
"style": "rule"
} | {
"index": "5520",
"split": "train"
} |
math | [
{
"content": "Find the largest integer value of $n$ such that $n^2-9n+18$ is negative. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "5",
"style": "rule"
} | {
"index": "5521",
"split": "train"
} |
math | [
{
"content": "A stone is dropped into a well and the report of the stone striking the bottom is heard $7.7$ seconds after it is dropped. Assume that the stone falls $16t^2$ feet in t seconds and that the velocity of sound is $1120$ feet per second. The depth of the well is: Please output the final answer within... | {
"ground_truth": "784",
"style": "rule"
} | {
"index": "5522",
"split": "train"
} |
math | [
{
"content": "In a right triangle \\( A B C \\) (with right angle at \\( C \\)), the medians \\( A M \\) and \\( B N \\) are drawn with lengths 19 and 22, respectively. Find the length of the hypotenuse of this triangle. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "29",
"style": "rule"
} | {
"index": "5523",
"split": "train"
} |
math | [
{
"content": "Given that $E, U, L, S, R, T$ represent the numbers $1, 2, 3, 4, 5, 6$ (with each letter representing a different number), and satisfying the following conditions:\n(1) $E + U + L = 6$\n(2) $S + R + U + T = 18$\n(3) $U \\times T = 15$\n(4) $S \\times L = 8$\n\nWhat is the six-digit number $\\overl... | {
"ground_truth": "132465",
"style": "rule"
} | {
"index": "5524",
"split": "train"
} |
math | [
{
"content": "Let \\(ABCD\\) be an isosceles trapezoid with \\(AB=1, BC=DA=5, CD=7\\). Let \\(P\\) be the intersection of diagonals \\(AC\\) and \\(BD\\), and let \\(Q\\) be the foot of the altitude from \\(D\\) to \\(BC\\). Let \\(PQ\\) intersect \\(AB\\) at \\(R\\). Compute \\(\\sin \\angle RPD\\). Please out... | {
"ground_truth": "\\frac{4}{5}",
"style": "rule"
} | {
"index": "5525",
"split": "train"
} |
math | [
{
"content": "This was a highly dangerous car rally. It began with a small and very narrow bridge, where one out of five cars would fall into the water. Then followed a terrifying sharp turn, where three out of ten cars would go off the road. Next, there was a dark and winding tunnel where one out of ten cars w... | {
"ground_truth": "69.76",
"style": "rule"
} | {
"index": "5526",
"split": "train"
} |
math | [
{
"content": "Given that the lengths of the three sides of $\\triangle ABC$ form an arithmetic sequence with a common difference of 2, and the sine of the largest angle is $\\frac{\\sqrt{3}}{2}$, the sine of the smallest angle of this triangle is \\_\\_\\_\\_\\_\\_. Please output the final answer within \\boxed... | {
"ground_truth": "\\frac{3\\sqrt{3}}{14}",
"style": "rule"
} | {
"index": "5527",
"split": "train"
} |
math | [
{
"content": "Find the units digit of the decimal expansion of $\\left(15 + \\sqrt{220}\\right)^{19} + \\left(15 + \\sqrt{220}\\right)^{82}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9",
"style": "rule"
} | {
"index": "5528",
"split": "train"
} |
math | [
{
"content": "In Yang's number theory class, Michael K, Michael M, and Michael R take a series of tests. Afterwards, Yang makes the following observations about the test scores:\n\n(a) Michael K had an average test score of $90$ , Michael M had an average test score of $91$ , and Michael R had an average test... | {
"ground_truth": "413",
"style": "rule"
} | {
"index": "5529",
"split": "train"
} |
math | [
{
"content": "A store sells jellybeans at a fixed price per gram. The price for 250 g of jellybeans is $\\$ 7.50$. What mass of jellybeans sells for $\\$ 1.80$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60 \\mathrm{~g}",
"style": "rule"
} | {
"index": "5530",
"split": "train"
} |
math | [
{
"content": "How many distinct sequences of five letters can be made from the letters in \"PROBLEMS\" if each letter can be used only once and each sequence must begin with \"S\" and not end with \"M\"? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "720",
"style": "rule"
} | {
"index": "5531",
"split": "train"
} |
math | [
{
"content": "Tessa has a unit cube, on which each vertex is labeled by a distinct integer between 1 and 8 inclusive. She also has a deck of 8 cards, 4 of which are black and 4 of which are white. At each step she draws a card from the deck, and if the card is black, she simultaneously replaces the number on ea... | {
"ground_truth": "42648",
"style": "rule"
} | {
"index": "5532",
"split": "train"
} |
math | [
{
"content": "In triangle $ABC,$ $D,$ $E,$ and $F$ are points on sides $\\overline{BC},$ $\\overline{AC},$ and $\\overline{AB},$ respectively, so that $BD:DC = CE:EA = AF:FB = 1:2.$\n\n[asy]\nunitsize(0.8 cm);\n\npair A, B, C, D, E, F, P, Q, R;\n\nA = (2,5);\nB = (0,0);\nC = (7,0);\nD = interp(B,C,1/3);\nE = in... | {
"ground_truth": "\\frac{1}{7}",
"style": "rule"
} | {
"index": "5533",
"split": "train"
} |
math | [
{
"content": "Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the length of the real a... | {
"ground_truth": "\\frac{5}{3}",
"style": "rule"
} | {
"index": "5534",
"split": "train"
} |
math | [
{
"content": "In a school event, there are 360 students participating. They need to be arranged in rows such that each row contains the same number of students. The arrangement must have at least 12 rows and at least 18 students in each row. Calculate the sum of all possible numbers of students per row. Please ... | {
"ground_truth": "92",
"style": "rule"
} | {
"index": "5535",
"split": "train"
} |
math | [
{
"content": "A circle with center $O$ is tangent to the coordinate axes and to the hypotenuse of the $30^\\circ$-$60^\\circ$-$90^\\circ$ triangle $ABC$ as shown, where $AB=1$. To the nearest hundredth, what is the radius of the circle? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2.37",
"style": "rule"
} | {
"index": "5536",
"split": "train"
} |
math | [
{
"content": "The square was cut into 25 smaller squares, of which exactly one has a side length different from 1 (each of the others has a side length of 1).\n\nFind the area of the original square. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "49",
"style": "rule"
} | {
"index": "5537",
"split": "train"
} |
math | [
{
"content": "Given the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), the left and right foci coincide with the symmetric points about the two asymptotes, respectively. Then, the eccentricity of the hyperbola is __________. Please output the final answer within \\boxed{}.",
"role": "... | {
"ground_truth": "\\sqrt{2}",
"style": "rule"
} | {
"index": "5538",
"split": "train"
} |
math | [
{
"content": "A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1979",
"style": "rule"
} | {
"index": "5539",
"split": "train"
} |
math | [
{
"content": "A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is: Ple... | {
"ground_truth": "2(n-1)",
"style": "rule"
} | {
"index": "5540",
"split": "train"
} |
math | [
{
"content": "Let $A_{11}$ denote the answer to problem 11. Determine the smallest prime $p$ such that the arithmetic sequence $p, p+A_{11}, p+2 A_{11}, \\ldots$ begins with the largest possible number of primes. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "5541",
"split": "train"
} |
math | [
{
"content": "If $8^x - 8^{x-1} = 60$, calculate the value of $(3x)^x$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "58.9",
"style": "rule"
} | {
"index": "5542",
"split": "train"
} |
math | [
{
"content": "(1) If $\\cos (\\frac{\\pi}{4}+x) = \\frac{3}{5}$, and $\\frac{17}{12}\\pi < x < \\frac{7}{4}\\pi$, find the value of $\\frac{\\sin 2x + 2\\sin^2 x}{1 - \\tan x}$.\n\n(2) Given the function $f(x) = 2\\sqrt{3}\\sin x\\cos x + 2\\cos^2 x - 1 (x \\in \\mathbb{R})$, if $f(x_0) = \\frac{6}{5}$, and $x_... | {
"ground_truth": "\\frac{3 - 4\\sqrt{3}}{10}",
"style": "rule"
} | {
"index": "5543",
"split": "train"
} |
math | [
{
"content": "Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "105",
"style": "rule"
} | {
"index": "5544",
"split": "train"
} |
math | [
{
"content": "A certain integer has $4$ digits when written in base $8$. The same integer has $d$ digits when written in base $2$. What is the sum of all possible values of $d$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "5545",
"split": "train"
} |
math | [
{
"content": "Suppose that $A$, $B$, and $C$ are non-zero distinct digits less than $6$, and suppose we have ${AB_6}+{C_6}={C}0_6$ and ${AB_6}+{BA_6}={CC_6}$. Find the three-digit number ${ABC}$. (Interpret $AB_6$ as a base-6 number with digits $A$ and $B$, not as $A$ times $B$. The other expressions should be ... | {
"ground_truth": "415",
"style": "rule"
} | {
"index": "5546",
"split": "train"
} |
math | [
{
"content": "Find the coefficient of $x^{90}$ in the expansion of\n\\[(x - 1)(x^2 - 2)(x^3 - 3) \\dotsm (x^{12} - 12)(x^{13} - 13).\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5547",
"split": "train"
} |
math | [
{
"content": "An octagon $ABCDEFGH$ is divided into eight smaller equilateral triangles, such as $\\triangle ABJ$ (where $J$ is the center of the octagon), shown in boldface in the diagram. By connecting every third vertex, we obtain a larger equilateral triangle $\\triangle ADE$, which is also shown in boldfac... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "5548",
"split": "train"
} |
math | [
{
"content": "An ordered pair $(n,p)$ is *juicy* if $n^{2} \\equiv 1 \\pmod{p^{2}}$ and $n \\equiv -1 \\pmod{p}$ for positive integer $n$ and odd prime $p$ . How many juicy pairs exist such that $n,p \\leq 200$ ?\n\n\nProposed by Harry Chen (Extile) Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "5549",
"split": "train"
} |
math | [
{
"content": "The number \\( n \\) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \\( n \\)? Please output the final ... | {
"ground_truth": "2891",
"style": "rule"
} | {
"index": "5550",
"split": "train"
} |
math | [
{
"content": "Given that a match between two people is played with a best-of-five-games format, where the winner is the first to win three games, and that the probability of person A winning a game is $\\dfrac{2}{3}$, calculate the probability that person A wins with a score of $3:1$. Please output the final an... | {
"ground_truth": "\\dfrac{8}{27}",
"style": "rule"
} | {
"index": "5551",
"split": "train"
} |
math | [
{
"content": "Given that $A$, $B$, and $C$ are three fixed points on the surface of a sphere with radius $1$, and $AB=AC=BC=1$, the vertex $P$ of a cone $P-ABC$ with a height of $\\frac{\\sqrt{6}}{2}$ is also located on the same spherical surface. Determine the area of the planar region enclosed by the trajecto... | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "5552",
"split": "train"
} |
math | [
{
"content": "Determine the value of $x$ that satisfies $\\sqrt[4]{x\\sqrt{x^4}}=4$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2^{8/3}",
"style": "rule"
} | {
"index": "5553",
"split": "train"
} |
math | [
{
"content": "From 3 male students and 2 female students, calculate the number of different election results in which at least one female student is elected. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "5554",
"split": "train"
} |
math | [
{
"content": "Simplify $$\\sqrt{6+4\\sqrt2}+\\sqrt{6-4\\sqrt2}.$$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5555",
"split": "train"
} |
math | [
{
"content": "Given a box contains $5$ shiny pennies and $6$ dull pennies, determine the sum of the numerator and denominator of the probability that it will take exactly six draws to get the fourth shiny penny. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "236",
"style": "rule"
} | {
"index": "5556",
"split": "train"
} |
math | [
{
"content": "Find the number of different numbers of the form $\\left\\lfloor\\frac{i^2}{2015} \\right\\rfloor$ , with $i = 1,2, ..., 2015$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2016",
"style": "rule"
} | {
"index": "5557",
"split": "train"
} |
math | [
{
"content": "How many numbers are in the list $ 4, 6, 8, \\ldots, 128, 130 ?$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "5558",
"split": "train"
} |
math | [
{
"content": "Lisa drew graphs of all functions of the form \\( y = ax + b \\), where \\( a \\) and \\( b \\) take all natural values from 1 to 100. How many of these graphs pass through the point \\((3, 333)\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "33",
"style": "rule"
} | {
"index": "5559",
"split": "train"
} |
math | [
{
"content": "Given that $m$ is a positive integer, and given that $\\mathop{\\text{lcm}}[40,m]=120$ and $\\mathop{\\text{lcm}}[m,45]=180$, what is $m$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "5560",
"split": "train"
} |
math | [
{
"content": "Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos? Please output the final answer within... | {
"ground_truth": "37/42",
"style": "rule"
} | {
"index": "5561",
"split": "train"
} |
math | [
{
"content": "Suppose $f(x) = 6x - 9$ and $g(x) = \\frac{x}{3} + 2$. Find $f(g(x)) - g(f(x))$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "5562",
"split": "train"
} |
math | [
{
"content": "Define a set \\( \\mathcal{T} \\) of distinct positive integers such that for every integer \\( y \\) in \\( \\mathcal{T}, \\) the geometric mean of the values obtained by omitting \\( y \\) from \\( \\mathcal{T} \\) remains a positive integer. In addition, assume that 1 is a member of \\( \\mathc... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5563",
"split": "train"
} |
math | [
{
"content": "Let $w = \\dfrac{\\sqrt{3} + i}{2}$ and $z = \\dfrac{-1 + i\\sqrt{3}}{2},$ where $i = \\sqrt{-1}.$ Find the number of ordered pairs $(r,s)$ of positive integers not exceeding $100$ that satisfy the equation $i \\cdot w^r = z^s.$ Please output the final answer within \\boxed{}.",
"role": "user"... | {
"ground_truth": "834",
"style": "rule"
} | {
"index": "5564",
"split": "train"
} |
math | [
{
"content": "Mathematical operation refers to the ability to solve mathematical problems based on clear operation objects and operation rules. Because of operations, the power of numbers is infinite; without operations, numbers are just symbols. Logarithmic operation and exponential operation are two important... | {
"ground_truth": "\\frac{1}{2}",
"style": "rule"
} | {
"index": "5565",
"split": "train"
} |
math | [
{
"content": "Simplify the expression $\\frac{15b^4 - 45b^3}{75b^2}$ when $b=2$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{2}{5}",
"style": "rule"
} | {
"index": "5566",
"split": "train"
} |
math | [
{
"content": "Calculate the volume of the tetrahedron with vertices at points \\( A_{1}, A_{2}, A_{3}, A_{4} \\) and its height dropped from the vertex \\( A_{4} \\) onto the face \\( A_{1} A_{2} A_{3} \\).\n\nGiven points:\n\\( A_{1}(1, -1, 1) \\)\n\n\\( A_{2}(-2, 0, 3) \\)\n\n\\( A_{3}(2, 1, -1) \\)\n\n\\( A_... | {
"ground_truth": "\\frac{33}{\\sqrt{101}}",
"style": "rule"
} | {
"index": "5567",
"split": "train"
} |
math | [
{
"content": "Given 6000 cards, each with a unique natural number from 1 to 6000 written on it. It is required to choose two cards such that the sum of the numbers on them is divisible by 100. In how many ways can this be done? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "179940",
"style": "rule"
} | {
"index": "5568",
"split": "train"
} |
math | [
{
"content": "A point $P$ lies in the same plane as a given square of side $1$. Let the vertices of the square, taken counterclockwise, be $A, B, C$ and $D$. Also, let the distances from $P$ to $A, B$ and $C$, respectively, be $u, v$ and $w$. What is the greatest distance that $P$ can be from $D$ if $u^2 + v^2 ... | {
"ground_truth": "$2 + \\sqrt{2}$",
"style": "rule"
} | {
"index": "5569",
"split": "train"
} |
math | [
{
"content": "Let $M = 123456789101112\\dots5354$ be the number that results from writing the integers from $1$ to $54$ consecutively. What is the remainder when $M$ is divided by $55$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "44",
"style": "rule"
} | {
"index": "5570",
"split": "train"
} |
math | [
{
"content": "Find all positive integers $n$ such that the inequality $$\\left( \\sum\\limits_{i=1}^n a_i^2\\right) \\left(\\sum\\limits_{i=1}^n a_i \\right) -\\sum\\limits_{i=1}^n a_i^3 \\geq 6 \\prod\\limits_{i=1}^n a_i$$ holds for any $n$ positive numbers $a_1, \\dots, a_n$. Please output the final answer wi... | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "5571",
"split": "train"
} |
math | [
{
"content": "In an isosceles trapezoid, the longer base \\(AB\\) is 24 units, the shorter base \\(CD\\) is 12 units, and each of the non-parallel sides has a length of 13 units. What is the length of the diagonal \\(AC\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{457}",
"style": "rule"
} | {
"index": "5572",
"split": "train"
} |
math | [
{
"content": "Mike has two containers. One container is a rectangular prism with width 2 cm, length 4 cm, and height 10 cm. The other is a right cylinder with radius 1 cm and height 10 cm. Both containers sit on a flat surface. Water has been poured into the two containers so that the height of the water in bot... | {
"ground_truth": "7.2",
"style": "rule"
} | {
"index": "5573",
"split": "train"
} |
math | [
{
"content": "What is the result if you add the largest odd two-digit number to the smallest even three-digit number? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "199",
"style": "rule"
} | {
"index": "5574",
"split": "train"
} |
math | [
{
"content": "The digits $0,1,2,3,4,5,6$ are randomly arranged in a sequence. What is the probability of obtaining a seven-digit number that is divisible by four? (The number cannot start with zero.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.25",
"style": "rule"
} | {
"index": "5575",
"split": "train"
} |
math | [
{
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $4\\cos C \\cdot \\sin^2 \\frac{C}{2} + \\cos 2C = 0$.\n\n(1) If $\\tan A = 2\\tan B$, find the value of $\\sin(A-B)$;\n\n(2) If $3ab = 25 - c^2$, find the maximum area of $... | {
"ground_truth": "\\frac{25\\sqrt{3}}{16}",
"style": "rule"
} | {
"index": "5576",
"split": "train"
} |
math | [
{
"content": "The function \\( f(n) \\) is defined on the set of natural numbers \\( N \\) as follows:\n\\[ f(n) = \\begin{cases} \nn - 3, & \\text{ if } n \\geqslant 1000, \\\\\nf[f(n + 7)], & \\text{ if } n < 1000. \n\\end{cases} \\]\n\nWhat is the value of \\( f(90) \\)? Please output the final answer within... | {
"ground_truth": "999",
"style": "rule"
} | {
"index": "5577",
"split": "train"
} |
math | [
{
"content": "Find the product of all possible real values for $k$ such that the system of equations $$ x^2+y^2= 80 $$ $$ x^2+y^2= k+2x-8y $$ has exactly one real solution $(x,y)$ .\n\n*Proposed by Nathan Xiong* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "960",
"style": "rule"
} | {
"index": "5578",
"split": "train"
} |
math | [
{
"content": "For positive integers $n$, the number of pairs of different adjacent digits in the binary (base two) representation of $n$ can be denoted as $D(n)$. Determine the number of positive integers less than or equal to 97 for which $D(n) = 2$. Please output the final answer within \\boxed{}.",
"role... | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "5579",
"split": "train"
} |
math | [
{
"content": "Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\\frac mn,$ where $m$ and $n$ are relatively prime positive int... | {
"ground_truth": "742",
"style": "rule"
} | {
"index": "5580",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\( AB \\parallel EF \\parallel DC \\). Given that \\( AC + BD = 250 \\), \\( BC = 100 \\), and \\( EC + ED = 150 \\), find \\( CF \\). Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "5581",
"split": "train"
} |
math | [
{
"content": "Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "87341",
"style": "rule"
} | {
"index": "5582",
"split": "train"
} |
math | [
{
"content": "Compute\n\\[e^{2 \\pi i/13} + e^{4 \\pi i/13} + e^{6 \\pi i/13} + \\dots + e^{24 \\pi i/13}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5583",
"split": "train"
} |
math | [
{
"content": "$(1)$ Calculate: $\\sqrt{12}-(-\\frac{1}{2})^{-1}-|\\sqrt{3}+3|+(2023-\\pi)^0$<br/>$(2)$ Simplify the algebraic expression $\\frac{3x-8}{x-1}-\\frac{x+1}{x}÷\\frac{{x}^{2}-1}{{x}^{2}-3x}$, and select appropriate integers from $0 \\lt x\\leqslant 3$ to substitute and find the value. Please output t... | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "5584",
"split": "train"
} |
math | [
{
"content": "Define an ordered quadruple of integers $(a, b, c, d)$ as captivating if $1 \\le a < b < c < d \\le 15$, and $a+d > 2(b+c)$. How many captivating ordered quadruples are there? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "200",
"style": "rule"
} | {
"index": "5585",
"split": "train"
} |
math | [
{
"content": "A quadrilateral has vertices at $(0,1)$, $(3,4)$, $(4,3)$ and $(3,0)$. Its perimeter can be expressed in the form $a\\sqrt2+b\\sqrt{10}$ with $a$ and $b$ integers. What is the sum of $a$ and $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "5586",
"split": "train"
} |
math | [
{
"content": "Find the integer \\( m \\) so that\n\\[ 10^{m} < \\frac{1}{2} \\times \\frac{3}{4} \\times \\frac{5}{6} \\times \\ldots \\times \\frac{99}{100} < 10^{m+1} \\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-2",
"style": "rule"
} | {
"index": "5587",
"split": "train"
} |
math | [
{
"content": "A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. [asy]\nsize(7cm... | {
"ground_truth": "\\dfrac{1001}{2003}",
"style": "rule"
} | {
"index": "5588",
"split": "train"
} |
math | [
{
"content": "Calculate the integrals:\n1) \\(\\int_{0}^{\\frac{\\pi}{2}} \\sin ^{3} x \\, dx\\);\n2) \\(\\int_{0}^{\\ln 2} \\sqrt{e^{x}-1} \\, dx\\);\n3) \\(\\int_{-a}^{a} x^{2} \\sqrt{a^{2}-x^{2}} \\, dx\\);\n4) \\(\\int_{1}^{2} \\frac{\\sqrt{x^{2}-1}}{x} \\, dx\\). Please output the final answer within \\box... | {
"ground_truth": "\\sqrt{3} - \\frac{\\pi}{3}",
"style": "rule"
} | {
"index": "5589",
"split": "train"
} |
math | [
{
"content": "Two skiers started from the same point one after another with an interval of 9 minutes. The second skier caught up with the first one 9 km from the starting point. After reaching the “27 km” mark, the second skier turned back and met the first skier at a distance of 2 km from the turning point. Fi... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "5590",
"split": "train"
} |
math | [
{
"content": "Acute triangle $A B C$ has circumcenter $O$. The bisector of $\\angle A B C$ and the altitude from $C$ to side $A B$ intersect at $X$. Suppose that there is a circle passing through $B, O, X$, and $C$. If $\\angle B A C=n^{\\circ}$, where $n$ is a positive integer, compute the largest possible val... | {
"ground_truth": "67",
"style": "rule"
} | {
"index": "5591",
"split": "train"
} |
math | [
{
"content": "How many positive integers \\( n \\) satisfy \\[ (n + 9)(n - 4)(n - 13) < 0 \\]? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "5592",
"split": "train"
} |
math | [
{
"content": "Given $a \\in \\mathbb{R}$, the function $f(x) = ax^3 - 3x^2$, and $x = 2$ is an extreme point of the function $y = f(x)$.\n\n1. Find the value of $a$.\n2. Find the extreme values of the function $f(x)$ in the interval $[-1, 5]$. Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "50",
"style": "rule"
} | {
"index": "5593",
"split": "train"
} |
math | [
{
"content": "Let the ellipse \\\\(C: \\dfrac{x^2}{a^2} + \\dfrac{y^2}{b^2} = 1, (a > b > 0)\\\\) have an eccentricity of \\\\(\\dfrac{2\\sqrt{2}}{3}\\\\), and it is inscribed in the circle \\\\(x^2 + y^2 = 9\\\\).\n\n\\\\((1)\\\\) Find the equation of ellipse \\\\(C\\\\).\n\n\\\\((2)\\\\) A line \\\\(l\\\\) (n... | {
"ground_truth": "-\\dfrac{9}{4}",
"style": "rule"
} | {
"index": "5594",
"split": "train"
} |
math | [
{
"content": "A pair of natural numbers is called \"good\" if one of the numbers is divisible by the other. The numbers from 1 to 30 are divided into 15 pairs. What is the maximum number of good pairs that could be formed? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "13",
"style": "rule"
} | {
"index": "5595",
"split": "train"
} |
math | [
{
"content": "Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that\n\\[\\mu(A_{k-1}A_... | {
"ground_truth": "2\\pi",
"style": "rule"
} | {
"index": "5596",
"split": "train"
} |
math | [
{
"content": "$ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{51}",
"style": "rule"
} | {
"index": "5597",
"split": "train"
} |
math | [
{
"content": "Tom, Dick, and Harry each flip a fair coin repeatedly until they get their first tail. Calculate the probability that all three flip their coins an even number of times and they all get their first tail on the same flip. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{63}",
"style": "rule"
} | {
"index": "5598",
"split": "train"
} |
math | [
{
"content": "Let $\\mathbf{a},$ $\\mathbf{b},$ $\\mathbf{c},$ $\\mathbf{d}$ be four distinct unit vectors in space such that\n\\[\\mathbf{a} \\cdot \\mathbf{b} = \\mathbf{a} \\cdot \\mathbf{c} = \\mathbf{b} \\cdot \\mathbf{c} =\\mathbf{b} \\cdot \\mathbf{d} = \\mathbf{c} \\cdot \\mathbf{d} = -\\frac{1}{11}.\\... | {
"ground_truth": "-\\frac{53}{55}",
"style": "rule"
} | {
"index": "5599",
"split": "train"
} |
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