data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The function \\( g \\), defined on the set of ordered pairs of positive integers, satisfies the following properties:\n\\[\n\\begin{align*}\ng(x, x) &= x, \\\\\ng(x, y) &= g(y, x), \\quad \\text{and} \\\\\n(x + 2y)g(x, y) &= yg(x, x + 2y).\n\\end{align*}\n\\]\nCalculate \\( g(1... | math | {
"style": "rule",
"ground_truth": "198"
} | {
"split": "train",
"index": 1500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f$ be a monic cubic polynomial satisfying $f(x)+f(-x)=0$ for all real numbers $x$. For all real numbers $y$, define $g(y)$ to be the number of distinct real solutions $x$ to the equation $f(f(x))=y$. Suppose that the set of possible values of $g(y)$ over all real numbers $... | math | {
"style": "rule",
"ground_truth": "970"
} | {
"split": "train",
"index": 1501,
"solution": "We claim that we must have $f(x)=x^{3}-3 x$. First, note that the condition $f(x)+f(-x)=0$ implies that $f$ is odd. Combined with $f$ being monic, we know that $f(x)=x^{3}+a x$ for some real number $a$. Note that $a$ must be negative; otherwise $f(x)$ and $f(f(x))$ ... |
deepscaler | [
{
"role": "user",
"content": "A jar contains $5$ different colors of gumdrops. $30\\%$ are blue, $20\\%$ are brown, $15\\%$ are red, $10\\%$ are yellow, and other $30$ gumdrops are green. If half of the blue gumdrops are replaced with brown gumdrops, how many gumdrops will be brown? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 1502,
"solution": "1. **Calculate the total number of gumdrops**:\n Given that $30\\%$ are blue, $20\\%$ are brown, $15\\%$ are red, $10\\%$ are yellow, and $30$ gumdrops are green. First, we calculate the percentage of gumdrops that are green:\n \\[\n 100\\% - (30\\% + 20\\% + ... |
deepscaler | [
{
"role": "user",
"content": "When $\\sqrt[3]{7200}$ is simplified, the result is $c\\sqrt[3]{d}$, where $c$ and $d$ are positive integers and $d$ is as small as possible. What is $c+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "452"
} | {
"split": "train",
"index": 1503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Below is a portion of the graph of a function, $y=h(x)$:\n\n[asy]\nimport graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.25,xmax=4.25,ymin=-7.25,ymax=6.25;\n\npen cqcqcq=rgb(0.75,0.75,0.75);\n\n/*grid*/ pen gs=li... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 1504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a mathematics class, the probability of earning an A is 0.6 times the probability of earning a B, and the probability of earning a C is 1.6 times the probability of earning a B. The probability of earning a D is 0.3 times the probability of earning a B. Assuming that all gra... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 1505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An arithmetic sequence with first term $1$ has a common difference of $6$. A second sequence begins with $4$ and has a common difference of $7$. In the range of $1$ to $100$, what is the largest number common to both sequences? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 1506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the largest multiple of 7 less than 50? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 1507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has $AB = 2$ , $BC = 3$ , $CA = 4$ , and circumcenter $O$ . If the sum of the areas of triangles $AOB$ , $BOC$ , and $COA$ is $\\tfrac{a\\sqrt{b}}{c}$ for positive integers $a$ , $b$ , $c$ , where $\\gcd(a, c) = 1$ and $b$ is not divisible by t... | math | {
"style": "rule",
"ground_truth": "152"
} | {
"split": "train",
"index": 1508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For natural numbers \\\\(m\\\\) greater than or equal to \\\\(2\\\\) and their powers of \\\\(n\\\\), the following decomposition formula is given: \n\\\\(2^{2}=1+3\\\\) \\\\(3^{2}=1+3+5\\\\) \\\\(4^{2}=1+3+5+7\\\\) \\\\(…\\\\) \n\\\\(2^{3}=3+5\\\\) \\\\(3^{3}=7+9+11... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 1509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When dividing the numbers 312837 and 310650 by some three-digit natural number, the remainders are the same. Find this remainder. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 1510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What number is one third of the way from $\\frac14$ to $\\frac34$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{12}"
} | {
"split": "train",
"index": 1511,
"solution": "\n#### Step-by-step Explanation:\n\n1. **Understanding the Problem:**\n We need to find the number that is one third of the way from $\\frac{1}{4}$ to $\\frac{3}{4}$.\n\n2. **Using the Concept of Weighted Average:**\n The number one third of the way from $\\frac... |
deepscaler | [
{
"role": "user",
"content": "Find the largest real \\( k \\) such that if \\( a, b, c, d \\) are positive integers such that \\( a + b = c + d \\), \\( 2ab = cd \\) and \\( a \\geq b \\), then \\(\\frac{a}{b} \\geq k\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 1512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular flag is divided into four triangles, labelled Left, Right, Top, and Bottom. Each triangle is to be colored one of red, white, blue, green, and purple such that no two triangles that share an edge are the same color. How many different flags can be made? Let's thin... | math | {
"style": "rule",
"ground_truth": "260"
} | {
"split": "train",
"index": 1513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer \\( n \\) cannot be divided by \\( 2 \\) or \\( 3 \\), and there do not exist non-negative integers \\( a \\) and \\( b \\) such that \\( |2^a - 3^b| = n \\). Find the smallest value of \\( n \\). Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 1514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that the 3 sides of a triangle are consecutive positive integers and the largest angle is twice the smallest angle. Find the perimeter of this triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 1515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the first 512 smallest positive integers written in base 8 use the digit 5 or 6 (or both)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"split": "train",
"index": 1516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the triangle with vertices $(-1,4),$ $(7,0),$ and $(11,5).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 1517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side length of an equilateral triangle ABC is 2. Calculate the area of the orthographic (isometric) projection of triangle ABC. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{4}"
} | {
"split": "train",
"index": 1518,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $(xOy)$, a pole is established at the origin $O$ with the non-negative semi-axis of the $x$-axis as the polar axis, forming a polar coordinate system. Given that the equation of line $l$ is $4ρ\\cos θ-ρ\\sin θ-25=0$, and the curve $W$ is defin... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{17}}{17}"
} | {
"split": "train",
"index": 1519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cylinder with a volume of 21 is inscribed in a cone. The plane of the upper base of this cylinder truncates the original cone, forming a frustum with a volume of 91. Find the volume of the original cone. Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "94.5"
} | {
"split": "train",
"index": 1520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct ordered pairs of positive integers $(m,n)$ are there so that the sum of the reciprocals of $m$ and $n$ is $\\frac14$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 1521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three planets orbit a star circularly in the same plane. Each moves in the same direction and moves at constant speed. Their periods are 60, 84, and 140 years. The three planets and the star are currently collinear. What is the fewest number of years from now that they will all... | math | {
"style": "rule",
"ground_truth": "105"
} | {
"split": "train",
"index": 1522,
"solution": "Denote the planets $A, B, C$ respectively. Let $a(t), b(t), c(t)$ denote the angle which each of the respective planets makes with its initial position after $t$ years. These are given by $a(t) = \\frac{t \\pi}{30}$, $b(t) = \\frac{t \\pi}{42}$, $c(t) = \\frac{t \\p... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=x- \\frac {1}{x}+2a\\ln x$ $(a\\in\\mathbb{R})$.\n\n$(1)$ Discuss the monotonicity of $f(x)$;\n\n$(2)$ If $f(x)$ has two extreme values $x_{1}$ and $x_{2}$, where $x_{2}\\in[e,+\\infty)$, find the minimum value of $f(x_{1})-f(x_{2})$. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac {4}{e}"
} | {
"split": "train",
"index": 1523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's call a year interesting if a person turns the number of years equal to the sum of the digits of the year of their birth in that year. A certain year turned out to be interesting for Ivan, who was born in the 20th century, and for Vovochka, who was born in the 21st century... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 1524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least five out of the nine natural numbers \\( 1 \\) through \\( 9 \\) can divide \\( N \\) evenly, then \\( N \\) is called a \"Five Sequential Number.\" What is the smallest \"Five Sequential Number\" greater than 2000? Let's think step b... | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 1525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the fifth term of the geometric sequence with first term $2$ and second term $\\frac{1}{4}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2048}"
} | {
"split": "train",
"index": 1526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\omega=-\\tfrac{1}{2}+\\tfrac{1}{2}i\\sqrt3.$ Let $S$ denote all points in the complex plane of the form $a+b\\omega+c\\omega^2,$ where $0\\leq a \\leq 1,0\\leq b\\leq 1,$ and $0\\leq c\\leq 1.$ What is the area of $S$? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}\\sqrt3"
} | {
"split": "train",
"index": 1527,
"solution": "1. **Identify $\\omega$ and $\\omega^2$:**\n Given $\\omega = -\\frac{1}{2} + \\frac{1}{2}i\\sqrt{3}$, we recognize that $\\omega$ is a cube root of unity, specifically $\\omega = e^{2\\pi i / 3}$. Therefore, $\\omega^2 = \\omega^* = -\\frac{1}{2} - \\frac{1}{2}i\... |
deepscaler | [
{
"role": "user",
"content": "Compute the expressions \n\\[ C = 3 \\times 4 + 5 \\times 6 + 7 \\times 8 + \\cdots + 43 \\times 44 + 45 \\]\nand \n\\[ D = 3 + 4 \\times 5 + 6 \\times 7 + \\cdots + 42 \\times 43 + 44 \\times 45 \\]\nand find the positive difference between integers $C$ and $D$. Let's think st... | math | {
"style": "rule",
"ground_truth": "882"
} | {
"split": "train",
"index": 1528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of consecutive positive integers starting from 10 that can be added together before the sum exceeds 500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 1529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the two-variable function\n$$\nf(a, b)=\\max _{x \\in[-1,1]}\\left\\{\\left|x^{2}-a x-b\\right|\\right\\},\n$$\nfind the minimum value of \\( f(a, b) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 1530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At Euler Middle School, $198$ students voted on two issues in a school referendum with the following results: $149$ voted in favor of the first issue and $119$ voted in favor of the second issue. If there were exactly $29$ students who voted against both issues, how many studen... | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 1531,
"solution": "1. **Define the Sets:**\n Let $A$ be the set of students who voted in favor of the first issue, and $B$ be the set of students who voted in favor of the second issue. We are given:\n - $|A| = 149$\n - $|B| = 119$\n - Total students, $|U| = 198$\n - Student... |
deepscaler | [
{
"role": "user",
"content": "How many natural five-digit numbers have the product of their digits equal to 2000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 1532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given angles $\\alpha$ and $\\beta$ have their vertices at the origin of coordinates, and their initial sides coincide with the positive half-axis of the x-axis, $\\alpha, \\beta \\in (0, \\pi)$. The terminal side of angle $\\beta$ intersects the unit circle at a point whose x-... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{65}"
} | {
"split": "train",
"index": 1533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a large square, points $P$, $Q$, $R$, and $S$ are midpoints of the sides. Inside the square, a triangle is formed by connecting point $P$ to the center of the square and point $Q$. If the area of the larger square is 80, what are the areas of the smaller square and the trian... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{16+8}{4-2}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 1535,
"solution": "1. **Evaluate the Numerator and Denominator Separately:**\n \\[\n \\text{Numerator: } 16 + 8 = 24\n \\]\n \\[\n \\text{Denominator: } 4 - 2 = 2\n \\]\n\n2. **Divide the Numerator by the Denominator:**\n \\[\n \\frac{24}{2} = 12\n \\]\n\n3. **Identi... |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be real numbers such that $x + y + z = 7$ and $x, y, z \\geq 2.$ Find the maximum value of\n\\[\\sqrt{2x + 3} + \\sqrt{2y + 3} + \\sqrt{2z + 3}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{69}"
} | {
"split": "train",
"index": 1536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point $M$ represents the number $9$ on the number line.<br/>$(1)$ If point $N$ is first moved $4$ units to the left and then $6$ units to the right to reach point $M$, then the number represented by point $N$ is ______.<br/>$(2)$ If point $M$ is moved $4$ units on th... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 1537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function:\n\\[ f(x) = \\left\\{ \n\\begin{aligned} \n3x + 1 & \\quad \\text{ if } x \\leq 2 \\\\\nx^2 & \\quad \\text{ if } x > 2 \n\\end{aligned} \n\\right.\\]\nThe function has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(0) + \\dots + f^{-1}(8) + ... | math | {
"style": "rule",
"ground_truth": "22 + 2\\sqrt{2}"
} | {
"split": "train",
"index": 1538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, let $f(n) = n^4 - 360n^2 + 400$. What is the sum of all values of $f(n)$ that are prime numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "802"
} | {
"split": "train",
"index": 1539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the isosceles trapezoid $ABCD$, $AD \\parallel BC$, $\\angle B = 45^\\circ$. Point $P$ is on the side $BC$. The area of $\\triangle PAD$ is $\\frac{1}{2}$, and $\\angle APD = 90^\\circ$. Find the minimum value of $AD$. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 1540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the unique pair of positive integers $(a, b)$ with $a<b$ for which $$\\frac{2020-a}{a} \\cdot \\frac{2020-b}{b}=2$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(505,1212)"
} | {
"split": "train",
"index": 1541,
"solution": "If either $a$ or $b$ is larger than 2020, then both must be for the product to be positive. However, the resulting product would be less than 1, so this case is impossible. Now, we see that $\\left(\\frac{2020-a}{a}, \\frac{2020-b}{b}\\right)$ must be in the form $\... |
deepscaler | [
{
"role": "user",
"content": "Solve\n\\[(x - 3)^4 + (x - 5)^4 = -8.\\]Enter all the solutions, separated by commas. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4 + i, 4 - i, 4 + i \\sqrt{5}, 4 - i \\sqrt{5}"
} | {
"split": "train",
"index": 1542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the shortest distance from a point on the ellipse $\\frac{y^2}{16} + \\frac{x^2}{9} = 1$ to the line $y = x + m$ is $\\sqrt{2}$, find the minimum value of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 1543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 5 different types of books, with at least 3 copies of each. If we want to buy 3 books to gift to 3 students, with each student receiving one book, how many different ways are there to do this? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"split": "train",
"index": 1544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jindra collects dice, all of the same size. Yesterday he found a box in which he started stacking the dice. He managed to fully cover the square base with one layer of dice. He similarly stacked five more layers, but he ran out of dice halfway through the next layer. Today, Jin... | math | {
"style": "rule",
"ground_truth": "234"
} | {
"split": "train",
"index": 1545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_{\\sqrt{6}} (216\\sqrt{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 1546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:\n$\\textbf{(A)}\\ 1:3 \\qquad \\textbf{(B)}\\ 3:4 \\qquad \\textbf{(C)}\\ 4:3 \\qquad \\textbf{(D)}\\ 3:... | math | {
"style": "rule",
"ground_truth": "3:1"
} | {
"split": "train",
"index": 1547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $2^{2^{2^2}}$ is divided by 500. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 1548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The local junior football team is deciding on their new uniforms. The team's ninth-graders will choose the color of the socks (options: red, green, or blue), and the tenth-graders will pick the color for the t-shirts (options: red, yellow, green, blue, or white). Neither group ... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{15}"
} | {
"split": "train",
"index": 1549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m\\in R$, the moving straight line passing through the fixed point $A$ with equation $x+my-2=0$ intersects the moving straight line passing through the fixed point $B$ with equation $mx-y+4=0$ at point $P\\left(x,y\\right)$. Find the maximum value of $|PA|\\cdot |PB|$. Let... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 1550,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Senya has three straight sticks, each 24 centimeters long. Senya broke one of them into two parts such that with the two pieces of this stick and the two whole sticks, he could form the contour of a right triangle. How many square centimeters is the area of this triangle? Let's... | math | {
"style": "rule",
"ground_truth": "216"
} | {
"split": "train",
"index": 1551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Among the following 4 propositions:\n\n① The converse of \"If $a$, $G$, $b$ form a geometric sequence, then $G^2=ab$\";\n\n② The negation of \"If $x^2+x-6\\geqslant 0$, then $x > 2$\";\n\n③ In $\\triangle ABC$, the contrapositive of \"If $A > B$, then $\\sin A > \\sin B$\";... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{17}}{2}"
} | {
"split": "train",
"index": 1552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $f(x)$ is a linear function satisfying the equation $f(x) = 4f^{-1}(x) + 6$. Given that $f(1) = 4$, find $f(2)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1553,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$ with the first term $\\frac{3}{2}$ and common ratio $- \\frac{1}{2}$, calculate the maximum value of the sum of the first $n$ terms, $S_n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 1554,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of all real numbers that are tripled when added to their reciprocals? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{2}"
} | {
"split": "train",
"index": 1555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe gardener Fedya has a miracle tree with seven branches in his garden. On each branch, there can either grow 6 apples, 5 pears, or 3 oranges. Fedya discovered that the tree has fruit of all types, with the most pears and the fewest apples.\n\nHow many fruits in total grew o... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 1556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $|\\omega^2 + 7\\omega + 40|$ if $\\omega = 4 + 3i$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15\\sqrt{34}"
} | {
"split": "train",
"index": 1557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $| \\overrightarrow{a}|=6$, $| \\overrightarrow{b}|=3$, and $\\overrightarrow{a} \\cdot \\overrightarrow{b}=-12$, find the projection of vector $\\overrightarrow{a}$ onto vector $\\overrightarrow{b}$. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 1558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "I ponder some numbers in bed, all products of three primes I've said, apply $\\phi$ they're still fun: $$n=37^{2} \\cdot 3 \\ldots \\phi(n)= 11^{3}+1 ?$$ now Elev'n cubed plus one. What numbers could be in my head? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "2007, 2738, 3122"
} | {
"split": "train",
"index": 1559,
"solution": "The numbers expressible as a product of three primes are each of the form $p^{3}, p^{2} q$, or $p q r$, where $p, q$, and $r$ are distinct primes. Now, $\\phi\\left(p^{3}\\right)=p^{2}(p-1), \\phi\\left(p^{2} q\\right)=$ $p(p-1)(q-1)$, and $\\phi(p q r)=(p-1)(q-1)(r... |
deepscaler | [
{
"role": "user",
"content": "Given $α∈(\\frac{π}{2},π)$, and $sin(α+\\frac{π}{3})=\\frac{12}{13}$, determine the value of $sin(\\frac{π}{6}-α)+sin(\\frac{2π}{3}-α)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{13}"
} | {
"split": "train",
"index": 1560,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $$(5x- \\frac {1}{ \\sqrt {x}})^{n}$$, the sum of the binomial coefficients in its expansion is 64. Find the constant term in the expansion. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "375"
} | {
"split": "train",
"index": 1561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given sets A = {x | $x^2 - ax + a^2 - 19 = 0$}, B = {x | $x^2 - 5x + 6 = 0$}, C = {x | $x^2 + 2x - 8 = 0$} such that $A \\cap B \\neq \\emptyset$ and $A \\cap C = \\emptyset$. Find the value of the real number $a$. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 1562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two stores have warehouses storing millet: the first warehouse has 16 tons more than the second. Every night, exactly at midnight, the owner of each store steals a quarter of their competitor's millet and moves it to their own warehouse. After 10 nights, the thieves were caught... | math | {
"style": "rule",
"ground_truth": "2^{-6}"
} | {
"split": "train",
"index": 1563,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$ABC$ is an isosceles triangle with base $AC$. $CD$ is the bisector of angle $C$, and $\\angle ADC = 150^\\circ$. Find $\\angle B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 1564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "After lunch, there are dark spots with a total area of $S$ on a transparent square tablecloth. It turns out that if the tablecloth is folded in half along any of the two lines connecting the midpoints of its opposite sides or along one of its two diagonals, the total visible ar... | math | {
"style": "rule",
"ground_truth": "2/3"
} | {
"split": "train",
"index": 1565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $g$ be a function defined on the positive integers, such that\n\\[g(xy) = g(x) + g(y)\\]\nfor all positive integers $x$ and $y$. Given $g(8) = 21$ and $g(18) = 26$, find $g(432)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"split": "train",
"index": 1566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A=\\{V, W, X, Y, Z, v, w, x, y, z\\}$. Find the number of subsets of the 2-configuration \\( \\{\\{V, W\\}, \\{W, X\\}, \\{X, Y\\}, \\{Y, Z\\}, \\{Z, V\\}, \\{v, x\\}, \\{v, y\\}, \\{w, y\\}, \\{w, z\\}, \\{x, z\\}, \\{V, v\\}, \\{W, w\\}, \\{X, x\\}, \\{Y, y\\}, \\{Z, z\\... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 1567,
"solution": "No more than two of the pairs \\( \\{v, x\\}, \\{v, y\\}, \\{w, y\\}, \\{w, z\\}, \\{x, z\\} \\) may be included in a 2-configuration of order 1, since otherwise at least one of \\( v, w, x, y, z \\) would occur more than once. If exactly one is included, say \\( \\... |
deepscaler | [
{
"role": "user",
"content": "Solve for $X$ and $Y$ such that\n\\[\\frac{Yx + 8}{x^2 - 11x + 30} = \\frac{X}{x - 5} + \\frac{7}{x - 6}\\].\nFind $X+Y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{22}{3}"
} | {
"split": "train",
"index": 1568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the polynomial $x^4 - 6x^3 + 16x^ 2 - 25x + 10$ is divided by $x^2 - 2x + k,$ the remainder is $x + a.$ Enter the ordered pair $(k,a).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(5,-5)"
} | {
"split": "train",
"index": 1569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people agreed to meet at a specific location between 12 PM and 1 PM. The condition is that the first person to arrive will wait for the second person for 15 minutes and then leave. What is the probability that these two people will meet if each of them chooses their moment ... | math | {
"style": "rule",
"ground_truth": "7/16"
} | {
"split": "train",
"index": 1570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When two fair dice are thrown once each, what is the probability that the upward-facing numbers are different and that one of them shows a 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 1571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A geometric sequence of positive integers has a first term of 3, and the fourth term is 243. Find the sixth term of the sequence. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"split": "train",
"index": 1572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $\\frac12\\cdot\\frac41\\cdot\\frac18\\cdot\\frac{16}{1} \\dotsm \\frac{1}{2048}\\cdot\\frac{4096}{1}$, and multiply the result by $\\frac34$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1536"
} | {
"split": "train",
"index": 1573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $p$, $q$, $r$, $s$, $t$, and $u$ are integers for which $343x^3+64 = (px^2 + qx + r)(sx^2 + tx + u)$ for all $x$, then what is $p^2+q^2+r^2+s^2+t^2+u^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3506"
} | {
"split": "train",
"index": 1574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$ , the diagonals intersect at $E$ , the area of $\\triangle ABE$ is 72 and the area of $\\triangle CDE$ is 50. What is the area of trapezoid $ABCD$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "242"
} | {
"split": "train",
"index": 1575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. When a die (with faces numbered 1 through 6) is thrown twice in succession, find the probability that the sum of the numbers facing up is at least 10.\n\n2. On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 1576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of triangle $PQR$ are tangent to a circle with center $C$ as shown. Given that $\\angle PQR = 63^\\circ$ and $\\angle QPR = 59^\\circ$, find $\\angle QRC$, in degrees.\n\n[asy]\n\nunitsize(1.0 cm);\n\npair Q, P, R, C;\n\nQ = (2.43,3.46);\n\nP = (0,0);\n\nR = (4.43,0);... | math | {
"style": "rule",
"ground_truth": "29^\\circ"
} | {
"split": "train",
"index": 1577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The function $y = x^2 + 2x - 1$ attains its minimum value on the interval $[0, 3]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 1578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the hyperbola $\\frac{x^{2}}{4-m} + \\frac{y^{2}}{m-2}=1$, find the value of $m$ if its asymptote equations are $y=± \\frac{1}{3}x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{4}"
} | {
"split": "train",
"index": 1579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a parallelogram, the lengths of the sides are given as $5$, $10y-2$, $3x+5$, and $12$. Determine the value of $x+y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{91}{30}"
} | {
"split": "train",
"index": 1580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows a large square divided into squares of three different sizes. What percentage of the large square is shaded? \n\nA) 61%\nB) 59%\nC) 57%\nD) 55%\nE) 53% Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59\\%"
} | {
"split": "train",
"index": 1581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1 , the value of each blue card is equal to ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 1582,
"solution": "If there are $B$ blue cards, then each red card contributes $1+2 B$ points (one for itself and two for each blue card) and each yellow card contributes $3 B$ points. Thus, if $B>1$, it is optimal to change all red cards to yellow cards. When $B=0$, the maximum numbe... |
deepscaler | [
{
"role": "user",
"content": "A rectangular prism has a volume of $12 \\mathrm{~cm}^{3}$. A new prism is formed by doubling the length, doubling the width, and tripling the height of the original prism. What is the volume of this new prism? Let's think step by step and put only the final answer within \\box... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 1583,
"solution": "Suppose that the original prism has length $\\ell \\mathrm{cm}$, width $w \\mathrm{~cm}$, and height $h \\mathrm{~cm}$. Since the volume of this prism is $12 \\mathrm{~cm}^{3}$, then $\\ell w h=12$. The new prism has length $2 \\ell \\mathrm{cm}$, width $2 w \\mathr... |
deepscaler | [
{
"role": "user",
"content": "There are four balls in a bag, each with the same shape and size, and their numbers are \\\\(1\\\\), \\\\(2\\\\), \\\\(3\\\\), and \\\\(4\\\\).\n\n\\\\((1)\\\\) Draw two balls randomly from the bag. Calculate the probability that the sum of the numbers on the balls drawn is no ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{13}{16}"
} | {
"split": "train",
"index": 1584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates 2, 0, and 18 in some order. Find the sum of all possible values of $A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1168"
} | {
"split": "train",
"index": 1585,
"solution": "More generally, suppose three vertices of the square lie on lines $y=y_{1}, y=y_{2}, y=y_{3}$. One of these vertices must be adjacent to two others. If that vertex is on $y=y_{1}$ and the other two are on $y=y_{2}$ and $y=y_{3}$, then we can use the Pythagorean theo... |
deepscaler | [
{
"role": "user",
"content": "The expressions $A$ = $1 \\times 2 + 3 \\times 4 + 5 \\times 6 + \\cdots + 37 \\times 38 + 39$ and $B$ = $1 + 2 \\times 3 + 4 \\times 5 + \\cdots + 36 \\times 37 + 38 \\times 39$ are obtained by writing multiplication and addition operators in an alternating pattern between suc... | math | {
"style": "rule",
"ground_truth": "722"
} | {
"split": "train",
"index": 1586,
"solution": "We have \\[|A-B|=|1+3(4-2)+5(6-4)+ \\cdots + 37(38-36)-39(1-38)|\\]\\[\\implies |2(1+3+5+7+ \\cdots +37)-1-39(37)|\\]\\[\\implies |361(2)-1-39(37)|=|722-1-1443|=|-722|\\implies \\boxed{722}\\]"
} |
deepscaler | [
{
"role": "user",
"content": "The numbers $1978^{n}$ and $1978^{m}$ have the same last three digits. Find the positive integers $n$ and $m$ such that $m+n$ is minimized, given that $n > m \\geq 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 1587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The side of a square is increased by $20\\%$. To keep the area of the square unchanged, what percentage must the other side be reduced? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"split": "train",
"index": 1588,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The \ffive-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest \ffive-digit number which is divisible by the sum of its digits Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99972"
} | {
"split": "train",
"index": 1589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Charlie is planning to construct a boundary around a rectangular playground using exactly 380 feet of fencing. Regulations specify that the length of the boundary must be at least 100 feet and the width must be at least 60 feet. Charlie wants to maximize the area enclosed by th... | math | {
"style": "rule",
"ground_truth": "9000"
} | {
"split": "train",
"index": 1590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \\sqrt{x^2 + 18x + 45}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 1591,
"solution": "Begin by completing the square on both sides of the equation, which gives \\[(x+9)^2-51=2\\sqrt{(x+3)(x+15)}\\] Now by substituting $y=x+9$, we get $y^2-51=2\\sqrt{(y-6)(y+6)}$, or \\[y^4-106y^2+2745=0\\] The solutions in $y$ are then \\[y=x+9=\\pm3\\sqrt{5},\\pm\\s... |
deepscaler | [
{
"role": "user",
"content": "Yesterday, Sasha cooked soup and added too little salt, requiring additional seasoning. Today, he added twice as much salt as yesterday, but still had to season the soup additionally, though with half the amount of salt he used for additional seasoning yesterday. By what factor... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 1592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rod with a length of four meters has weights attached as follows: $20 \\mathrm{~kg}$ at one end, and at distances of one, two, and three meters from that end, weights of $30, 40, 50$ $\\mathrm{kg}$ respectively. Additionally, a weight of $60 \\mathrm{~kg}$ is attached at the ... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 1593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A drawer contains a mixture of red socks and blue socks, at most $1991$ in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $\\frac{1}{2}$ that both are red or both are blue. What is the largest possible number... | math | {
"style": "rule",
"ground_truth": "990"
} | {
"split": "train",
"index": 1594,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be a point on the circle $x^2 + y^2 - 12x + 31 = 0,$ and let $B$ be a point on the parabola $y^2 = 4x.$ Find the smallest possible distance $AB.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 1595,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum $\\sum_{d=1}^{2012}\\left\\lfloor\\frac{2012}{d}\\right\\rfloor$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15612"
} | {
"split": "train",
"index": 1596,
"solution": "Note that the number of integers between 1 and 2012 that have $n$ as a divisor is $\\left\\lfloor\\frac{2012}{n}\\right\\rfloor$. Therefore, if we sum over the possible divisors, we see that the sum is equivalent to $\\sum_{d=1}^{2012} \\left\\lfloor\\frac{2012}{d}\... |
deepscaler | [
{
"role": "user",
"content": "If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\\binom{x}{k}$ is defined by the formula\n\\[\n \\binom{x}{k} = \\frac{x(x - 1)(x - 2) \\dots (x - k + 1)}{k!} \\, .\n\\]Compute the value of\n\\[\n \\frac{\\binom{1/2}{2014} \\cdo... | math | {
"style": "rule",
"ground_truth": "-\\frac{1} { 4027}"
} | {
"split": "train",
"index": 1597,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A chess tournament is held with the participation of boys and girls. The girls are twice as many as boys. Each player plays against each other player exactly once. By the end of the tournament, there were no draws and the ratio of girl winnings to boy winnings was $7/9$ . How ... | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 1598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A, B, C, D, E, F$ be 6 points on a circle in that order. Let $X$ be the intersection of $AD$ and $BE$, $Y$ is the intersection of $AD$ and $CF$, and $Z$ is the intersection of $CF$ and $BE$. $X$ lies on segments $BZ$ and $AY$ and $Y$ lies on segment $CZ$. Given that $AX=3,... | math | {
"style": "rule",
"ground_truth": "\\frac{77}{6}"
} | {
"split": "train",
"index": 1599,
"solution": "Let $XY=z, YZ=x$, and $ZX=y$. By Power of a Point, we have that $3(z+10)=2(y+16), 4(x+12)=10(z+3), \\text{ and } 12(x+4)=16(y+2)$. Solving this system gives $XY=\\frac{11}{3}$ and $YZ=\\frac{14}{3}$ and $ZX=\\frac{9}{2}$. Therefore, the answer is $XY+YZ+ZX=\\frac{77... |
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