data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The sum of the areas of all triangles whose vertices are also vertices of a $1$ by $1$ by $1$ cube is $m + \\sqrt{n} + \\sqrt{p},$ where $m, n,$ and $p$ are integers. Find $m + n + p.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "348"
} | {
"split": "train",
"index": 5300,
"solution": "[asy] size(120); import three; unitsize(1cm); draw((0,0,0)--(1,0,0)--(1,1,0)--(0,1,0)--(0,0,0)--(0,0,1)--(0,1,1)--(1,1,1)--(1,0,1)--(0,0,1)); draw((0,1,1)--(0,1,0)); draw((1,1,1)--(1,1,0)); draw((1,0,1)--(1,0,0)); draw((1,1,1)--(1,0,0)--(1,1,0)--(1,1,1), blue); [/as... |
deepscaler | [
{
"role": "user",
"content": "Points $A,B,C,D,E$ and $F$ lie, in that order, on $\\overline{AF}$, dividing it into five segments, each of length 1. Point $G$ is not on line $AF$. Point $H$ lies on $\\overline{GD}$, and point $J$ lies on $\\overline{GF}$. The line segments $\\overline{HC}, \\overline{JE},$ a... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 5301,
"solution": "1. **Identify Key Points and Relationships**: \n - Points $A, B, C, D, E,$ and $F$ are collinear on line $\\overline{AF}$, and each segment between consecutive points is of length 1.\n - Point $G$ is not on line $AF$, and points $H$ and $J$ lie on lines $\\overl... |
deepscaler | [
{
"role": "user",
"content": "The constant term in the expansion of the binomial \\\\((x \\sqrt {x}- \\dfrac {1}{x})^{5}\\\\) is \\_\\_\\_\\_\\_\\_ . (Answer with a number) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 5302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $a$ is a nonzero constant for which the equation $ax^2+20x+7=0$ has only one solution. Find this solution. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{10}"
} | {
"split": "train",
"index": 5303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "10 times 0.1 equals to ____, 10 times 0.01 equals to ____, 10 times 0.001 equals to ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.01"
} | {
"split": "train",
"index": 5304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive number is called $n$-primable if it is divisible by $n$ and each of its digits is a one-digit prime number. How many 5-primable positive integers are there that are less than 500? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 5305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the absolute value of the difference between single-digit integers $C$ and $D$ such that:\n$$ \\begin{array}{c@{}c@{\\;}c@{}c@{}c@{}c}\n & & & D& D & C_6\\\\\n & & & \\mathbf{5} & \\mathbf{2} & D_6\\\\\n & & + & C & \\mathbf{2} & \\mathbf{4_6}\\\\\n \\cline{2-6}\n & &... | math | {
"style": "rule",
"ground_truth": "1_6"
} | {
"split": "train",
"index": 5306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n$ !. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1, 2, 3, 4"
} | {
"split": "train",
"index": 5307,
"solution": "Answer: $1,2,3,4$ Solution 1. First, note that clearly there are no composite positive integers less than 2 !, and no 3 consecutive composite positive integers less than 3 !. The only composite integers less than 4 ! are $$4,6,8,9,10,12,14,15,16,18,20,21,22$$ and it... |
deepscaler | [
{
"role": "user",
"content": "The lengths of the sides of a triangle with positive area are $\\log_{10}12$, $\\log_{10}75$, and $\\log_{10}n$, where $n$ is a positive integer. Find the number of possible values for $n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "893"
} | {
"split": "train",
"index": 5308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $m(n)$ to be the greatest proper natural divisor of $n\\in \\mathbb{N}$ . Find all $n \\in \\mathbb{N} $ such that $n+m(n) $ is a power of $10$ .\n\n*N. Agakhanov* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 5309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the fourth grade, there are 20 boys and 26 girls. The percentage of the number of boys to the number of girls is %. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "76.9"
} | {
"split": "train",
"index": 5310,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the positive real numbers \\( u, v, \\) and \\( w \\) are all not equal to 1, if \\(\\log _{u} (v w)+\\log _{v} w=5\\) and \\(\\log _{v} u+\\log _{w} v=3\\), then find the value of \\(\\log _{w} u\\). Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"split": "train",
"index": 5311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Jenna collects stamps. She puts the same number of stamps on each page and then inserts each page into one of her two stamp books. One of her stamp books has a total of 840 stamps. The other has 1008. What is the largest number of stamps that Jenna could be putting on each ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 5312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, the parametric equation of curve $C$ is $$\\begin{cases} x=3\\cos \\theta \\\\ y=2\\sin \\theta \\end{cases} (\\theta \\text{ is the parameter}),$$ and the parametric equation of the line $l$ is $$\\begin{cases} x=t-1 \\\\ y=2t-a-1 \\en... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}-2\\sqrt{2}"
} | {
"split": "train",
"index": 5313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the fractional equation $\\frac{3}{{x-2}}+1=\\frac{m}{{4-2x}}$ has a root, then the value of $m$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 5314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sides of a triangle have lengths $6.5$, $10$, and $s$, where $s$ is a whole number. What is the smallest possible value of $s$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 5315,
"solution": "To find the smallest possible value of $s$ such that the sides $6.5$, $10$, and $s$ can form a triangle, we must apply the triangle inequality theorem. The triangle inequality theorem states that the sum of the lengths of any two sides of a triangle must be greater ... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\sin(\\omega x+\\varphi)$ is monotonically increasing on the interval ($\\frac{π}{6}$,$\\frac{{2π}}{3}$), and the lines $x=\\frac{π}{6}$ and $x=\\frac{{2π}}{3}$ are the two axes of symmetry of the graph of the function $y=f(x)$, evaluate the value of $... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 5316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "6.67"
} | {
"split": "train",
"index": 5317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( p(x) = x^{4} + a x^{3} + b x^{2} + c x + d \\), where \\( a, b, c, d \\) are constants, and \\( p(1) = 1993 \\), \\( p(2) = 3986 \\), \\( p(3) = 5979 \\). Calculate \\( \\frac{1}{4}[p(11) + p(-7)] \\). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "5233"
} | {
"split": "train",
"index": 5318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the ordered pair of positive integers $(x,y)$ such that\n\n\\begin{align*}\nx^y+1&=y^x,\\\\\n2x^y&=y^x+7.\n\\end{align*} Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(2,3)"
} | {
"split": "train",
"index": 5319,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a moving point \\( P \\) on the \\( x \\)-axis, \\( M \\) and \\( N \\) lie on the circles \\((x-1)^{2}+(y-2)^{2}=1\\) and \\((x-3)^{2}+(y-4)^{2}=3\\) respectively. Find the minimum value of \\( |PM| + |PN| \\). Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{10} - \\sqrt{3} - 1"
} | {
"split": "train",
"index": 5320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven identical bowling balls weigh the same as three identical canoes. If one of the canoes weighs a total of 28 pounds, how many pounds does one of the bowling balls weigh? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 5321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angles $A$ and $B$ are supplementary. If the measure of angle $A$ is $8$ times angle $B$, what is the measure of angle A? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 5322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 6 red balls and 8 white balls. If 5 balls are randomly placed into Box $A$ and the remaining 9 balls are placed into Box $B$, what is the probability that the sum of the number of white balls in Box $A$ and the number of red balls in Box $B$ is not a prime number... | math | {
"style": "rule",
"ground_truth": "213/1001"
} | {
"split": "train",
"index": 5323,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a four-digit natural number $\\overline{abcd}$ has digits that are all different and not equal to $0$, and satisfies $\\overline{ab}-\\overline{bc}=\\overline{cd}$, then this four-digit number is called a \"decreasing number\". For example, the four-digit number $4129$, sinc... | math | {
"style": "rule",
"ground_truth": "8165"
} | {
"split": "train",
"index": 5324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with an edge length of 2 Chinese feet is cut 4 times horizontally and then 5 times vertically. What is the total surface area of all the small blocks after cutting, in square Chinese feet? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 5325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $c$: \n$$\\sqrt{9+\\sqrt{27+9c}} + \\sqrt{3+\\sqrt{3+c}} = 3+3\\sqrt{3}$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "33"
} | {
"split": "train",
"index": 5326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $3x + y = 10$ and $x + 3y = 14$, find $10x^2 + 12xy + 10y^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "296"
} | {
"split": "train",
"index": 5327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The areas of three squares are 16, 49 and 169. What is the average (mean) of their side lengths? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 5328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive value of $t$ that satisfies $ab = t-2i$ given $|a|=2$ and $|b|=\\sqrt{26}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 5329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A parking lot in Flower Town is a square with $7 \\times 7$ cells, each of which can accommodate a car. The parking lot is enclosed by a fence, and one of the corner cells has an open side (this is the gate). Cars move along paths that are one cell wide. Neznaika was asked to p... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 5330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure drawn is not to scale. Which of the five segments shown is the longest? [asy]\npair A = (-3,0), B=(0,2), C=(3,0), D=(0,-1);\ndraw(D(MP(\"A\", A, W))--D(MP(\"B\", B, N))--D(MP(\"C\", C, E))--D(MP(\"D\", D, S))--A);\ndraw(B--D);\nMP(\"55^\\circ\", (0,-0.75), NW);\nMP(\... | math | {
"style": "rule",
"ground_truth": "CD"
} | {
"split": "train",
"index": 5331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD$, $AB = 4$ and $BC = 8$. The rectangle is folded so that points $A$ and $C$ coincide, forming the pentagon $ABEFD$. What is the length of segment $EF$? Express your answer in simplest radical form. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{5}"
} | {
"split": "train",
"index": 5332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the list 7, 9, 10, 11, 18, which number is the average (mean) of the other four numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 5333,
"solution": "The average of the numbers \\(7, 9, 10, 11\\) is \\(\\frac{7+9+10+11}{4} = \\frac{37}{4} = 9.25\\), which is not equal to 18, which is the fifth number. The average of the numbers \\(7, 9, 10, 18\\) is \\(\\frac{7+9+10+18}{4} = \\frac{44}{4} = 11\\), which is equal ... |
deepscaler | [
{
"role": "user",
"content": "Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 5334,
"solution": "1. **Define Variables:**\n Let $p$ be Pete's current age and $c$ be Claire's current age.\n\n2. **Translate the Problem into Equations:**\n - From the statement \"Two years ago Pete was three times as old as his cousin Claire,\" we have:\n \\[\n p - 2 = ... |
deepscaler | [
{
"role": "user",
"content": "\nThe chord \\( A B \\) subtends an arc of the circle equal to \\( 120^{\\circ} \\). Point \\( C \\) lies on this arc, and point \\( D \\) lies on the chord \\( A B \\). Additionally, \\( A D = 2 \\), \\( B D = 1 \\), and \\( D C = \\sqrt{2} \\).\n\nFind the area of triangle \\... | math | {
"style": "rule",
"ground_truth": "\\frac{3 \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 5335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x^{2017} - 2x + 1 = 0$ and $x \\neq 1.$ Find the value of\n\\[x^{2016} + x^{2015} + \\dots + x + 1.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 5336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in square ABCD, AE = 3EC and BF = 2FB, and G is the midpoint of CD, find the ratio of the area of triangle EFG to the area of square ABCD. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 5337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Eight distinct pieces of candy are to be distributed among three bags: red, blue, and white, with each bag receiving at least one piece of candy. Determine the total number of arrangements possible. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "846720"
} | {
"split": "train",
"index": 5338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "31 cars simultaneously started from one point on a circular track: the first car at a speed of 61 km/h, the second at 62 km/h, and so on (the 31st at 91 km/h). The track is narrow, and if one car overtakes another on a lap, they collide, both go off the track, and are out of th... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 5339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a polar coordinate system, the equation of curve C<sub>1</sub> is given by $\\rho^2 - 2\\rho(\\cos\\theta - 2\\sin\\theta) + 4 = 0$. With the pole as the origin and the polar axis in the direction of the positive x-axis, a Cartesian coordinate system is established using the... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 5340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with center $D$ and radius four feet is tangent at $E$ to a circle with center $F$, as shown. If point $F$ is on the small circle, what is the area of the shaded region? Express your answer in terms of $\\pi$.\n\n[asy]\nfilldraw(circle((0,0),8),gray,linewidth(2));\nfil... | math | {
"style": "rule",
"ground_truth": "48\\pi"
} | {
"split": "train",
"index": 5341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $p$, let $c(p)$ denote the unique positive integer $k$ such that $|k - \\sqrt[3]{p}| < \\frac{1}{2}$. For example, $c(8)=2$ and $c(27)=3$. Find $T = \\sum_{p=1}^{1728} c(p)$. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "18252"
} | {
"split": "train",
"index": 5342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence $\\{a_{n}\\}$, where $a_{n}$ are integers, and for $n \\geq 3, n \\in \\mathbf{N}$, the relation $a_{n} = a_{n-1} - a_{n-2}$ holds. If the sum of the first 1985 terms of the sequence is 1000, and the sum of the first 1995 terms is 4000, then what is the sum o... | math | {
"style": "rule",
"ground_truth": "3000"
} | {
"split": "train",
"index": 5343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular quadrilateral pyramid $S-ABCD$, with a base side length of $2$ and a volume of $\\frac{{4\\sqrt{3}}}{3}$, the length of the lateral edge of this quadrilateral pyramid is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 5344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate plane $(xOy)$, the focus of the parabola $y^{2}=2x$ is $F$. Let $M$ be a moving point on the parabola, then the maximum value of $\\frac{MO}{MF}$ is _______. 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": 5345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "First, find the derivative of the following functions and calculate the derivative at \\\\(x=\\pi\\\\).\n\n\\\\((1) f(x)=(1+\\sin x)(1-4x)\\\\) \\\\((2) f(x)=\\ln (x+1)-\\dfrac{x}{x+1}\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{\\pi}{(\\pi+1)^{2}}"
} | {
"split": "train",
"index": 5346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{3^4+3^2}{3^3-3}$ . Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{15}{4}"
} | {
"split": "train",
"index": 5347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn the prism \\( A B C - A_{1} B_{1} C_{1} \\), \\( A B \\) is perpendicular to the lateral face \\( B B_{1} C_{1} C \\). Point \\( E \\) lies on edge \\( C C_{1} \\) such that \\( E \\ne C \\) and \\( E \\ne C_{1} \\). Given that \\( E A \\perp E B_1 \\), \\( A B = \\sqrt{2}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 5348,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The total in-store price for a blender is $\\textdollar 129.95$. A television commercial advertises the same blender for four easy payments of $\\textdollar 29.99$ and a one-time shipping and handling charge of $\\textdollar 14.95$. Calculate the number of cents saved by purcha... | math | {
"style": "rule",
"ground_truth": "496"
} | {
"split": "train",
"index": 5349,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers \\(k\\) such that the inequality\n\\[ \na^{3} + b^{3} + c^{3} + d^{3} + 1 \\geq k(a + b + c + d)\n\\]\nholds for any \\(a, b, c, d \\in [-1, +\\infty)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 5350,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\(ABCD\\) be a convex trapezoid such that \\(\\angle BAD = \\angle ADC = 90^{\\circ}\\), \\(AB = 20\\), \\(AD = 21\\), and \\(CD = 28\\). Point \\(P \\neq A\\) is chosen on segment \\(AC\\) such that \\(\\angle BPD = 90^{\\circ}\\). Compute \\(AP\\). Let's think step by st... | math | {
"style": "rule",
"ground_truth": "143/5"
} | {
"split": "train",
"index": 5351,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify first, then evaluate: $(1-\\frac{2}{{m+1}})\\div \\frac{{{m^2}-2m+1}}{{{m^2}-m}}$, where $m=\\tan 60^{\\circ}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3-\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 5352,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The state income tax where Kristin lives is levied at the rate of $p\\%$ of the first $\\$28000$ of annual income plus $(p + 2)\\%$ of any amount above $\\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\\%$ of her annual income. What was her ... | math | {
"style": "rule",
"ground_truth": "32000"
} | {
"split": "train",
"index": 5353,
"solution": "Let $A$ represent Kristin's annual income and $T$ represent the total income tax she paid. The tax rate is $p\\%$ for the first $\\textdollar 28000$ and $(p + 2)\\%$ for any amount above $\\textdollar 28000$. We are given that the total tax paid is $(p + 0.25)\\%$ o... |
deepscaler | [
{
"role": "user",
"content": "Let \\( x_{1}, x_{2}, \\ldots, x_{60} \\) be natural numbers greater than 1 (not necessarily distinct). In a \\( 60 \\times 60 \\) table, numbers are placed as follows: in the intersection of the \\( i \\)-th row and the \\( k \\)-th column, the number written is \\( \\log _{x_... | math | {
"style": "rule",
"ground_truth": "-7200"
} | {
"split": "train",
"index": 5354,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$AL$ and $BM$ are the angle bisectors of triangle $ABC$. The circumcircles of triangles $ALC$ and $BMC$ intersect again at point $K$, which lies on side $AB$. Find the measure of angle $ACB$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 5355,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of\n\\[y = \\tan \\left( x + \\frac{2 \\pi}{3} \\right) - \\tan \\left( x + \\frac{\\pi}{6} \\right) + \\cos \\left( x + \\frac{\\pi}{6} \\right)\\]for $-\\frac{5 \\pi}{12} \\le x \\le -\\frac{\\pi}{3}.$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{11 \\sqrt{3}}{6}"
} | {
"split": "train",
"index": 5356,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store increased the original price of a shirt by a certain percent and then lowered the new price by the same amount. Given that the resulting price was $84\\%$ of the original price, by what percent was the price increased and decreased? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 5357,
"solution": "Let the original price of the shirt be denoted as $P$. We are given that the final price is $84\\%$ of the original price, which can be expressed mathematically as:\n\\[ 0.84P \\]\n\nLet the percentage by which the price was increased and then decreased be $x\\%$. T... |
deepscaler | [
{
"role": "user",
"content": "Let $F_{1}$ and $F_{2}$ be the left and right foci of the hyperbola $\\frac{x^{2}}{a^{2}} - \\frac{y^{2}}{b^{2}} = 1$, respectively. Point $P$ is on the right branch of the hyperbola, and $|PF_{2}| = |F_{1}F_{2}|$. The distance from $F_{2}$ to the line $PF_{1}$ is equal to the ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{3}"
} | {
"split": "train",
"index": 5358,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 20-quart container is fully filled with water. Five quarts are removed and replaced with pure antifreeze liquid. Then, five quarts of the mixture are removed and replaced with pure antifreeze. This process is repeated three more times (for a total of five times). Determine th... | math | {
"style": "rule",
"ground_truth": "\\frac{243}{1024}"
} | {
"split": "train",
"index": 5359,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ways in which the nine numbers $$1,12,123,1234, \\ldots, 123456789$$ can be arranged in a row so that adjacent numbers are relatively prime. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 5360,
"solution": "The six numbers $12,123,12345,123456,12345678$, and 123456789 are divisible by 3 , so they cannot be adjacent. However, arranging six numbers in a row with no two adjacent requires at least 11 numbers, which is impossible."
} |
deepscaler | [
{
"role": "user",
"content": "In a regular tetrahedron \\( ABCD \\) with side length \\( \\sqrt{2} \\), it is known that \\( \\overrightarrow{AP} = \\frac{1}{2} \\overrightarrow{AB} \\), \\( \\overrightarrow{AQ} = \\frac{1}{3} \\overrightarrow{AC} \\), and \\( \\overrightarrow{AR} = \\frac{1}{4} \\overright... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{36}"
} | {
"split": "train",
"index": 5361,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The horse walks a distance that is half of the previous day's distance each day, and after walking for 7 days, it has covered a total distance of 700 li. Calculate the total distance it will cover from the 8th day to the 14th day. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac {175}{32}"
} | {
"split": "train",
"index": 5362,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integer divisors of $2004^{2004}$ are divisible by exactly 2004 positive integers?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 5363,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We regularly transport goods from city $A$ to city $B$, which is $183 \\mathrm{~km}$ away. City $A$ is $33 \\mathrm{~km}$ from the river, while city $B$ is built on the riverbank. The cost of transportation per kilometer is half as much on the river as on land. Where should we ... | math | {
"style": "rule",
"ground_truth": "11\\sqrt{3}"
} | {
"split": "train",
"index": 5364,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Zhang Hua, Li Liang, and Wang Min each sent out x, y, z New Year's cards, respectively. If it is known that the least common multiple of x, y, z is 60, the greatest common divisor of x and y is 4, and the greatest common divisor of y and z is 3, then how many New Year's cards d... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5365,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The equation of the asymptotes of the hyperbola \\\\(x^{2}- \\frac {y^{2}}{2}=1\\\\) is \\_\\_\\_\\_\\_\\_; the eccentricity equals \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 5366,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of ways to arrange the letters of the word PERSEVERANCE. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19,958,400"
} | {
"split": "train",
"index": 5367,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\left\\lfloor |{-34.1}|\\right\\rfloor$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 5368,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a parallelepiped \\( A B C D A_{1} B_{1} C_{1} D_{1} \\). On edge \\( A_{1} D_{1} \\), point \\( X \\) is selected, and on edge \\( B C \\), point \\( Y \\) is selected. It is known that \\( A_{1} X = 5 \\), \\( B Y = 3 \\), and \\( B_{1} C_{1} = 14 \\). The plane \\( C_{... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5369,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Elena intends to buy 7 binders priced at $\\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\\textdollar 5$ rebate for purchases over $\\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of ... | math | {
"style": "rule",
"ground_truth": "10.25"
} | {
"split": "train",
"index": 5370,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For natural numbers, when $P$ is divided by $D$, the quotient is $Q$ and the remainder is $R$. When $Q$ is divided by $D'$, the quotient is $Q'$ and the remainder is $R'$. Then, when $P$ is divided by $DD'$, the remainder is: Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "R+R'D"
} | {
"split": "train",
"index": 5371,
"solution": "1. **Given Equations**: We start with the given relationships:\n - $P = QD + R$ (where $P$ is divided by $D$)\n - $Q = D'Q' + R'$ (where $Q$ is divided by $D'$)\n\n2. **Substitute $Q$ in the first equation**: Replace $Q$ in the first equation with its expression... |
deepscaler | [
{
"role": "user",
"content": "Let $p,$ $q,$ $r,$ $s$ be real numbers such that\n\\[\\frac{(p - q)(r - s)}{(q - r)(s - p)} = \\frac{3}{4}.\\]Find the sum of all possible values of\n\\[\\frac{(p - r)(q - s)}{(p - q)(r - s)}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 5372,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of the divisors of \\(72\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72^6"
} | {
"split": "train",
"index": 5373,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many pairs of real numbers $(x, y)$ satisfy the equation $y^{4}-y^{2}=x y^{3}-x y=x^{3} y-x y=x^{4}-x^{2}=0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 5374,
"solution": "We can see that if they solve the first and fourth equations, they are automatically solutions to the second and third equations. Hence, the solutions are just the $3^{2}=9$ points where $x, y$ can be any of $-1,0,1$."
} |
deepscaler | [
{
"role": "user",
"content": "Given a regular pentagon of area 1, a pivot line is a line not passing through any of the pentagon's vertices such that there are 3 vertices of the pentagon on one side of the line and 2 on the other. A pivot point is a point inside the pentagon with only finitely many non-pivo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}(7-3 \\sqrt{5})"
} | {
"split": "train",
"index": 5375,
"solution": "Let the pentagon be labeled $ABCDE$. First, no pivot point can be on the same side of $AC$ as vertex $B$. Any such point $P$ has the infinite set of non-pivot lines within the hourglass shape formed by the acute angles between lines $PA$ and $PC$. Similar logic can ... |
deepscaler | [
{
"role": "user",
"content": "Given that the hyperbola $C_2$ and the ellipse $C_1$: $$\\frac {x^{2}}{4} + \\frac {y^{2}}{3} = 1$$ have the same foci, the eccentricity of the hyperbola $C_2$ when the area of the quadrilateral formed by their four intersection points is maximized is ______. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\sqrt {2}"
} | {
"split": "train",
"index": 5376,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $x$ for which $(2008+x)^2=x^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1004"
} | {
"split": "train",
"index": 5377,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Vanessa set a school record for most points in a single basketball game when her team scored $48$ points. The six other players on her team averaged $3.5$ points each. How many points did Vanessa score to set her school record? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 5378,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the first thirteen terms of an arithmetic progression is $50\\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the rat... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 5379,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pete liked the puzzle; he decided to glue it together and hang it on the wall. In one minute, he glued together two pieces (initial or previously glued). As a result, the entire puzzle was assembled into one complete picture in 2 hours. How much time would it take to assemble t... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 5380,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$ and $(5,0)$. The slope of the line through the origin that divides the area of this region exactly in half is Let's think ste... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{9}"
} | {
"split": "train",
"index": 5381,
"solution": "1. **Identify the vertices and shape of the region**: The vertices of the L-shaped region are given as $A=(0,0)$, $B=(0,3)$, $C=(3,3)$, $D=(3,1)$, $E=(5,1)$, and $F=(5,0)$. The region is composed of two rectangles, one with vertices $A, B, C, D$ and the other with v... |
deepscaler | [
{
"role": "user",
"content": "For some integer $m$, the polynomial $x^3 - 2011x + m$ has the three integer roots $a$, $b$, and $c$. Find $|a| + |b| + |c|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "98"
} | {
"split": "train",
"index": 5382,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the triangle with vertices $(0,7,10),$ $(-1,6,6),$ and $(-4,9,6).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 5383,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers \\(a\\) and \\(b\\) that satisfy \\(0 \\leqslant a, b \\leqslant 8\\) and \\(b^2 = 16 + a^2\\), find the sum of the maximum and minimum values of \\(b - a\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12 - 4\\sqrt{3}"
} | {
"split": "train",
"index": 5384,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\sin 300^\\circ$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 5385,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There exist constants $b_1,$ $b_2,$ $b_3,$ $b_4,$ $b_5,$ $b_6,$ $b_7$ such that\n\\[\\cos^7 \\theta = b_1 \\cos \\theta + b_2 \\cos 2 \\theta + b_3 \\cos 3 \\theta + b_4 \\cos 4 \\theta + b_5 \\cos 5 \\theta + b_6 \\cos 6 \\theta + b_7 \\cos 7 \\theta\\]for all angles $\\theta.... | math | {
"style": "rule",
"ground_truth": "\\frac{1555}{4096}"
} | {
"split": "train",
"index": 5386,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_{1}=1$, and let $a_{n}=\\left\\lfloor n^{3} / a_{n-1}\\right\\rfloor$ for $n>1$. Determine the value of $a_{999}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"split": "train",
"index": 5387,
"solution": "We claim that for any odd $n, a_{n}=n$. The proof is by induction. To get the base cases $n=1$, 3, we compute $a_{1}=1, a_{2}=\\left\\lfloor 2^{3} / 1\\right\\rfloor=8, a_{3}=\\left\\lfloor 3^{3} / 8\\right\\rfloor=3$. And if the claim holds for odd $n \\geq 3$, the... |
deepscaler | [
{
"role": "user",
"content": "The math scores of a high school's college entrance examination, denoted as $\\xi$, approximately follow a normal distribution $N(100, 5^2)$. Given that $P(\\xi < 110) = 0.98$, find the value of $P(90 < \\xi < 100)$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "0.48"
} | {
"split": "train",
"index": 5388,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What number is formed from five consecutive digits (not necessarily in order) such that the number formed by the first two digits, when multiplied by the middle digit, gives the number formed by the last two digits? (For example, if we take the number 12896, 12 multiplied by 8 ... | math | {
"style": "rule",
"ground_truth": "13452"
} | {
"split": "train",
"index": 5389,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five points are chosen uniformly at random on a segment of length 1. What is the expected distance between the closest pair of points? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{24}"
} | {
"split": "train",
"index": 5390,
"solution": "Choose five points arbitrarily at $a_{1}, a_{2}, a_{3}, a_{4}, a_{5}$ in increasing order. Then the intervals $\\left(a_{2}-x, a_{2}\\right),\\left(a_{3}-x, a_{3}\\right),\\left(a_{4}-x, a_{4}\\right),\\left(a_{5}-x, a_{5}\\right)$ must all be unoccupied. The probab... |
deepscaler | [
{
"role": "user",
"content": "When the greatest common divisor and least common multiple of two integers are multiplied, their product is 200. How many different values could be the greatest common divisor of the two integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 5391,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f\\left(x\\right)=\\left(1+2x\\right)^{n}$, where the sum of the binomial coefficients in the expansion is $64$, and ${\\left(1+2x\\right)^n}={a_0}+{a_1}x+{a_2}{x^2}+…+{a_n}{x^n}$. \n$(1)$ Find the value of $a_{2}$; \n$(2)$ Find the term with the largest binomial coeff... | math | {
"style": "rule",
"ground_truth": "2916"
} | {
"split": "train",
"index": 5392,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(This question is worth 14 points.)\n\nA newspaper stand in a city buys the \"Evening News\" from the newspaper office at a price of 0.20 yuan per copy and sells it at 0.30 yuan per copy. The unsold newspapers can be returned to the newspaper office at a price of 0.05 yuan per ... | math | {
"style": "rule",
"ground_truth": "825"
} | {
"split": "train",
"index": 5393,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One fair die is rolled; let $a$ denote the number that comes up. We then roll $a$ dice; let the sum of the resulting $a$ numbers be $b$. Finally, we roll $b$ dice, and let $c$ be the sum of the resulting $b$ numbers. Find the expected (average) value of $c$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "343/8"
} | {
"split": "train",
"index": 5394,
"solution": "$343 / 8$. The expected result of an individual die roll is $(1+2+3+4+5+6) / 6=7 / 2$. For any particular value of $b$, if $b$ dice are rolled independently, then the expected sum is $(7 / 2) b$. Likewise, when we roll $a$ dice, the expected value of their sum $b$ i... |
deepscaler | [
{
"role": "user",
"content": "Suppose the state of Georgia uses a license plate format \"LLDLLL\", and the state of Nebraska uses a format \"LLDDDDD\". Assuming all 10 digits are equally likely to appear in the numeric positions, and all 26 letters are equally likely to appear in the alpha positions, how ma... | math | {
"style": "rule",
"ground_truth": "21902400"
} | {
"split": "train",
"index": 5395,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The non-negative difference between two numbers \\(a\\) and \\(b\\) is \\(a-b\\) or \\(b-a\\), whichever is greater than or equal to 0. For example, the non-negative difference between 24 and 64 is 40. In the sequence \\(88, 24, 64, 40, 24, \\ldots\\), each number after the sec... | math | {
"style": "rule",
"ground_truth": "760"
} | {
"split": "train",
"index": 5396,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the following expression in $x$: \\[2x+8x^2+9-(4-2x-8x^2).\\] Express your answer in the form $ax^2 +bx+c$, where $a$, $b$, and $c$ are numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16x^2+4x+5"
} | {
"split": "train",
"index": 5397,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What percent of the positive integers less than or equal to $100$ have no remainders when divided by $5?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 5398,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two positive numbers is 9. The reciprocal of one of these numbers is 4 times the reciprocal of the other number. What is the sum of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{2}"
} | {
"split": "train",
"index": 5399,
"solution": "1. **Define the variables**: Let the two numbers be $x$ and $y$.\n\n2. **Set up the equations**:\n - The product of the two numbers is given by:\n \\[\n xy = 9\n \\]\n - The reciprocal relationship is given by:\n \\[\n \\frac{1}{x} = 4 \\left... |
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