data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "For every subset $T$ of $U = \\{ 1,2,3,\\ldots,18 \\}$, let $s(T)$ be the sum of the elements of $T$, with $s(\\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\\frac{m}{n}$, where $m$ and $n... | math | {
"style": "rule",
"ground_truth": "683"
} | {
"split": "train",
"index": 7400,
"solution": "Consider the elements of $U$ modulo $3.$\n Ignore the $0$'s because we're gonna multiply $\\binom{6}{0}+..+\\binom{6}{6}=2^6$ at the end. Let $a$ be the $1's$ and $b$ be the $2's$. The key here is that $2 \\equiv -1 \\pmod{3}$ so the difference between the number of... |
deepscaler | [
{
"role": "user",
"content": "Given the decimal representation of $\\frac{1}{30^{30}}$, determine how many zeros immediately follow the decimal point. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 7401,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a real number \\( x \\), let \\( [x] \\) denote the greatest integer that does not exceed \\( x \\). For a certain integer \\( k \\), there exist exactly 2008 positive integers \\( n_{1}, n_{2}, \\cdots, n_{2008} \\), such that \\( k=\\left[\\sqrt[3]{n_{1}}\\right]=\\left[\... | math | {
"style": "rule",
"ground_truth": "668"
} | {
"split": "train",
"index": 7402,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2ax- \\frac {3}{2}x^{2}-3\\ln x$, where $a\\in\\mathbb{R}$ is a constant, \n$(1)$ If $f(x)$ is a decreasing function on $x\\in[1,+\\infty)$, find the range of the real number $a$; \n$(2)$ If $x=3$ is an extremum point of $f(x)$, find the maximum value... | math | {
"style": "rule",
"ground_truth": "\\frac {33}{2}-3\\ln 3"
} | {
"split": "train",
"index": 7403,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively. Given that $b^2=ac$ and $a^2-c^2=ac-bc$, find the value of $$\\frac{c}{b\\sin B}$$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 7404,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Positive integers $a$, $b$, $c$, and $d$ satisfy $a > b > c > d$, $a + b + c + d = 2010$, and $a^2 - b^2 + c^2 - d^2 = 2010$. Find the number of possible values of $a.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 7405,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a quadrilateral $WXYZ$, the angles satisfy $\\angle W = 3\\angle X = 4\\angle Y = 6\\angle Z$. Determine the exact degree measure of $\\angle W$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1440}{7}"
} | {
"split": "train",
"index": 7406,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 6 teachers who will be allocated to two classes, where the maximum number of teachers in each class is 4, determine the number of different arrangements. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 7407,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are five students: A, B, C, D, and E;\n(1) If these five students line up in a row, in how many ways can A not stand in the first position?\n(2) If these five students line up in a row, and A and B must be next to each other while C and D must not be next to each other, i... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 7408,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The average age of $5$ people in a room is $30$ years. An $18$-year-old person leaves the room. What is the average age of the four remaining people? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 7409,
"solution": "1. **Calculate the total age of all people initially in the room**: \n Given that the average age of 5 people is 30 years, the total age of all people can be calculated by multiplying the average age by the number of people:\n \\[\n \\text{Total age} = 30 \\t... |
deepscaler | [
{
"role": "user",
"content": "If the cotangents of the three interior angles \\(A, B, C\\) of triangle \\(\\triangle ABC\\), denoted as \\(\\cot A, \\cot B, \\cot C\\), form an arithmetic sequence, then the maximum value of angle \\(B\\) is \\(\\frac{\\pi}{3}\\). Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 7410,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "15. If \\( a = 1.69 \\), \\( b = 1.73 \\), and \\( c = 0.48 \\), find the value of\n$$\n\\frac{1}{a^{2} - a c - a b + b c} + \\frac{2}{b^{2} - a b - b c + a c} + \\frac{1}{c^{2} - a c - b c + a b}.\n$$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7411,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer less than or equal to\\[\\frac{3^{100}+2^{100}}{3^{96}+2^{96}}?\\]\n$\\textbf{(A) }80\\qquad \\textbf{(B) }81 \\qquad \\textbf{(C) }96 \\qquad \\textbf{(D) }97 \\qquad \\textbf{(E) }625\\qquad$\n Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 7412,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For all triples \\((x, y, z)\\) satisfying the system\n\n$$\n\\left\\{\n\\begin{array}{l}\n2 \\sin x = \\tan y \\\\\n2 \\cos y = \\cot z \\\\\n\\sin z = \\tan x\n\\end{array}\n\\right.\n$$\n\nfind the minimum value of the expression \\(\\cos x - \\sin z\\). Let's think step by ... | math | {
"style": "rule",
"ground_truth": "-\\frac{5\\sqrt{3}}{6}"
} | {
"split": "train",
"index": 7413,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Analogical reasoning is an important method of reasoning. Based on the similarity of two things in some characteristics, conclusions can be drawn that they may be similar in other characteristics. Reading perception: In addition and subtraction of fractions with different denom... | math | {
"style": "rule",
"ground_truth": "-\\frac{1011}{2023}"
} | {
"split": "train",
"index": 7414,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the side \\( CD \\) of the trapezoid \\( ABCD \\) with \\( AD \\parallel BC \\), a point \\( M \\) is marked. From vertex \\( A \\), a perpendicular \\( AH \\) is dropped to the segment \\( BM \\). It is known that \\( AD = HD \\). Find the length of segment \\( AD \\), give... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7415,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse $\\Gamma: \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1 (a > b > 0)$, its right focus is $F(3,0)$, and its top and bottom vertices are $A$ and $B$ respectively. The line $AF$ intersects $\\Gamma$ at another point $M$. If the line $BM$ intersects the $x$-axis at poin... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 7416,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( x, y, z \\in \\mathbf{R}_{+} \\) and \\( x^{2} + y^{2} + z^{2} = 1 \\), find the value of \\( z \\) when \\(\\frac{(z+1)^{2}}{x y z} \\) reaches its minimum. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} - 1"
} | {
"split": "train",
"index": 7417,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On the Cartesian plane $\\mathbb{R}^{2}$, a circle is said to be nice if its center is at the origin $(0,0)$ and it passes through at least one lattice point (i.e. a point with integer coordinates). Define the points $A=(20,15)$ and $B=(20,16)$. How many nice circles intersect ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7418,
"solution": "The square of the radius of a nice circle is the sum of the square of two integers. The nice circle of radius $r$ intersects (the open segment) $\\overline{A B}$ if and only if a point on $\\overline{A B}$ is a distance $r$ from the origin. $\\overline{A B}$ consist... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\begin{cases} \\log_{10} x, & x > 0 \\\\ x^{-2}, & x < 0 \\end{cases}$, if $f(x\\_0) = 1$, find the value of $x\\_0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7419,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A five-digit integer is chosen at random from all possible positive five-digit integers. What is the probability that the number's units digit is an even number and less than 6? Express your answer as a common fraction. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 7420,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $2^{2005}$ is divided by 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 7421,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty people, including \\( A, B, \\) and \\( C \\), sit randomly at a round table. What is the probability that at least two of \\( A, B, \\) and \\( C \\) sit next to each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17/57"
} | {
"split": "train",
"index": 7422,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 3 is inscribed in a hemisphere with radius 8 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{55}"
} | {
"split": "train",
"index": 7423,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\ln{x}-\\frac{1}{x}-2x$.\n(1) Find the monotonic intervals of $f(x)$;\n(2) If the line $l: y=ax+b$ is a tangent line of the function $F(x)=f(x)+2x$ and $a,b\\in\\mathbb{R}$, find the minimum value of $a+b$. Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 7424,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\\circ}, 180^{\\circ},$ and $270^{\\circ}$ counterclockwise around the origin, reflection ac... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7425,
"solution": "We are given a triangle $T$ with vertices at $(0,0), (4,0),$ and $(0,3)$ and asked to determine how many sequences of three transformations from the set of rotations by $90^\\circ, 180^\\circ, 270^\\circ$ counterclockwise around the origin, and reflections across th... |
deepscaler | [
{
"role": "user",
"content": "In the triangular prism \\(P-ABC\\), \\(\\triangle ABC\\) is an equilateral triangle with side length \\(2\\sqrt{3}\\), \\(PB = PC = \\sqrt{5}\\), and the dihedral angle \\(P-BC-A\\) is \\(45^\\circ\\). Find the surface area of the circumscribed sphere around the triangular pri... | math | {
"style": "rule",
"ground_truth": "25\\pi"
} | {
"split": "train",
"index": 7426,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An $a \\times b \\times c$ rectangular box is built from $a \\cdot b \\cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1 \\times b \\times c$ parallel to the $(b \\times c)$ faces of the box contains exactly $9$ red cubes, ex... | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 7427,
"solution": "The total number of green cubes is given by $12a=20b\\Longrightarrow a=\\frac{5}{3}b$.\n Let $r$ be the number of red cubes on each one of the $b$ layers then the total number of red cubes is $9a=br$. Substitute $a=\\frac{5}{3}b$ gives $r=15$.\n Repeating the proced... |
deepscaler | [
{
"role": "user",
"content": "Dizzy Daisy is standing on the point $(0,0)$ on the $xy$-plane and is trying to get to the point $(6,6)$. She starts facing rightward and takes a step 1 unit forward. On each subsequent second, she either takes a step 1 unit forward or turns 90 degrees counterclockwise then tak... | math | {
"style": "rule",
"ground_truth": "131922"
} | {
"split": "train",
"index": 7428,
"solution": "Because Daisy can only turn in one direction and never goes to the same square twice, we see that she must travel in an increasing spiral about the origin. Clearly, she must arrive at $(6,6)$ coming from below. To count her paths, it therefore suffices to consider t... |
deepscaler | [
{
"role": "user",
"content": "Given the graph of $y = mx + 2$ passes through no lattice point with $0 < x \\le 100$ for all $m$ such that $\\frac{1}{2} < m < a$, find the maximum possible value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{50}{99}"
} | {
"split": "train",
"index": 7429,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rhombus $ABCD$ has side length $2$ and $\\angle B = 120^\\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. What is the area of $R$? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 7430,
"solution": "1. **Identify the region $R$:** \n Region $R$ consists of all points inside rhombus $ABCD$ that are closer to vertex $B$ than to any other vertex $A$, $C$, or $D$. \n\n2. **Use of perpendicular bisectors:**\n Let $\\ell_{BC}$, $\\ell_{BD}$, and $\\ell_{BA}$ be t... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ is a positive multiple of $3$. If $x$ cubed is less than $1000$, what is the greatest possible value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 7431,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $[(6^{6} \\div 6^{5})^3 \\cdot 8^3] \\div 4^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"split": "train",
"index": 7432,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. A positive integer is called prime-minister if it has a primest number of distinct primest factors.... | math | {
"style": "rule",
"ground_truth": "378000"
} | {
"split": "train",
"index": 7433,
"solution": "An estimate of $E>0$ earns \\left\\lfloor 20 \\min \\left(\\frac{N}{E}, \\frac{E}{N}\\right)\\right\\rfloor$ points. One heuristic for estimating the answer is that numbers of the form $p^{q} r^{s}$ for primes $p, q, r, s$ with $p \\neq r, q \\neq s$ are primest. Th... |
deepscaler | [
{
"role": "user",
"content": "How many numbers between $1$ and $2005$ are integer multiples of $3$ or $4$ but not $12$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1002"
} | {
"split": "train",
"index": 7434,
"solution": "To solve this problem, we need to find the number of integers between $1$ and $2005$ that are multiples of $3$ or $4$ but not multiples of $12$. We will use the principle of inclusion-exclusion and then subtract the multiples of $12$.\n\n1. **Count multiples of $3$:... |
deepscaler | [
{
"role": "user",
"content": "Find [the decimal form of] the largest prime divisor of $100111011_6$.\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"split": "train",
"index": 7435,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, where AB = 24 and BC = 18, find the largest possible value of $\\tan A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{7}}{7}"
} | {
"split": "train",
"index": 7436,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In diagram square $ABCD$, four triangles are removed resulting in rectangle $PQRS$. Two triangles at opposite corners ($SAP$ and $QCR$) are isosceles with each having area $120 \\text{ m}^2$. The other two triangles ($SDR$ and $BPQ$) are right-angled at $D$ and $B$ respectively... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{15}"
} | {
"split": "train",
"index": 7437,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the line $l:3x-\\sqrt{3}y-6=0$, and the curve $C:\\rho -4\\sin \\theta =0$ in the polar coordinate system with the origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis.\n\n(I) Write the line $l$ in parametric form $\\begin{cases} x=2+t\\cos \\alpha \... | math | {
"style": "rule",
"ground_truth": "6+2\\sqrt{3}"
} | {
"split": "train",
"index": 7438,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of the two $99$-digit numbers\n$303,030,303,...,030,303$ and $505,050,505,...,050,505$\nhas thousands digit $A$ and units digit $B$. What is the sum of $A$ and $B$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 7439,
"solution": "To solve the problem, we need to determine the units digit and the thousands digit of the product of two $99$-digit numbers:\n\\[ 303030303\\ldots030303 \\quad \\text{and} \\quad 505050505\\ldots050505. \\]\n\n1. **Identify the pattern in each number:**\n - The fi... |
deepscaler | [
{
"role": "user",
"content": "LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B... | math | {
"style": "rule",
"ground_truth": "\\frac{B - A}{2}"
} | {
"split": "train",
"index": 7440,
"solution": "1. **Total Expenses Calculation**:\n Let's denote the total amount of money spent by both LeRoy and Bernardo as $T$. Thus, we have:\n \\[\n T = A + B\n \\]\n\n2. **Equal Share Calculation**:\n Since they agreed to share the costs equally, each should pay h... |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a square and a regular decagon that share an edge. One side of the square is extended to meet an extended edge of the decagon. What is the value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 7441,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has a perimeter of 120 inches and each side has an even integer length. How many non-congruent rectangles meet these criteria? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7442,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a convex quadrilateral \\(ABCD\\), side \\(AB\\) is equal to diagonal \\(BD\\), \\(\\angle A=65^\\circ\\), \\(\\angle B=80^\\circ\\), and \\(\\angle C=75^\\circ\\). What is \\(\\angle CAD\\) (in degrees)? Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 7443,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the three-digit numbers formed by the digits 0, 1, 2, 3, 4, 5 without repetition, there are a total of numbers whose digits sum up to 9 (answer in digits). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7444,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7445,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=x^{3}+ax^{2}+bx+2$ has an extremum of $7$ at $x=-1$.<br/>$(1)$ Find the intervals where $f\\left(x\\right)$ is monotonic;<br/>$(2)$ Find the extremum of $f\\left(x\\right)$ on $\\left[-2,4\\right]$. Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 7446,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\\\(f(x)=2x^{5}+3x^{3}-2x^{2}+x-1\\\\), when calculating the value of the function at \\\\(x=2\\\\) using the Horner's method, find \\\\(v_{3}=\\\\) \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 7447,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It takes Pearl 7 days to dig 4 holes. It takes Miguel 3 days to dig 2 holes. If they work together and each continues digging at these same rates, how many holes in total will they dig in 21 days? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 7448,
"solution": "Since Pearl digs 4 holes in 7 days and $\\frac{21}{7}=3$, then in 21 days, Pearl digs $3 \\cdot 4=12$ holes. Since Miguel digs 2 holes in 3 days and $\\frac{21}{3}=7$, then in 21 days, Miguel digs $7 \\cdot 2=14$ holes. In total, they dig $12+14=26$ holes in 21 days... |
deepscaler | [
{
"role": "user",
"content": "Consider a cube where each pair of opposite faces sums to 8 instead of the usual 7. If one face shows 1, the opposite face will show 7; if one face shows 2, the opposite face will show 6; if one face shows 3, the opposite face will show 5. Calculate the largest sum of three num... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the degree measure of an angle whose complement is 25% of its supplement. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 7450,
"solution": "\nLet the angle be $x$ degrees. Then, its complement is $90^\\circ - x$ and its supplement is $180^\\circ - x$.\n\nAccording to the problem, the complement of the angle is 25% (or $\\frac{1}{4}$) of its supplement. We can set up the equation:\n\\[ 90^\\circ - x = \\... |
deepscaler | [
{
"role": "user",
"content": "Find the coordinates of the center of the hyperbola \\[\\frac{(2y-2)^2}{5^2} - \\frac{(3x-4)^2}{4^2} = 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left(\\frac43, 1\\right)"
} | {
"split": "train",
"index": 7451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle $DEF$ has $DE = DF = 5\\sqrt{3}$, and a circle with radius $3\\sqrt{3}$ is tangent to line $DE$ at $E$ and to line $DF$ at $F$. What is the area of the circle that passes through vertices $D$, $E$, and $F$?\nA) $63\\pi$\nB) $48\\pi$\nC) $72\\pi$\nD) $36\\p... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"split": "train",
"index": 7452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine all rational numbers \\(a\\) for which the matrix \\(\\left(\\begin{array}{cccc} a & -a & -1 & 0 \\\\ a & -a & 0 & -1 \\\\ 1 & 0 & a & -a \\\\ 0 & 1 & a & -a \\end{array}\\right)\\) is the square of a matrix with all rational entries. Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "a=0"
} | {
"split": "train",
"index": 7453,
"solution": "We will show that the only such number is \\(a=0\\). Let \\(A=\\left(\\begin{array}{cccc} a & -a & -1 & 0 \\\\ a & -a & 0 & -1 \\\\ 1 & 0 & a & -a \\\\ 0 & 1 & a & -a \\end{array}\\right)\\) and suppose that \\(A=B^{2}\\). It is easy to compute the characteristic po... |
deepscaler | [
{
"role": "user",
"content": "If $m+1=\frac{n-2}{3}$, what is the value of $3 m-n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 7454,
"solution": "Since $m+1=\frac{n-2}{3}$, then $3(m+1)=n-2$. This means that $3 m+3=n-2$ and so $3 m-n=-2-3=-5$."
} |
deepscaler | [
{
"role": "user",
"content": "The solutions to the equation $x^2 - 3|x| - 2 = 0$ are. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-3 - \\sqrt{17}}{2}"
} | {
"split": "train",
"index": 7455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( r(\\theta) = \\frac{1}{1-2\\theta} \\). Calculate \\( r(r(r(r(r(r(10)))))) \\) (where \\( r \\) is applied 6 times). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7456,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\log_{7}{2400}$ rounded to the nearest integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 7457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $x$ is divided by each of $4$, $5$, and $6$, remainders of $3$, $4$, and $5$ (respectively) are obtained. What is the smallest possible positive integer value of $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 7458,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A multiple choice examination consists of $20$ questions. The scoring is $+5$ for each correct answer, $-2$ for each incorrect answer, and $0$ for each unanswered question. John's score on the examination is $48$. What is the maximum number of questions he could have answere... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 7459,
"solution": "Let $c$ be the number of questions John answered correctly, $w$ be the number of questions he answered incorrectly, and $b$ be the number of questions he left blank. We know from the problem statement that:\n1. The total number of questions is 20:\n \\[\n c + w ... |
deepscaler | [
{
"role": "user",
"content": "Let \\( x, y, z \\) be positive real numbers such that \\( xyz = 3 \\). Compute the minimum value of\n\\[ x^2 + 4xy + 12y^2 + 8yz + 3z^2. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "162"
} | {
"split": "train",
"index": 7460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "34560"
} | {
"split": "train",
"index": 7461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "27 identical dice were glued together to form a $3 \\times 3 \\times 3$ cube in such a way that any two adjacent small dice have the same number of dots on the touching faces. How many dots are there on the surface of the large cube? Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "189"
} | {
"split": "train",
"index": 7462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A carton contains milk that is $2\\%$ fat, an amount that is $40\\%$ less fat than the amount contained in a carton of whole milk. What is the percentage of fat in whole milk? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 7463,
"solution": "1. **Understanding the Problem:**\n The problem states that a carton of milk contains 2% fat, which is 40% less than the fat content in a carton of whole milk. We need to find the percentage of fat in the whole milk.\n\n2. **Setting Up the Equation:**\n Let $x$ ... |
deepscaler | [
{
"role": "user",
"content": "The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from the base to the top) at certain angles, with the sum of these three angles being ... | math | {
"style": "rule",
"ground_truth": "374"
} | {
"split": "train",
"index": 7464,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a coordinate plane where at each lattice point, there is a circle with radius $\\frac{1}{8}$ and a square with sides of length $\\frac{1}{4}$, whose sides are parallel to the coordinate axes. A line segment runs from $(0,0)$ to $(729, 243)$. Determine how many of these... | math | {
"style": "rule",
"ground_truth": "972"
} | {
"split": "train",
"index": 7465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: (i) All the squares are congruent. (ii) If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. (iii) Each square touches exactly three othe... | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 7466,
"solution": "We will prove that there is no tri-connected collection if $n$ is odd, and that tri-connected collections exist for all even $n \\geq 38$. Since there are 501 even numbers in the range from 2018 to 3018, this yields 501 as the answer. For any two different squares $... |
deepscaler | [
{
"role": "user",
"content": "There are 6 people including A, B, and C standing in a row for a photo, where A cannot stand at either end, and B and C must stand next to each other. How many such arrangements are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 7467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of all real solutions to the equation \\[\\sqrt{x} + \\sqrt{\\frac{4}{x}} + \\sqrt{x + \\frac{4}{x}} = 6.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{64}{9}"
} | {
"split": "train",
"index": 7468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"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}\\). Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 7469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $(xOy)$, given vectors $\\overrightarrow{AB}=(6,1)$, $\\overrightarrow{BC}=(x,y)$, $\\overrightarrow{CD}=(-2,-3)$, and $\\overrightarrow{AD}$ is parallel to $\\overrightarrow{BC}$.\n(1) Find the relationship between $x$ and $y$;\n(2) If $\\over... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 7470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $2018$ people (call them $A, B, C, \\ldots$ ) stand in a line with each permutation equally likely. Given that $A$ stands before $B$ , what is the probability that $C$ stands after $B$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/3"
} | {
"split": "train",
"index": 7471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer $n$ for which $\\frac{n-13}{5n+6}$ is a non-zero reducible fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "84"
} | {
"split": "train",
"index": 7472,
"solution": "1. **Identify the condition for reducibility**: The fraction $\\frac{n-13}{5n+6}$ is reducible if and only if the greatest common divisor (GCD) of the numerator $n-13$ and the denominator $5n+6$ is greater than 1.\n\n2. **Apply the Euclidean algorithm**: To find the... |
deepscaler | [
{
"role": "user",
"content": "Let $ S $ be the set of all sides and diagonals of a regular pentagon. A pair of elements of $ S $ are selected at random without replacement. What is the probability that the two chosen segments have the same length? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "\\tfrac{4}{9}"
} | {
"split": "train",
"index": 7473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $168$ primes below $1000$ . Then sum of all primes below $1000$ is, Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76127"
} | {
"split": "train",
"index": 7474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A hotel has 5 distinct rooms, each with single beds for up to 2 people. The hotel has no other guests, and 5 friends want to stay there for the night. In how many ways can the 5 friends choose their rooms? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "2220"
} | {
"split": "train",
"index": 7475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR$, $\\angle Q=90^\\circ$, $PQ=9$ and $QR=12$. Points $S$ and $T$ are on $\\overline{PR}$ and $\\overline{QR}$, respectively, and $\\angle PTS=90^\\circ$. If $ST=6$, then what is the length of $PS$? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 7476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.\n\nAt the end of the lesson, $N$ seventh graders said, \"I r... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 7477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all values of $t$ such that $t^2 = 36$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-36"
} | {
"split": "train",
"index": 7478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each square in an $8 \\times 8$ grid is to be painted either white or black. The goal is to ensure that for any $2 \\times 3$ or $3 \\times 2$ rectangle selected from the grid, there are at least two adjacent squares that are black. What is the minimum number of squares that ne... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 7479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system xOy, the parametric equation of curve C is $$\\begin{cases} x=3\\cos\\theta \\\\ y=3\\sin\\theta \\end{cases}$$ (θ is the parameter). Establish a polar coordinate system with the coordinate origin as the pole and the positive semi-axis of th... | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {2}}{8}"
} | {
"split": "train",
"index": 7480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be the smallest positive integer such that the sum of its digits is 2011. How many digits does \\( n \\) have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224"
} | {
"split": "train",
"index": 7481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The focus of a vertically oriented, rotational paraboloid-shaped tall vessel is at a distance of 0.05 meters above the vertex. If a small amount of water is poured into the vessel, what angular velocity $\\omega$ is needed to rotate the vessel around its axis so that the water ... | math | {
"style": "rule",
"ground_truth": "9.9"
} | {
"split": "train",
"index": 7482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a regular polygon of $n$ sides, $n>4$, are extended to form a star. The number of degrees at each point of the star is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{(n-2)180}{n}"
} | {
"split": "train",
"index": 7483,
"solution": "1. **Identify the internal angle of the regular polygon**: \n The internal angle of a regular polygon with $n$ sides can be calculated using the formula:\n \\[\n \\text{Internal angle} = \\frac{(n-2) \\times 180^\\circ}{n}\n \\]\n\n2. **Calculate the externa... |
deepscaler | [
{
"role": "user",
"content": "Triangles \\(ABC\\) and \\(ABD\\) are inscribed in a semicircle with diameter \\(AB = 5\\). A perpendicular from \\(D\\) to \\(AB\\) intersects segment \\(AC\\) at point \\(Q\\), ray \\(BC\\) at point \\(R\\), and segment \\(AB\\) at point \\(P\\). It is known that \\(PR = \\fr... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 7484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "132009 students are taking a test which comprises ten true or false questions. Find the minimum number of answer scripts required to guarantee two scripts with at least nine identical answers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "513"
} | {
"split": "train",
"index": 7485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the dis... | math | {
"style": "rule",
"ground_truth": "0.5"
} | {
"split": "train",
"index": 7486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A train arrives randomly sometime between 3:00 and 4:00 PM and waits for 15 minutes before leaving. If John arrives randomly between 3:15 and 4:15 PM, what is the probability that the train will still be there when John arrives? Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{16}"
} | {
"split": "train",
"index": 7487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "While Steve and LeRoy are fishing 1 mile from shore, their boat springs a leak, and water comes in at a constant rate of 10 gallons per minute. The boat will sink if it takes in more than 30 gallons of water. Steve starts rowing towards the shore at a constant rate of 4 miles p... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 7488,
"solution": "1. **Calculate the time to reach the shore**: \n Steve rows at a speed of 4 miles per hour. The distance from the shore is 1 mile. The time \\( t \\) required to reach the shore is given by the formula:\n \\[\n t = \\frac{\\text{distance}}{\\text{speed}} = \\f... |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}=(\\cos 25^{\\circ},\\sin 25^{\\circ})$ and $\\overrightarrow{b}=(\\sin 20^{\\circ},\\cos 20^{\\circ})$, let $t$ be a real number and $\\overrightarrow{u}=\\overrightarrow{a}+t\\overrightarrow{b}$. Determine the minimum value of $|\\overrightar... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 7489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different rectangles with sides parallel to the grid can be formed by connecting four of the dots in a $5\\times 5$ square array of dots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 7490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a solid with six triangular faces and five vertices. Andrew wants to write an integer at each of the vertices so that the sum of the numbers at the three vertices of each face is the same. He has already written the numbers 1 and 5 as shown. What is the sum of... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 7491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that Lucas makes a batch of lemonade using 200 grams of lemon juice, 100 grams of sugar, and 300 grams of water. If there are 25 calories in 100 grams of lemon juice and 386 calories in 100 grams of sugar, and water has no calories, determine the total number of calories ... | math | {
"style": "rule",
"ground_truth": "145"
} | {
"split": "train",
"index": 7492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each edge of a cube is colored either red or black. Every face of the cube has at least one black edge. The smallest number possible of black edges is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 7493,
"solution": "1. **Understanding the Problem**: We need to color the edges of a cube using two colors, red and black, such that every face of the cube has at least one black edge. We aim to find the minimum number of black edges required.\n\n2. **Analyzing the Cube's Geometry**: ... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive odd number that has the same number of divisors as 360? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3465"
} | {
"split": "train",
"index": 7494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base and one side of a triangle are 30 and 14, respectively. Find the area of this triangle if the median drawn to the base is 13. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 7495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Twenty distinct points are marked on a circle and labeled $1$ through $20$ in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original $20$ points. Let'... | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 7496,
"solution": "Let $a$, $b$, and $c$ be the vertex of a triangle that satisfies this problem, where $a > b > c$. \\[a - b = p_1\\] \\[b - c = p_2\\] \\[a - c = p_3\\]\n$p_3 = a - c = a - b + b - c = p_1 + p_2$. Because $p_3$ is the sum of two primes, $p_1$ and $p_2$, $p_1$ or $p_2... |
deepscaler | [
{
"role": "user",
"content": "What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 7497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the vertices of a regular 100-gon, 100 chips numbered $1, 2, \\ldots, 100$ are placed in exactly that order in a clockwise direction. During each move, it is allowed to swap two chips placed at adjacent vertices if the numbers on these chips differ by no more than $k$. What ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 7498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what smallest positive value of \\(a\\) is the inequality \\(\\frac{\\sqrt[3]{\\sin ^{2} x} - \\sqrt[3]{\\cos ^{2} x}}{\\sqrt[3]{\\tan ^{2} x} - \\sqrt[3]{\\cot ^{2} x}} < \\frac{a}{2}\\) satisfied for all permissible \\(x \\in \\left(\\frac{3 \\pi}{2}, 2 \\pi\\right)\\)? R... | math | {
"style": "rule",
"ground_truth": "0.79"
} | {
"split": "train",
"index": 7499,
"solution": ""
} |
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