data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "With square tiles of a side length of an exact number of units, a room with a surface area of 18,144 square units has been tiled in the following manner: on the first day one tile was placed, the second day two tiles, the third day three tiles, and so on. How many tiles were ne... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 30749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An isosceles triangle with a base of $\\sqrt{2}$ has medians intersecting at a right angle. Calculate the area of this triangle. 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": 30750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\) be a positive integer with at least four different positive divisors. Let the four smallest of these divisors be \\( d_{1}, d_{2}, d_{3}, d_{4} \\). Find all such numbers \\( n \\) for which\n\\[ d_{1}^{2}+d_{2}^{2}+d_{3}^{2}+d_{4}^{2}=n \\] Let's think step by st... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 30751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be positive real numbers. Find the maximum value of\n\\[2(a - x)(x + \\sqrt{x^2 + b^2})\\]in terms of $a$ and $b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "a^2 + b^2"
} | {
"split": "train",
"index": 30752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the square roots of a number are $2a+3$ and $a-18$, then this number is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"split": "train",
"index": 30753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $7\\left(\\frac{1}{5}\\right)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{125}"
} | {
"split": "train",
"index": 30754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\\sqrt{19}$. Find the distance from the ball's point of first contact with a wall to the nearest vertex. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 30755,
"solution": "Consider the diagram above, where $M$ is the midpoint of $BC$. Then $AM$ is perpendicular to $BC$ since $ABC$ is equilateral, so by the Pythagorean theorem $AM = \\frac{5 \\sqrt{3}}{2}$. Then, using the Pythagorean theorem again, we see that $MY = \\frac{1}{2}$, so... |
deepscaler | [
{
"role": "user",
"content": "A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. The smallest value of $n$ is Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 30756,
"solution": "1. **Define the variables**: Let the price of the item in cents be $x$. Since the sales tax is 4%, the total cost including tax is $x + 0.04x = 1.04x$ cents.\n\n2. **Convert the total cost to dollars**: Since $1$ dollar = $100$ cents, the total cost in dollars is $... |
deepscaler | [
{
"role": "user",
"content": "Given that Ron incorrectly calculated the product of two positive integers $a$ and $b$ by reversing the digits of the three-digit number $a$, and that His wrong product totaled $468$, determine the correct value of the product of $a$ and $b$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "1116"
} | {
"split": "train",
"index": 30757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\begin{cases} a+\\ln x,x > 0 \\\\ g(x)-x,x < 0\\\\ \\end{cases}$, which is an odd function, and $g(-e)=0$, find the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1-e"
} | {
"split": "train",
"index": 30758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is $\\sqrt[4]{81} \\cdot \\sqrt[3]{27} \\cdot \\sqrt{9}$ expressed as a positive integer? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 30759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum number of bishops that we can place in a $8 \\times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop. \n\nNote: A bishop threatens another one, if both are placed in different cells, in... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 30760,
"solution": "To solve this problem, we need to determine the maximum number of bishops that can be placed on an \\(8 \\times 8\\) chessboard such that each bishop is threatened by at most one other bishop. The key here is to understand the attacking capability of bishops on a c... |
deepscaler | [
{
"role": "user",
"content": "Let \\[P(x) = 24x^{24} + \\sum_{j = 1}^{23}(24 - j)(x^{24 - j} + x^{24 + j}).\\] Let $z_{1},z_{2},\\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2} = a_{k} + b_{k}i$ for $k = 1,2,\\ldots,r,$ where $a_{k}$ and $b_{k}$ are real numbers. Let\n$\\sum_{k = 1}^{r}|b_... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30761,
"solution": "As in Solution 1, we find that the roots of $P(x)$ we care about are the 24th roots of unity except $1$. Therefore, the squares of these roots are the 12th roots of unity. In particular, every 12th root of unity is counted twice, except for $1$, which is only count... |
deepscaler | [
{
"role": "user",
"content": "Given two equiangular polygons $P_1$ and $P_2$ with different numbers of sides; each angle of $P_1$ is $x$ degrees and each angle of $P_2$ is $kx$ degrees, where $k$ is an integer greater than $1$. The number of possibilities for the pair $(x, k)$ is: Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30762,
"solution": "1. **Understanding the angle formula for equiangular polygons**: For any equiangular polygon with $n$ sides, each interior angle is given by the formula:\n \\[\n \\text{Angle} = \\frac{180(n-2)}{n} = 180 - \\frac{360}{n}\n \\]\n This formula arises from the... |
deepscaler | [
{
"role": "user",
"content": "How many minutes are needed at least to finish these tasks: washing rice for 2 minutes, cooking porridge for 10 minutes, washing vegetables for 3 minutes, and chopping vegetables for 5 minutes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\binom{18}{8}=31824$, $\\binom{18}{9}=48620$, and $\\binom{18}{10}=43758$, calculate $\\binom{20}{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "172822"
} | {
"split": "train",
"index": 30764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\frac{ax+b}{{x}^{2}+4}$ attains a maximum value of $1$ at $x=-1$, find the minimum value of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{4}"
} | {
"split": "train",
"index": 30765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten positive integers are arranged around a circle. Each number is one more than the greatest common divisor of its two neighbors. What is the sum of the ten numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 30766,
"solution": "First note that all the integers must be at least 2, because the greatest common divisor of any two positive integers is at least 1. Let $n$ be the largest integer in the circle. The greatest common divisor of its two neighbors is $n-1$. Therefore, each of the two ... |
deepscaler | [
{
"role": "user",
"content": "The storage capacity of two reservoirs, A and B, changes over time. The relationship between the storage capacity of reservoir A (in hundred tons) and time $t$ (in hours) is: $f(t) = 2 + \\sin t$, where $t \\in [0, 12]$. The relationship between the storage capacity of reservoi... | math | {
"style": "rule",
"ground_truth": "6.721"
} | {
"split": "train",
"index": 30767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A retailer purchases a gadget at $50$ minus $10%$. He aims to sell the gadget at a gain of $25%$ on his cost price after offering a $15%$ discount on the marked price. Determine the gadget's marked price in dollars. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "66.18"
} | {
"split": "train",
"index": 30768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer $x$ for which $\\frac79 > \\frac{x}{13}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 30769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A road of 1500 meters is being repaired. In the first week, $\\frac{5}{17}$ of the total work was completed, and in the second week, $\\frac{4}{17}$ was completed. What fraction of the total work was completed in these two weeks? And what fraction remains to complete the entire... | math | {
"style": "rule",
"ground_truth": "\\frac{8}{17}"
} | {
"split": "train",
"index": 30770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid with $\\overline{BC} \\parallel \\overline{AD}$ and $AB=CD$. Points $X$ and $Y$ lie on diagonal $\\overline{AC}$ with $X$ between $A$ and $Y$. Suppose $\\angle AXD = \\angle BYC = 90^\\circ$, $AX = 3$, $XY = 1$, and $YC = 2$. What is the are... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{35}"
} | {
"split": "train",
"index": 30771,
"solution": "1. **Setting up the coordinate system**: Place $X$ at the origin $(0,0)$, align $AC$ along the $x$-axis, and $DX$ along the $y$-axis. This gives us:\n - $X = (0,0)$\n - $A = (3,0)$ (since $AX = 3$)\n - $Y = (-1,0)$ (since $XY = 1$)\n - $C = (-3,0)$ (since $... |
deepscaler | [
{
"role": "user",
"content": "In parallelogram \\(ABCD\\), points \\(A_{1}, A_{2}, A_{3}, A_{4}\\) and \\(C_{1}, C_{2}, C_{3}, C_{4}\\) are respectively the quintisection points of \\(AB\\) and \\(CD\\). Points \\(B_{1}, B_{2}\\) and \\(D_{1}, D_{2}\\) are respectively the trisection points of \\(BC\\) and ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 30772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then calculate the ratio of the ocean area in the southern hemisphere to the ocean area in the northern hemisphere. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "171:113"
} | {
"split": "train",
"index": 30773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( b = \\frac{\\pi}{2010} \\). Find the smallest positive integer \\( m \\) such that\n\\[\n2\\left[\\cos(b) \\sin(b) + \\cos(4b) \\sin(2b) + \\cos(9b) \\sin(3b) + \\cdots + \\cos(m^2b) \\sin(mb)\\right]\n\\]\nis an integer. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 30774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n = 2^3 \\cdot 5^6 \\cdot 8^9 \\cdot 10^{10} \\). How many natural-number factors does \\( n \\) have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "697"
} | {
"split": "train",
"index": 30775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "396"
} | {
"split": "train",
"index": 30776,
"solution": "After testing various values of $m$ in $f(m)$ of solution 1 to determine $m$ for which $f(m) = 1992$, we find that $m \\in \\{7980, 7981, 7982, 7983, 7984\\}$. WLOG, we select $7980$. Furthermore, note that every time $k$ reaches a multiple of $25$, $k!$ will gain ... |
deepscaler | [
{
"role": "user",
"content": "In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \\( S \\) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \\... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 30777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the middle of the school year, $40\\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how ... | math | {
"style": "rule",
"ground_truth": "170"
} | {
"split": "train",
"index": 30778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A primary school conducted a height survey. For students with heights not exceeding 130 cm, there are 99 students with an average height of 122 cm. For students with heights not less than 160 cm, there are 72 students with an average height of 163 cm. The average height of stud... | math | {
"style": "rule",
"ground_truth": "621"
} | {
"split": "train",
"index": 30779,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangular array of $2016$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. What is the sum of the digits of $N$? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 30780,
"solution": "1. **Identify the formula for the sum of the first $N$ natural numbers**: The sum of the first $N$ natural numbers is given by the formula:\n \\[\n S = 1 + 2 + 3 + \\cdots + N = \\frac{N(N+1)}{2}\n \\]\n We need to find $N$ such that this sum equals $2016$.... |
deepscaler | [
{
"role": "user",
"content": "How many functions $f:\\{1,2, \\ldots, 2013\\} \\rightarrow\\{1,2, \\ldots, 2013\\}$ satisfy $f(j)<f(i)+j-i$ for all integers $i, j$ such that $1 \\leq i<j \\leq 2013$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{4025}{2013}"
} | {
"split": "train",
"index": 30781,
"solution": "Note that the given condition is equivalent to $f(j)-j<f(i)-i$ for all $1 \\leq i<j \\leq$ 2013. Let $g(i)=f(i)-i$, so that the condition becomes $g(j)<g(i)$ for $i<j$ and $1-i \\leq g(i) \\leq 2013-i$. However, since $g$ is decreasing, we see by induction that $g(... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be an equilateral triangle with $AB=1.$ Let $M$ be the midpoint of $BC,$ and let $P$ be on segment $AM$ such that $AM/MP=4.$ Find $BP.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{7}}{5}"
} | {
"split": "train",
"index": 30782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Choose any four distinct digits $w, x, y, z$ and form the four-digit numbers $wxyz$ and $zyxw$. What is the greatest common divisor of the numbers of the form $wxyz + zyxw + wxyz \\cdot zyxw$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 30783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points. Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 30784,
"solution": "\nConsider that there are 2009 distinct points in the plane, and each point is colored either blue or red. The objective is to determine the greatest possible number of blue points under the condition that every blue-centered unit circle contains exactly two red po... |
deepscaler | [
{
"role": "user",
"content": "In isosceles $\\vartriangle ABC, AB = AC, \\angle BAC$ is obtuse, and points $E$ and $F$ lie on sides $AB$ and $AC$ , respectively, so that $AE = 10, AF = 15$ . The area of $\\vartriangle AEF$ is $60$ , and the area of quadrilateral $BEFC$ is $102$ . Find $BC$ ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 30785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(3p^3 - 5p + 6) + (4 - 6p^2 + 2p)$. Express your answer in the form $Ap^3 + Bp^2 + Cp +D$, where $A$, $B$, $C$, and $D$ are numbers (possibly negative). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3p^3 - 6p^2 - 3p + 10"
} | {
"split": "train",
"index": 30786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "You have 68 coins of different weights. Using 100 weighings on a balance scale without weights, find the heaviest and the lightest coin. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 30787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Participation in the local soccer league is $10\\%$ higher this year than it was last year. Also, from last year to this year, the number of males increased by $5\\%$ and the number of females increased by $20\\%$. There were $20$ males participating in the league last year. Wh... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 30788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$ , what is the largest possible area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 30789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiao Hua needs to attend an event at the Youth Palace at 2 PM, but his watch gains 4 minutes every hour. He reset his watch at 10 AM. When Xiao Hua arrives at the Youth Palace according to his watch at 2 PM, how many minutes early is he actually? Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m, n > 2$ be integers. One of the angles of a regular $n$-gon is dissected into $m$ angles of equal size by $(m-1)$ rays. If each of these rays intersects the polygon again at one of its vertices, we say $n$ is $m$-cut. Compute the smallest positive integer $n$ that is bot... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 30791,
"solution": "For the sake of simplicity, inscribe the regular polygon in a circle. Note that each interior angle of the regular $n$-gon will subtend $n-2$ of the $n$ arcs on the circle. Thus, if we dissect an interior angle into $m$ equal angles, then each must be represented b... |
deepscaler | [
{
"role": "user",
"content": "Let point O be a point inside triangle ABC with an area of 6, and it satisfies $$\\overrightarrow {OA} + \\overrightarrow {OB} + 2\\overrightarrow {OC} = \\overrightarrow {0}$$, then the area of triangle AOC is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "\\frac {3}{2}"
} | {
"split": "train",
"index": 30792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate\n\\[\\begin{vmatrix} y + 1 & y & y \\\\ y & y + 1 & y \\\\ y & y & y + 1 \\end{vmatrix}.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3y + 1"
} | {
"split": "train",
"index": 30793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\{a, b, c, d, e, f, g, h\\}$ be a permutation of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ . What is the probability that $\\overline{abc} +\\overline{def}$ is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3/7"
} | {
"split": "train",
"index": 30794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( S_{n} = \\sum_{i=1}^{10} i|n-i| \\left( n \\in \\mathbf{Z}_{+} \\right) \\), find the minimum value of \\( S_{n} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 30795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider all four-digit numbers (including leading zeros) from $0000$ to $9999$. A number is considered balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. Calculate the total number of such balanced four-digit numbers. Let's think step by... | math | {
"style": "rule",
"ground_truth": "670"
} | {
"split": "train",
"index": 30796,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit positive integers are multiples of 7? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1286"
} | {
"split": "train",
"index": 30797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A standard deck of 54 playing cards (with four cards of each of thirteen ranks, as well as two Jokers) is shuffled randomly. Cards are drawn one at a time until the first queen is reached. What is the probability that the next card is also a queen? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{27}"
} | {
"split": "train",
"index": 30798,
"solution": "Since the four queens are equivalent, we can compute the probability that a specific queen, say the queen of hearts, is right after the first queen. Remove the queen of hearts; then for every ordering of the 53 other cards, there are 54 locations for the queen of h... |
deepscaler | [
{
"role": "user",
"content": "Distribute 5 students into three groups: A, B, and C. Group A must have at least two students, while groups B and C must have at least one student each. Determine the number of different distribution schemes. Let's think step by step and put only the final answer within \\boxed... | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 30799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $$ A=\\lim _{n \\rightarrow \\infty} \\sum_{i=0}^{2016}(-1)^{i} \\cdot \\frac{\\binom{n}{i}\\binom{n}{i+2}}{\\binom{n}{i+1}^{2}} $$ Find the largest integer less than or equal to $\\frac{1}{A}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 30800,
"solution": "Note $$ \\sum_{i=0}^{2016}(-1)^{i} \\cdot \\frac{\\binom{n}{i}\\binom{n}{i+2}}{\\binom{n}{i+1}^{2}}=\\sum_{i=0}^{2016}(-1)^{i} \\cdot \\frac{(i+1)(n-i-1)}{(i+2)(n-i)} $$ So $$ \\lim _{n \\rightarrow \\infty} \\sum_{i=0}^{2016}(-1)^{i} \\cdot \\frac{\\binom{n}{i}\\b... |
deepscaler | [
{
"role": "user",
"content": "Given $A=a^{2}-2ab+b^{2}$, $B=a^{2}+2ab+b^{2}$, where $a\\neq b$. <br/>$(1)$ Determine the sign of $A+B$ and explain the reason; <br/>$(2)$ If $ab$ are reciprocals of each other, find the value of $A-B$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 30801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sphere with radius 1 and center $(0,0,1)$ rests on the $xy$-plane. A light source is at $P = (0,-1,2).$ Then the boundary of the shadow of the sphere can be expressed in the form $y = f(x),$ for some function $f(x).$ Find the function $f(x).$ Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{x^2}{4} - 1"
} | {
"split": "train",
"index": 30802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest natural number ending in the digit 4 that becomes 4 times larger when its last digit is moved to the beginning of the number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "102564"
} | {
"split": "train",
"index": 30803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Euler's Bridge: The following figure is the graph of the city of Konigsburg in 1736 - vertices represent sections of the cities, edges are bridges. An Eulerian path through the graph is a path which moves from vertex to vertex, crossing each edge exactly once. How many ways cou... | math | {
"style": "rule",
"ground_truth": "13023"
} | {
"split": "train",
"index": 30804,
"solution": "The number of ways to destroy bridges to create an Eulerian path depends on ensuring that exactly 0 or 2 vertices have an odd degree. The specific graph of Konigsburg can be analyzed to find the number of such configurations, resulting in 13023 ways."
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, where $A > B > C$, if $2 \\cos 2B - 8 \\cos B + 5 = 0$, $\\tan A + \\tan C = 3 + \\sqrt{3}$, and the height $CD$ from $C$ to $AB$ is $2\\sqrt{3}$, then find the area of $\\triangle ABC$. 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": 30805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "George and Henry started a race from opposite ends of the pool. After a minute and a half, they passed each other in the center of the pool. If they lost no time in turning and maintained their respective speeds, how many minutes after starting did they pass each other the seco... | math | {
"style": "rule",
"ground_truth": "4\\frac{1}{2}"
} | {
"split": "train",
"index": 30806,
"solution": "1. **Understanding the Problem:**\n George and Henry start from opposite ends of the pool and meet in the center after $1.5$ minutes. This means each of them has traveled half the length of the pool in $1.5$ minutes.\n\n2. **Analyzing the Speeds:**\n Since they... |
deepscaler | [
{
"role": "user",
"content": "How many degrees are there in the measure of angle $P?$\n\n[asy]\nsize (5cm,5cm);\npair A,B,C,D,E;\n\nA=(0,1.1);\nB=(4.5,0);\nC=(6.4,1.7);\nD=(4.2,5);\nE=(0.5,4.2);\n\ndraw (A--B--C--D--E--A,linewidth(1));\n\nlabel(\"$P$\",A,SW);\nlabel(\"$128^\\circ$\",shift(0,0.6)*B);\n\nlabe... | math | {
"style": "rule",
"ground_truth": "96^\\circ"
} | {
"split": "train",
"index": 30807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an integer $k\\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\\cdots x_k!$ for all positive integers $x_1,x_2,\\cdots x_k$.\n\n$Albania$ Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "f(n) = n"
} | {
"split": "train",
"index": 30808,
"solution": "\nConsider the function \\( f \\) from the positive integers into themselves. We need to determine all such functions such that for every collection of positive integers \\( x_1, x_2, \\ldots, x_k \\), the expression\n\n\\[\nf(x_1)! + f(x_2)! + \\cdots + f(x_k)!\n\... |
deepscaler | [
{
"role": "user",
"content": "Heather compares the price of a new computer at two different stores. Store $A$ offers $15\\%$ off the sticker price followed by a $\\$90$ rebate, and store $B$ offers $25\\%$ off the same sticker price with no rebate. Heather saves $\\$15$ by buying the computer at store $A$ i... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 30809,
"solution": "\n#### Step-by-step Analysis:\n\n1. **Define the Variables:**\n Let the sticker price of the computer be \\( x \\) dollars.\n\n2. **Calculate the Final Prices:**\n - At store \\( A \\), the price after a \\( 15\\% \\) discount is \\( 0.85x \\). Then, a \\( \\$9... |
deepscaler | [
{
"role": "user",
"content": "If $a$ is a constant such that $9x^2 + 24x + a$ is the square of a binomial, then what is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 30810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of \\( x + y \\), given that \\( x^2 + y^2 - 3y - 1 = 0 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{26}+3}{2}"
} | {
"split": "train",
"index": 30811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the graphs of $y=\\sin (\\frac{1}{2}x-\\frac{\\pi }{6})$, determine the horizontal shift required to obtain the graph of $y=\\sin \\frac{1}{2}x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 30812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven distinct integers are picked at random from $\\{1,2,3,\\ldots,12\\}$. What is the probability that, among those selected, the third smallest is $4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{33}"
} | {
"split": "train",
"index": 30813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among four people, A, B, C, and D, they pass a ball to each other. The first pass is from A to either B, C, or D, and the second pass is from the receiver to any of the other three. This process continues for several passes. Calculate the number of ways the ball can be passed s... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 30814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many sequences of $0$s and $1$s of length $19$ are there that begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 30815,
"solution": "To solve this problem, we need to determine the number of valid sequences of length 19 that satisfy the given conditions. We define $f(n)$ as the number of valid sequences of length $n$ that meet the criteria.\n\n#### Step 1: Understanding the Problem\nThe sequence... |
deepscaler | [
{
"role": "user",
"content": "Given $m+n=2$ and $mn=-2$. Find the value of:\n\n1. $2^{m}\\cdot 2^{n}-(2^{m})^{n}$\n2. $(m-4)(n-4)$\n3. $(m-n)^{2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 30816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer less than or equal to \\[\\frac{5^{50} + 3^{50}}{5^{45} + 3^{45}}?\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3124"
} | {
"split": "train",
"index": 30817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a regular octagon with side length 3, inside of which eight semicircles lie such that their diameters coincide with the sides of the octagon. Determine the area of the shaded region, which is the area inside the octagon but outside all of the semicircles.\nA) $54 + 18\... | math | {
"style": "rule",
"ground_truth": "54 + 36\\sqrt{2} - 9\\pi"
} | {
"split": "train",
"index": 30818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A mother gives pocket money to her children sequentially: 1 ruble to Anya, 2 rubles to Borya, 3 rubles to Vitya, then 4 rubles to Anya, 5 rubles to Borya, and so on until Anya receives 202 rubles, and Borya receives 203 rubles. How many more rubles will Anya receive compared to... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"split": "train",
"index": 30819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bob Barker went back to school for a PhD in math, and decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answe... | math | {
"style": "rule",
"ground_truth": "292864"
} | {
"split": "train",
"index": 30820,
"solution": "$15!/\\left(3^{4} \\cdot 5^{3} \\cdot 7^{2} \\cdot 9\\right)=292864$. These are Standard Young Tableaux."
} |
deepscaler | [
{
"role": "user",
"content": "The inhabitants of the island of Jumble use the standard Kobish alphabet ($20$ letters, A through T). Each word in their language is $4$ letters or less, and for some reason, they insist that all words contain the letter A at least once. How many words are possible? Let's thin... | math | {
"style": "rule",
"ground_truth": "30860"
} | {
"split": "train",
"index": 30821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair standard six-sided dice is tossed four times. Given that the sum of the first three tosses equals the fourth toss, what is the probability that at least one \"3\" is tossed?\nA) $\\frac{1}{6}$\nB) $\\frac{6}{17}$\nC) $\\frac{9}{17}$\nD) $\\frac{1}{2}$\nE) $\\frac{1}{3}$ ... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{17}"
} | {
"split": "train",
"index": 30822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertical axis indicates the number of employees, but the scale was accidentally omitted from this graph. What percent of the employees at the Gauss company have worked there for $5$ years or more? Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "30 \\%"
} | {
"split": "train",
"index": 30823,
"solution": "To solve this problem, we need to determine the percentage of employees who have worked at the Gauss company for 5 years or more based on the given information.\n\n1. **Assign Variables:**\n Let $x$ represent the number of employees each $\\text{X}$ on the graph ... |
deepscaler | [
{
"role": "user",
"content": "Numbers from 1 to 100 are written in a vertical row in ascending order. Fraction bars of different sizes are inserted between them. The calculation starts with the smallest fraction bar and ends with the largest one, for example, $\\frac{1}{\\frac{5}{3}}=\\frac{15}{4}$. What is... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 30824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle is inscribed in a regular hexagon. A smaller hexagon has two non-adjacent sides coinciding with the sides of the larger hexagon and the remaining vertices touching the circle. What percentage of the area of the larger hexagon is the area of the smaller hexagon? Assume ... | math | {
"style": "rule",
"ground_truth": "25\\%"
} | {
"split": "train",
"index": 30825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can two distinct squares be chosen from an $8 \\times 8$ chessboard such that the midpoint of the line segment connecting their centers is also the center of a square on the board? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 30826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( x \\) and \\( y \\) are real numbers greater than 10, the leading digit of \\( \\lg x \\) is \\( a \\) and the trailing digit is \\( b \\); the leading digit of \\( \\lg y \\) is \\( c \\) and the trailing digit is \\( d \\). Additionally, it is known that \\( |1... | math | {
"style": "rule",
"ground_truth": "10^7"
} | {
"split": "train",
"index": 30827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, square $ABCD$ has sides of length $4,$ and $\\triangle ABE$ is equilateral. Line segments $BE$ and $AC$ intersect at $P.$ Point $Q$ is on $BC$ so that $PQ$ is perpendicular to $BC$ and $PQ=x.$ [asy]\npair A, B, C, D, E, P, Q;\nA=(0,0);\nB=(4,0);\nC=(4,-4);\nD=(... | math | {
"style": "rule",
"ground_truth": "105^\\circ"
} | {
"split": "train",
"index": 30828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin(2x-\\frac{\\pi}{3})-1$, find the probability that a real number $a$ randomly selected from the interval $\\left[0,\\frac{\\pi}{2}\\right]$ satisfies $f(a) > 0$. 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": 30829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle with side length \\( n \\), divided into unit triangles as illustrated, let \\( f(n) \\) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a commo... | math | {
"style": "rule",
"ground_truth": "2011!"
} | {
"split": "train",
"index": 30830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an equilateral triangle ∆ABC with side length 6, where all three vertices lie on the surface of sphere O with O as the center, and the angle between OA and plane ABC is 45°, find the surface area of sphere O. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "96\\pi"
} | {
"split": "train",
"index": 30831,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute and simplify the following expressions:\n1. $(1)(1 \\frac{1}{2})^{0}-(1-0.5^{-2})÷(\\frac{27}{8})^{\\frac{2}{3}}$\n2. $\\sqrt{2 \\sqrt{2 \\sqrt{2}}}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2^{\\frac{7}{8}}"
} | {
"split": "train",
"index": 30832,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x+1) = x^2 - 1$, \n(1) Find $f(x)$. \n(2) Find the maximum or minimum value of $f(x)$ and the corresponding value of $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 30833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In writing the integers from 10 through 99 inclusive, how many times is the digit 7 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 30834,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 2010, the ages of a brother and sister were 16 and 10 years old, respectively. In what year was the brother's age twice that of the sister's? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"split": "train",
"index": 30835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six different digits from the set \\(\\{ 1,2,3,4,5,6,7,8,9\\}\\) are placed in the squares in the figure shown so that the sum of the entries in the vertical column is 23 and the sum of the entries in the horizontal row is 12. The sum of the six digits used is Let's think step ... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 30836,
"solution": "1. **Identify the structure of the problem**: We have a vertical column and a horizontal row intersecting at one square. The vertical column has three squares, and the horizontal row has four squares, with one square shared between them.\n\n2. **Set up the equation... |
deepscaler | [
{
"role": "user",
"content": "Rectangle \\(ABCD\\) has length 9 and width 5. Diagonal \\(AC\\) is divided into 5 equal parts at \\(W, X, Y\\), and \\(Z\\). Determine the area of the shaded region. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 30837,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "500"
} | {
"split": "train",
"index": 30838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the land of Draconia, there are red, green, and blue dragons. Each dragon has three heads, and each head always tells the truth or always lies. Additionally, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of th... | math | {
"style": "rule",
"ground_truth": "176"
} | {
"split": "train",
"index": 30839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A B C D$ be a convex quadrilateral whose diagonals $A C$ and $B D$ meet at $P$. Let the area of triangle $A P B$ be 24 and let the area of triangle $C P D$ be 25 . What is the minimum possible area of quadrilateral $A B C D ?$ Let's think step by step and put only the fina... | math | {
"style": "rule",
"ground_truth": "49+20 \\sqrt{6}"
} | {
"split": "train",
"index": 30840,
"solution": "Note that $\\angle A P B=180^{\\circ}-\\angle B P C=\\angle C P D=180^{\\circ}-\\angle D P A$ so $4[B P C][D P A]=(P B \\cdot P C \\cdot \\sin B P C)(P D \\cdot P A \\cdot \\sin D P A)=(P A \\cdot P B \\cdot \\sin A P B)(P C \\cdot P D \\cdot \\sin C P D)=4[A P B][... |
deepscaler | [
{
"role": "user",
"content": "Let $x$ and $y$ be two distinct positive real numbers. We define three sequences $(A_n),$ $(G_n),$ and $(H_n)$ as follows. First, $A_1,$ $G_1,$ and $H_1$ are the arithmetic mean, geometric mean, and harmonic mean of $x$ and $y,$ respectively. Then for $n \\ge 2,$ $A_n,$ $G_n... | math | {
"style": "rule",
"ground_truth": "273"
} | {
"split": "train",
"index": 30841,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $20^{21} = 2^a5^b = 4^c5^d = 8^e5^f$ for positive integers $a,b,c,d,e,$ and $f$ . Find $\\frac{100bdf}{ace}$ .\n\n*Proposed by Andrew Wu* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 30842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the 4 PM show, all the seats in the theater were taken, and 65 percent of the audience was children. At the 6 PM show, again, all the seats were taken, but this time only 50 percent of the audience was children. Of all the people who attended either of the shows, 57 percent ... | math | {
"style": "rule",
"ground_truth": "520"
} | {
"split": "train",
"index": 30843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a\\_n\\}$, it is known that $a\\_1+a\\_2+a\\_3=1$, $a\\_4+a\\_5+a\\_6=-2$, find the sum of the first 15 terms of the sequence, denoted as $S\\_{15}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 30844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A number $n$ is $b a d$ if there exists some integer $c$ for which $x^{x} \\equiv c(\\bmod n)$ has no integer solutions for $x$. Find the number of bad integers between 2 and 42 inclusive. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 30845,
"solution": "Call a number good if it is not bad. We claim all good numbers are products of distinct primes, none of which are equivalent to 1 modulo another. We first show that all such numbers are good. Consider $n=p_{1} p_{2} \\ldots p_{k}$, and let $x$ be a number satisfyin... |
deepscaler | [
{
"role": "user",
"content": "Denote $S$ as the subset of $\\{1,2,3,\\dots,1000\\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "501"
} | {
"split": "train",
"index": 30846,
"solution": "\nDenote \\( S \\) as a subset of \\( \\{ 1, 2, 3, \\ldots, 1000 \\} \\) with the property that no sum of two different elements in \\( S \\) is itself an element of \\( S \\). We wish to find the maximum number of elements in \\( S \\).\n\nTo address this problem,... |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR,$ $PQ = 12,$ $QR = 13,$ $PR = 15,$ and point $H$ is the intersection of the medians. Points $P',$ $Q',$ and $R',$ are the images of $P,$ $Q,$ and $R,$ respectively, after a $180^\\circ$ rotation about $H.$ What is the area of the union of the two regions enclos... | math | {
"style": "rule",
"ground_truth": "20\\sqrt{14}"
} | {
"split": "train",
"index": 30847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Johnny has 7 different colored marbles in his bag. In how many ways can he choose three different marbles from his bag to play a game? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 30848,
"solution": ""
} |
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