data_source large_stringclasses 1
value | prompt listlengths 1 1 | reward_model dict | extra_info dict |
|---|---|---|---|
math | [
{
"content": "Evaluate $\\left\\lceil\\left(-\\frac{5}{3}\\right)^2\\right\\rceil$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "6100",
"split": "train"
} |
math | [
{
"content": "Chicks hatch on the night from Sunday to Monday. For two weeks, a chick sits with its beak open, during the third week it silently grows feathers, and during the fourth week it flies out of the nest. Last week, there were 20 chicks in the nest sitting with their beaks open, and 14 growing feathers... | {
"ground_truth": "165",
"style": "rule"
} | {
"index": "6101",
"split": "train"
} |
math | [
{
"content": "Ryan is learning number theory. He reads about the *Möbius function* $\\mu : \\mathbb N \\to \\mathbb Z$ , defined by $\\mu(1)=1$ and\n\\[ \\mu(n) = -\\sum_{\\substack{d\\mid n d \\neq n}} \\mu(d) \\]\nfor $n>1$ (here $\\mathbb N$ is the set of positive integers).\nHowever, Ryan doesn't li... | {
"ground_truth": "14013",
"style": "rule"
} | {
"index": "6102",
"split": "train"
} |
math | [
{
"content": "Determine which of the following numbers is smallest in value: $54 \\sqrt{3}, 144,108 \\sqrt{6}-108 \\sqrt{2}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$54 \\sqrt{3}$",
"style": "rule"
} | {
"index": "6103",
"split": "train"
} |
math | [
{
"content": "How many two-digit prime numbers have the property that both digits are also primes? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6104",
"split": "train"
} |
math | [
{
"content": "An electrician was called to repair a garland of four light bulbs connected in series, one of which has burned out. It takes 10 seconds to unscrew any bulb from the garland and 10 seconds to screw it back in. The time spent on other actions is negligible. What is the minimum time in which the elec... | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "6105",
"split": "train"
} |
math | [
{
"content": "Given the function $f(x) = -\\frac{1}{3}x^3 + x^2 + ax + b$ reaches an extreme value of $4$ at $x=3$, determine the maximum value of $f(x)$ on the interval $[-2,1]$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-\\frac{4}{3}",
"style": "rule"
} | {
"index": "6106",
"split": "train"
} |
math | [
{
"content": "Given that $\\binom{21}{13}=20349$, $\\binom{21}{14}=11628$, and $\\binom{23}{15}=490314$, find $\\binom{22}{15}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "458337",
"style": "rule"
} | {
"index": "6107",
"split": "train"
} |
math | [
{
"content": "What is the maximum number of colours that can be used to paint an $8 \\times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?\n\n(A Shapovalov) Please output the final ... | {
"ground_truth": "16",
"style": "rule"
} | {
"index": "6108",
"split": "train"
} |
math | [
{
"content": "A chord $AB$ that makes an angle of $\\frac{\\pi}{6}$ with the horizontal passes through the left focus $F_1$ of the hyperbola $x^{2}- \\frac{y^{2}}{3}=1$. \n$(1)$ Find $|AB|$; \n$(2)$ Find the perimeter of $\\triangle F_{2}AB$ ($F_{2}$ is the right focus). Please output the final answer within ... | {
"ground_truth": "3+3\\sqrt{3}",
"style": "rule"
} | {
"index": "6109",
"split": "train"
} |
math | [
{
"content": "Rodney is now guessing a secret number based on these clues:\n- It is a two-digit integer.\n- The tens digit is even.\n- The units digit is odd.\n- The number is greater than 50. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{10}",
"style": "rule"
} | {
"index": "6110",
"split": "train"
} |
math | [
{
"content": "Eight numbers \\( a_{1}, a_{2}, a_{3}, a_{4} \\) and \\( b_{1}, b_{2}, b_{3}, b_{4} \\) satisfy the following equations:\n\n$$\n\\left\\{\\begin{array}{c}\na_{1} b_{1}+a_{2} b_{3}=1 \\\\\na_{1} b_{2}+a_{2} b_{4}=0 \\\\\na_{3} b_{1}+a_{4} b_{3}=0 \\\\\na_{3} b_{2}+a_{4} b_{4}=1\n\\end{array}\\right... | {
"ground_truth": "-6",
"style": "rule"
} | {
"index": "6111",
"split": "train"
} |
math | [
{
"content": "Vasya has a stick that is 22 cm long. He wants to break it into three pieces with integer lengths such that the pieces can form a triangle. In how many ways can he do this? (Ways that result in identical triangles are considered the same). Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6112",
"split": "train"
} |
math | [
{
"content": "In rectangle \\( ABCD \\), a circle \\(\\omega\\) is constructed using side \\( AB \\) as its diameter. Let \\( P \\) be the second intersection point of segment \\( AC \\) with circle \\(\\omega\\). The tangent to \\(\\omega\\) at point \\( P \\) intersects segment \\( BC \\) at point \\( K \\) a... | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6113",
"split": "train"
} |
math | [
{
"content": "ABCD is a square. BDEF is a rhombus with A, E, and F collinear. Find ∠ADE. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6114",
"split": "train"
} |
math | [
{
"content": "In a certain base $b$, the square of $22_b$ is $514_b$. What is $b$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "7",
"style": "rule"
} | {
"index": "6115",
"split": "train"
} |
math | [
{
"content": "In the diagram, \\( AB \\) is the diameter of circle \\( O \\) with a length of 6 cm. One vertex \\( E \\) of square \\( BCDE \\) is on the circumference of the circle, and \\( \\angle ABE = 45^\\circ \\). Find the difference in area between the non-shaded region of circle \\( O \\) and the non-sh... | {
"ground_truth": "10.26",
"style": "rule"
} | {
"index": "6116",
"split": "train"
} |
math | [
{
"content": "Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? Pleas... | {
"ground_truth": "\\[\n\\boxed{\\frac{n+1}{4n-2}}\n\\]",
"style": "rule"
} | {
"index": "6117",
"split": "train"
} |
math | [
{
"content": "Given the fraction $\\frac{987654321}{2^{24}\\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point needed to express it as a decimal. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6118",
"split": "train"
} |
math | [
{
"content": "Let $p,$ $q,$ $r,$ $s$ be distinct real numbers such that the roots of $x^2 - 12px - 13q = 0$ are $r$ and $s,$ and the roots of $x^2 - 12rx - 13s = 0$ are $p$ and $q.$ Calculate the value of $p + q + r + s.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2028",
"style": "rule"
} | {
"index": "6119",
"split": "train"
} |
math | [
{
"content": "Let $\\{a_{n}\\}$ be an arithmetic sequence with a common difference of $d$, and $d \\gt 1$. Define $b_{n}=\\frac{{n}^{2}+n}{{a}_{n}}$, and let $S_{n}$ and $T_{n}$ be the sums of the first $n$ terms of the sequences $\\{a_{n}\\}$ and $\\{b_{n}\\}$, respectively. \n$(1)$ If $3a_{2}=3a_{1}+a_{3}$ a... | {
"ground_truth": "\\frac{51}{50}",
"style": "rule"
} | {
"index": "6120",
"split": "train"
} |
math | [
{
"content": "Find the distance between the foci of the hyperbola $x^2 - 4x - 9y^2 - 18y = 45.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{40}{3}",
"style": "rule"
} | {
"index": "6121",
"split": "train"
} |
math | [
{
"content": "If\n\\[\n\\sum_{n=1}^{\\infty}\\frac{\\frac11 + \\frac12 + \\dots + \\frac 1n}{\\binom{n+100}{100}} = \\frac pq\n\\]\nfor relatively prime positive integers $p,q$ , find $p+q$ .\n\n*Proposed by Michael Kural* Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "9901",
"style": "rule"
} | {
"index": "6122",
"split": "train"
} |
math | [
{
"content": "Lines $l$ and $k$ are parallel to each other. $m\\angle A = 120^\\circ$, and $m\\angle C = 80^\\circ$. What is the number of degrees in $m\\angle B$?\n\n[asy]\nsize(100); real h = 1.2; currentpen = fontsize(10pt);\ndraw(Label(\"$l$\",Relative(1)),(0,0)--(1,0),E);\ndraw(Label(\"$k$\",Relative(1)),(... | {
"ground_truth": "160^\\circ",
"style": "rule"
} | {
"index": "6123",
"split": "train"
} |
math | [
{
"content": "Given that $O$ is the origin of coordinates, and $M$ is a point on the ellipse $\\frac{x^2}{2} + y^2 = 1$. Let the moving point $P$ satisfy $\\overrightarrow{OP} = 2\\overrightarrow{OM}$.\n- (I) Find the equation of the trajectory $C$ of the moving point $P$;\n- (II) If the line $l: y = x + m (m \... | {
"ground_truth": "2\\sqrt{2}",
"style": "rule"
} | {
"index": "6124",
"split": "train"
} |
math | [
{
"content": "Except for the first two terms, each term of the sequence $2000, y, 2000 - y,\\ldots$ is obtained by subtracting the preceding term from the one before that. The last term of the sequence is the first negative term encountered. What positive integer $y$ produces a sequence of maximum length? Pleas... | {
"ground_truth": "1236",
"style": "rule"
} | {
"index": "6125",
"split": "train"
} |
math | [
{
"content": "Let \\( S \\) be a set of size 11. A random 12-tuple \\((s_1, s_2, \\ldots, s_{12})\\) of elements of \\( S \\) is chosen uniformly at random. Moreover, let \\(\\pi: S \\rightarrow S\\) be a permutation of \\( S \\) chosen uniformly at random. The probability that \\( s_{i+1} \\neq \\pi(s_i) \\) f... | {
"ground_truth": "1000000000004",
"style": "rule"
} | {
"index": "6126",
"split": "train"
} |
math | [
{
"content": "Lily is riding her bicycle at a constant rate of 15 miles per hour and Leo jogs at a constant rate of 9 miles per hour. If Lily initially sees Leo 0.75 miles in front of her and later sees him 0.75 miles behind her, determine the duration of time, in minutes, that she can see Leo. Please output th... | {
"ground_truth": "15",
"style": "rule"
} | {
"index": "6127",
"split": "train"
} |
math | [
{
"content": "Find the real solution(s) to the equation $(x+y)^{2}=(x+1)(y-1)$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(-1,1)",
"style": "rule"
} | {
"index": "6128",
"split": "train"
} |
math | [
{
"content": "How many ordered pairs $(m,n)$ of positive integers are solutions to\n\\[\\frac{4}{m}+\\frac{2}{n}=1?\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "4",
"style": "rule"
} | {
"index": "6129",
"split": "train"
} |
math | [
{
"content": "There exist constants $a$ and $b$ so that\n\\[\\cos^3 \\theta = a \\cos 3 \\theta + b \\cos \\theta\\]for all angles $\\theta.$ Enter the ordered pair $(a,b).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\left( \\frac{1}{4}, \\frac{3}{4} \\right)",
"style": "rule"
} | {
"index": "6130",
"split": "train"
} |
math | [
{
"content": "A number $x$ is randomly chosen from the interval $[-1, 1]$. What is the probability that the value of $\\cos \\frac{\\pi x}{2}$ lies between $0$ and $\\frac{1}{2}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$\\frac{1}{3}$",
"style": "rule"
} | {
"index": "6131",
"split": "train"
} |
math | [
{
"content": "Let $ABC$ be a right triangle, right at $B$ , and let $M$ be the midpoint of the side $BC$ . Let $P$ be the point in\nbisector of the angle $ \\angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$ ). Determine the triangle area $ABC$ if $PM = 1$ an... | {
"ground_truth": "120",
"style": "rule"
} | {
"index": "6132",
"split": "train"
} |
math | [
{
"content": "A rectangle has positive integer side lengths and an area of 24. What perimeter of the rectangle cannot be? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "36",
"style": "rule"
} | {
"index": "6133",
"split": "train"
} |
math | [
{
"content": "Let \\(\\left(x^{2}+2x-2\\right)^{6}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\\cdots+a_{12}(x+2)^{12}\\), where \\(a_{i} (i=0,1,2,\\ldots,12)\\) are real constants. Determine the value of \\(a_{0}+a_{1}+2a_{2}+3a_{3}+\\cdots+12a_{12}\\). Please output the final answer within \\boxed{}.",
"role": "user... | {
"ground_truth": "64",
"style": "rule"
} | {
"index": "6134",
"split": "train"
} |
math | [
{
"content": "Given the arithmetic sequence $\\{a_n\\}$, it is given that $a_2+a_8-a_{12}=0$ and $a_{14}-a_4=2$. Let $s_n=a_1+a_2+\\ldots+a_n$, then determine the value of $s_{15}$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "30",
"style": "rule"
} | {
"index": "6135",
"split": "train"
} |
math | [
{
"content": "In the parallelogram $ABCD$ , a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$ . If $F$ is the midpoint of $AB$ and the area of $\\vartriangle BEC$ is $100$ , find the area of the quadrilateral $AFED$ . Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "250",
"style": "rule"
} | {
"index": "6136",
"split": "train"
} |
math | [
{
"content": "Find all nonnegative integer solutions $(x,y,z,w)$ of the equation\\[2^x\\cdot3^y-5^z\\cdot7^w=1.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "(1, 1, 1, 0), (2, 2, 1, 1), (1, 0, 0, 0), (3, 0, 0, 1)",
"style": "rule"
} | {
"index": "6137",
"split": "train"
} |
math | [
{
"content": "Let $a,$ $b,$ $c$ be positive real numbers such that $a + b + c = 1.$ Find the minimum value of\n\\[\\frac{1}{a + 2b} + \\frac{1}{b + 2c} + \\frac{1}{c + 2a}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "3",
"style": "rule"
} | {
"index": "6138",
"split": "train"
} |
math | [
{
"content": "In the numbers from 100 to 999, how many numbers have digits in strictly increasing or strictly decreasing order?\n\n(From the 41st American High School Mathematics Exam, 1990) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "204",
"style": "rule"
} | {
"index": "6139",
"split": "train"
} |
math | [
{
"content": "Suppose that $P(z)$, $Q(z)$, and $R(z)$ are polynomials with real coefficients, having degrees $2$, $3$, and $6$, respectively, and constant terms $1$, $2$, and $3$, respectively. Let $N$ be the number of distinct complex numbers $z$ that satisfy the equation $P(z) \\cdot Q(z) = R(z)$. What is the... | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6140",
"split": "train"
} |
math | [
{
"content": "For any $h = 2^{r}$ ($r$ is a non-negative integer), find all $k \\in \\mathbb{N}$ which satisfy the following condition: There exists an odd natural number $m > 1$ and $n \\in \\mathbb{N}$, such that $k \\mid m^{h} - 1, m \\mid n^{\\frac{m^{h}-1}{k}} + 1$. Please output the final answer within \\... | {
"ground_truth": "2^{r+1}",
"style": "rule"
} | {
"index": "6141",
"split": "train"
} |
math | [
{
"content": "Let $f(x) : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[\\frac{f(x) f(y) - f(xy)}{3} = x + y + 2\\]for all $x,$ $y \\in \\mathbb{R}.$ Find $f(x).$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "x + 3",
"style": "rule"
} | {
"index": "6142",
"split": "train"
} |
math | [
{
"content": "For some integers that are not palindromes, like 91, a person can create a palindrome by repeatedly reversing the number and adding the original number to its reverse. For example, $91 + 19 = 110$. Then $110+011 = 121$, which is a palindrome, so 91 takes two steps to become a palindrome. Of all po... | {
"ground_truth": "176",
"style": "rule"
} | {
"index": "6143",
"split": "train"
} |
math | [
{
"content": "Xiao Ming throws a die with uniform density three times and observes the number of points on the upper face each time. It is known that the numbers of points in the three throws are all different. Calculate the probability that the sum of the three numbers of points does not exceed $8$. Please out... | {
"ground_truth": "\\frac{1}{5}",
"style": "rule"
} | {
"index": "6144",
"split": "train"
} |
math | [
{
"content": "Solve for $x$: $3^{2x} = \\sqrt{27}$. Express your answer as a common fraction. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{3}{4}",
"style": "rule"
} | {
"index": "6145",
"split": "train"
} |
math | [
{
"content": "Triangle $ABC$ has side lengths $AB=7, BC=8,$ and $CA=9.$ Circle $\\omega_1$ passes through $B$ and is tangent to line $AC$ at $A.$ Circle $\\omega_2$ passes through $C$ and is tangent to line $AB$ at $A.$ Let $K$ be the intersection of circles $\\omega_1$ and $\\omega_2$ not equal to $A.$ Then $A... | {
"ground_truth": "11",
"style": "rule"
} | {
"index": "6146",
"split": "train"
} |
math | [
{
"content": "If the perimeter of a rectangle is $p$ and its diagonal is $d$, the difference between the length and width of the rectangle is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac {\\sqrt {8d^2 - p^2}}{2}",
"style": "rule"
} | {
"index": "6147",
"split": "train"
} |
math | [
{
"content": "The sum of two numbers $x$ and $y$ is 399, and the value of the fraction $\\frac{x}{y}$ is 0.9. What is the value of $y - x$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "21",
"style": "rule"
} | {
"index": "6148",
"split": "train"
} |
math | [
{
"content": "In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.\nBrian: \"Mike and I are differ... | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "6149",
"split": "train"
} |
math | [
{
"content": "In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\\{(x, y) \\mid(|x|+|3 y|-6)(|3 x|+|y|-6) \\leq 0\\}$ is ________. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "24",
"style": "rule"
} | {
"index": "6150",
"split": "train"
} |
math | [
{
"content": "Let $A B C$ be a triangle with $A B=13, B C=14, C A=15$. Let $O$ be the circumcenter of $A B C$. Find the distance between the circumcenters of triangles $A O B$ and $A O C$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{91}{6}",
"style": "rule"
} | {
"index": "6151",
"split": "train"
} |
math | [
{
"content": "Given that the point $(4,7)$ is on the graph of $y=f(x)$, there is one point that must be on the graph of $2y=3f(4x)+5$. What is the sum of the coordinates of that point? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "6152",
"split": "train"
} |
math | [
{
"content": "In \\( \\triangle ABC \\), \\( AB = AC = 26 \\) and \\( BC = 24 \\). Points \\( D, E, \\) and \\( F \\) are on sides \\( \\overline{AB}, \\overline{BC}, \\) and \\( \\overline{AC}, \\) respectively, such that \\( \\overline{DE} \\) and \\( \\overline{EF} \\) are parallel to \\( \\overline{AC} \\) ... | {
"ground_truth": "52",
"style": "rule"
} | {
"index": "6153",
"split": "train"
} |
math | [
{
"content": "Let a binary operation $\\star$ on ordered pairs of integers be defined by $(a,b)\\star (c,d)=(a-c,b+d)$. Then, if $(3,3)\\star (0,0)$ and $(x,y)\\star (3,2)$ represent identical pairs, $x$ equals: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "$6$",
"style": "rule"
} | {
"index": "6154",
"split": "train"
} |
math | [
{
"content": "If $3+x=5$ and $-3+y=5$, what is the value of $x+y$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10",
"style": "rule"
} | {
"index": "6155",
"split": "train"
} |
math | [
{
"content": "What work is required to stretch a spring by \\(0.06 \\, \\text{m}\\), if a force of \\(1 \\, \\text{N}\\) stretches it by \\(0.01 \\, \\text{m}\\)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "0.18",
"style": "rule"
} | {
"index": "6156",
"split": "train"
} |
math | [
{
"content": "Given that points P1 and P2 are two adjacent centers of symmetry for the curve $y= \\sqrt {2}\\sin ωx-\\cos ωx$ $(x\\in\\mathbb{R})$, if the tangents to the curve at points P1 and P2 are perpendicular to each other, determine the value of ω. Please output the final answer within \\boxed{}.",
"... | {
"ground_truth": "\\frac{\\sqrt{3}}{3}",
"style": "rule"
} | {
"index": "6157",
"split": "train"
} |
math | [
{
"content": "The ten-letter code $\\text{BEST OF LUCK}$ represents the ten digits $0-9$, in order. What 4-digit number is represented by the code word $\\text{CLUE}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "8671",
"style": "rule"
} | {
"index": "6158",
"split": "train"
} |
math | [
{
"content": "Find the volume of the region in space defined by\n\\[ |x + y + 2z| + |x + y - 2z| \\le 12 \\]\nand $x, y, z \\ge 0.$ Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "6159",
"split": "train"
} |
math | [
{
"content": "Given sets $A=\\{1, a, b\\}$ and $B=\\{a, a^2, ab\\}$. If $A=B$, find the value of $a+b$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "-1",
"style": "rule"
} | {
"index": "6160",
"split": "train"
} |
math | [
{
"content": "A certain number of books are distributed among children. If each child gets $m$ books, there are 14 books left. If each child gets 9 books, the last child only gets 6 books. How many children are there in total? And how many books are there? Please output the final answer within \\boxed{}.",
... | {
"ground_truth": "150",
"style": "rule"
} | {
"index": "6161",
"split": "train"
} |
math | [
{
"content": "Let $x$, $y$ and $z$ all exceed $1$ and let $w$ be a positive number such that $\\log_x w = 24$, $\\log_y w = 40$ and $\\log_{xyz} w = 12$. Find $\\log_z w$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "60",
"style": "rule"
} | {
"index": "6162",
"split": "train"
} |
math | [
{
"content": "Given the positive sequence $\\{a_n\\}$, where $a_1=2$, $a_2=1$, and $\\frac {a_{n-1}-a_{n}}{a_{n}a_{n-1}}= \\frac {a_{n}-a_{n+1}}{a_{n}a_{n+1}}(n\\geqslant 2)$, find the value of the 2016th term of this sequence. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{1}{1008}",
"style": "rule"
} | {
"index": "6163",
"split": "train"
} |
math | [
{
"content": "The inclination angle $\\alpha$ of the line $l: \\sqrt{3}x+3y+1=0$ is $\\tan^{-1}\\left( -\\frac{\\sqrt{3}}{3} \\right)$. Calculate the value of the angle $\\alpha$. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{5\\pi}{6}",
"style": "rule"
} | {
"index": "6164",
"split": "train"
} |
math | [
{
"content": "In a set of five consecutive integers, the largest integer is less than twice the average of the five integers. What is the smallest integer that could be in the set? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "1",
"style": "rule"
} | {
"index": "6165",
"split": "train"
} |
math | [
{
"content": "There are three novel series Peter wishes to read. Each consists of 4 volumes that must be read in order, but not necessarily one after the other. Let \\( N \\) be the number of ways Peter can finish reading all the volumes. Find the sum of the digits of \\( N \\). (Assume that he must finish a vo... | {
"ground_truth": "18",
"style": "rule"
} | {
"index": "6166",
"split": "train"
} |
math | [
{
"content": "Determine the coefficient of the term containing $x^3$ in the expansion of ${(1+2x)}^{5}$. (The result should be represented as a number.) Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "80",
"style": "rule"
} | {
"index": "6167",
"split": "train"
} |
math | [
{
"content": "A positive integer $n$ is known as an [i]interesting[/i] number if $n$ satisfies\n\\[{\\ \\{\\frac{n}{10^k}} \\} > \\frac{n}{10^{10}} \\] \nfor all $k=1,2,\\ldots 9$.\nFind the number of interesting numbers. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "999989991",
"style": "rule"
} | {
"index": "6168",
"split": "train"
} |
math | [
{
"content": "Alice, Bob, and Carol play a game in which each of them chooses a real number between 0 and 1. The winner of the game is the one whose number is between the numbers chosen by the other two players. Alice announces that she will choose her number uniformly at random from all the numbers between 0 a... | {
"ground_truth": "\\frac{13}{24}",
"style": "rule"
} | {
"index": "6169",
"split": "train"
} |
math | [
{
"content": "How many distinct products can you obtain by multiplying two or more distinct elements from the set $\\{1, 2, 3, 5, 7, 11\\}$? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "26",
"style": "rule"
} | {
"index": "6170",
"split": "train"
} |
math | [
{
"content": "On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day. Please output the final answer within \\boxed{}.",
"ro... | {
"ground_truth": "46656",
"style": "rule"
} | {
"index": "6171",
"split": "train"
} |
math | [
{
"content": "The Chinese mathematician Qin Jiushao (circa 1202-1261) from the Southern Song Dynasty proposed Qin Jiushao's algorithm for polynomial evaluation in his work \"Mathematical Book in Nine Chapters.\" The provided diagram illustrates an example of using Qin Jiushao's algorithm to evaluate a polynomia... | {
"ground_truth": "$2^{5}+2^{4}+2^{3}+2^{2}+2+1$",
"style": "rule"
} | {
"index": "6172",
"split": "train"
} |
math | [
{
"content": "In triangle \\( \\triangle ABC \\), the angles are \\( \\angle B = 30^\\circ \\) and \\( \\angle A = 90^\\circ \\). Point \\( K \\) is marked on side \\( AC \\), and points \\( L \\) and \\( M \\) are marked on side \\( BC \\) such that \\( KL = KM \\) (point \\( L \\) lies on segment \\( BM \\)).... | {
"ground_truth": "14",
"style": "rule"
} | {
"index": "6173",
"split": "train"
} |
math | [
{
"content": "Consider numbers of the form $1a1$ , where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome?\n\n*Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$ , $91719$ .* Please output the final... | {
"ground_truth": "55",
"style": "rule"
} | {
"index": "6174",
"split": "train"
} |
math | [
{
"content": "Find the number of real solutions to the equation\n\\[\\frac{1}{x - 1} + \\frac{2}{x - 2} + \\frac{3}{x - 3} + \\dots + \\frac{100}{x - 100} = x.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "101",
"style": "rule"
} | {
"index": "6175",
"split": "train"
} |
math | [
{
"content": "I live on the ground floor of a ten-story building. Each friend of mine lives on a different floor. One day, I put the numbers $1, 2, \\ldots, 9$ into a hat and drew them randomly, one by one. I visited my friends in the order in which I drew their floor numbers. On average, how many meters did I ... | {
"ground_truth": "440/3",
"style": "rule"
} | {
"index": "6176",
"split": "train"
} |
math | [
{
"content": "Let $s(\\theta) = \\frac{1}{2 - \\theta}$. What is $s(s(s(s(s(s(s(s(s(\\frac{1}{2})))))))))$ (where $s$ is applied 9 times)? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\frac{13}{15}",
"style": "rule"
} | {
"index": "6177",
"split": "train"
} |
math | [
{
"content": "Let $x,$ $y,$ $z$ be real numbers such that $x + y + z = 1,$ and $x \\ge -\\frac{1}{3},$ $y \\ge -1,$ and $z \\ge -\\frac{5}{3}.$ Find the maximum value of\n\\[\\sqrt{3x + 1} + \\sqrt{3y + 3} + \\sqrt{3z + 5}.\\] Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "6",
"style": "rule"
} | {
"index": "6178",
"split": "train"
} |
math | [
{
"content": "We have $ 23^2 = 529 $ ordered pairs $ (x, y) $ with $ x $ and $ y $ positive integers from 1 to 23, inclusive. How many of them have the property that $ x^2 + y^2 + x + y $ is a multiple of 6? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "225",
"style": "rule"
} | {
"index": "6179",
"split": "train"
} |
math | [
{
"content": "Let $f(x)=x^{2}+x^{4}+x^{6}+x^{8}+\\cdots$, for all real $x$ such that the sum converges. For how many real numbers $x$ does $f(x)=x$ ? Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "2",
"style": "rule"
} | {
"index": "6180",
"split": "train"
} |
math | [
{
"content": "Let $\\left(x_{1}, y_{1}\\right), \\ldots,\\left(x_{k}, y_{k}\\right)$ be the distinct real solutions to the equation $$\\left(x^{2}+y^{2}\\right)^{6}=\\left(x^{2}-y^{2}\\right)^{4}=\\left(2 x^{3}-6 x y^{2}\\right)^{3}$$ Then $\\sum_{i=1}^{k}\\left(x_{i}+y_{i}\\right)$ can be expressed as $\\frac{... | {
"ground_truth": "516",
"style": "rule"
} | {
"index": "6181",
"split": "train"
} |
math | [
{
"content": "Karl's car uses a gallon of gas every $35$ miles, and his gas tank holds $14$ gallons when it is full. One day, Karl started with a full tank of gas, drove $350$ miles, bought $8$ gallons of gas, and continued driving to his destination. When he arrived, his gas tank was half full. How many miles ... | {
"ground_truth": "525",
"style": "rule"
} | {
"index": "6182",
"split": "train"
} |
math | [
{
"content": "An ant moves on the following lattice, beginning at the dot labeled $A$. Each minute he moves to one of the dots neighboring the dot he was at, choosing from among its neighbors at random. What is the probability that after 5 minutes he is at the dot labeled $B$? [asy]\ndraw((-2,0)--(2,0));\ndraw(... | {
"ground_truth": "\\frac{1}{4}",
"style": "rule"
} | {
"index": "6183",
"split": "train"
} |
math | [
{
"content": "$A B C D$ is a cyclic quadrilateral in which $A B=4, B C=3, C D=2$, and $A D=5$. Diagonals $A C$ and $B D$ intersect at $X$. A circle $\\omega$ passes through $A$ and is tangent to $B D$ at $X . \\omega$ intersects $A B$ and $A D$ at $Y$ and $Z$ respectively. Compute $Y Z / B D$. Please output the... | {
"ground_truth": "\\frac{115}{143}",
"style": "rule"
} | {
"index": "6184",
"split": "train"
} |
math | [
{
"content": "A square piece of paper has sides of length $100$. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at a distance $\\sqrt{17}$ from the corner, and they meet on the diagonal at an angle of $60^{\\circ}$ (see the figure below). The paper... | {
"ground_truth": "871",
"style": "rule"
} | {
"index": "6185",
"split": "train"
} |
math | [
{
"content": "Given $\\tan ( \\frac {π}{4}+x)=- \\frac {1}{2}$, find the value of $\\tan 2x$ (Part 1) and simplify the trigonometric expression $\\sqrt { \\frac {1+\\sin x}{1-\\sin x}}+ \\sqrt { \\frac {1-\\sin x}{1+\\sin x}}$ if $x$ is an angle in the second quadrant. Then, find its value (Part 2). Please outp... | {
"ground_truth": "2\\sqrt {10}",
"style": "rule"
} | {
"index": "6186",
"split": "train"
} |
math | [
{
"content": "At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain ... | {
"ground_truth": "88",
"style": "rule"
} | {
"index": "6187",
"split": "train"
} |
math | [
{
"content": "Given a square pyramid \\(M-ABCD\\) with a square base such that \\(MA = MD\\), \\(MA \\perp AB\\), and the area of \\(\\triangle AMD\\) is 1, find the radius of the largest sphere that can fit into this square pyramid. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "\\sqrt{2} - 1",
"style": "rule"
} | {
"index": "6188",
"split": "train"
} |
math | [
{
"content": "On the grid shown, Jane starts at dot $A$. She tosses a fair coin to determine which way to move. If she tosses a head, she moves up one dot. If she tosses a tail, she moves right one dot. After four tosses of the coin, Jane will be at one of the dots $P, Q, R, S$, or $T$. What is the probability ... | {
"ground_truth": "$\\frac{3}{8}$",
"style": "rule"
} | {
"index": "6189",
"split": "train"
} |
math | [
{
"content": "A domino is a 1-by-2 or 2-by-1 rectangle. A domino tiling of a region of the plane is a way of covering it (and only it) completely by nonoverlapping dominoes. For instance, there is one domino tiling of a 2-by-1 rectangle and there are 2 tilings of a 2-by-2 rectangle (one consisting of two horizo... | {
"ground_truth": "89",
"style": "rule"
} | {
"index": "6190",
"split": "train"
} |
math | [
{
"content": "Cars A and B travel the same distance. Car A travels half that distance at $u$ miles per hour and half at $v$ miles per hour. Car B travels half the time at $u$ miles per hour and half at $v$ miles per hour. The average speed of Car A is $x$ miles per hour and that of Car B is $y$ miles per hour. ... | {
"ground_truth": "$x \\leq y$",
"style": "rule"
} | {
"index": "6191",
"split": "train"
} |
math | [
{
"content": "In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are $$\\begin{cases} x=2 \\sqrt {3}+at \\\\ y=4+ \\sqrt {3}t\\end{cases}$$ (where $t$ is the parameter), and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ ... | {
"ground_truth": "\\frac {1}{2}",
"style": "rule"
} | {
"index": "6192",
"split": "train"
} |
math | [
{
"content": "There are three spheres and a cube. The first sphere is tangent to each face of the cube, the second sphere is tangent to each edge of the cube, and the third sphere passes through each vertex of the cube. What is the ratio of the surface areas of these three spheres? Please output the final answe... | {
"ground_truth": "1:2:3",
"style": "rule"
} | {
"index": "6193",
"split": "train"
} |
math | [
{
"content": "Find the greatest positive integer $A$ with the following property: For every permutation of $\\{1001,1002,...,2000\\}$ , the sum of some ten consecutive terms is great than or equal to $A$ . Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "10055",
"style": "rule"
} | {
"index": "6194",
"split": "train"
} |
math | [
{
"content": "Determine all positive integers $n$ with at least $4$ factors such that $n$ is the sum the squares of its $4$ smallest factors. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "130",
"style": "rule"
} | {
"index": "6195",
"split": "train"
} |
math | [
{
"content": "In the diagram, triangle \\(ABC\\) is isosceles, with \\(AB = AC\\). If \\(\\angle ABC = 50^\\circ\\) and \\(\\angle DAC = 60^\\circ\\), the value of \\(x\\) is: Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "70",
"style": "rule"
} | {
"index": "6196",
"split": "train"
} |
math | [
{
"content": "If a 3'' by 3'' square is added at each successive stage, what will be the area of the rectangle at Stage 6, in square inches?\n\n[asy]size(250); real textsize = 10pt;\ndraw(unitsquare);\ndraw(shift(1.5*right)*unitsquare);\ndraw(shift(2.5*right)*unitsquare);\ndraw(shift(4*right)*unitsquare);\ndraw... | {
"ground_truth": "54",
"style": "rule"
} | {
"index": "6197",
"split": "train"
} |
math | [
{
"content": "Given that the simplest quadratic radical $\\sqrt{m+1}$ and $\\sqrt{8}$ are of the same type of quadratic radical, the value of $m$ is ______. Please output the final answer within \\boxed{}.",
"role": "user"
}
] | {
"ground_truth": "m = 1",
"style": "rule"
} | {
"index": "6198",
"split": "train"
} |
math | [
{
"content": "Let $\\{a_{n}\\}$ be an integer sequence such that for any $n \\in \\mathbf{N}^{*}$, the condition \\((n-1) a_{n+1} = (n+1) a_{n} - 2 (n-1)\\) holds. Additionally, \\(2008 \\mid a_{2007}\\). Find the smallest positive integer \\(n \\geqslant 2\\) such that \\(2008 \\mid a_{n}\\). Please output the... | {
"ground_truth": "501",
"style": "rule"
} | {
"index": "6199",
"split": "train"
} |
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