data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Let $A B C$ be a triangle with $A B=6, A C=7, B C=8$. Let $I$ be the incenter of $A B C$. Points $Z$ and $Y$ lie on the interior of segments $A B$ and $A C$ respectively such that $Y Z$ is tangent to the incircle. Given point $P$ such that $$\\angle Z P C=\\angle Y P B=90^{\\ci... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{30}}{2}"
} | {
"split": "train",
"index": 35249,
"solution": "Solution 1. Let $P U, P V$ tangent from $P$ to the incircle. We will invoke the dual of the Desargues Involution Theorem, which states the following: Given a point $P$ in the plane and four lines $\\ell_{1}, \\ell_{2}, \\ell_{3}, \\ell_{4}$, consider the set of con... |
deepscaler | [
{
"role": "user",
"content": "Convert the point $\\left( 8, \\frac{\\pi}{4}, \\sqrt{3} \\right)$ in cylindrical coordinates to rectangular coordinates. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(4 \\sqrt{2}, 4 \\sqrt{2}, \\sqrt{3})"
} | {
"split": "train",
"index": 35250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Five points on a circle are numbered 1,2,3,4, and 5 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves one point, and if it is on an even-numbered point, it moves two points. If the b... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 35251,
"solution": "1. **Identify the movement pattern**: The bug starts at point 5. According to the rules, since 5 is odd, it moves one point to point 1.\n\n2. **Continue the pattern**:\n - From point 1 (odd), it moves one point to point 2.\n - From point 2 (even), it moves two ... |
deepscaler | [
{
"role": "user",
"content": "When rolling a fair 6-sided die, what is the probability of a 2 or 4 being rolled? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"split": "train",
"index": 35252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a structure formed by joining eight unit cubes where one cube is at the center, and each face of the central cube is shared with one additional cube, calculate the ratio of the volume to the surface area in cubic units to square units. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 35253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Segments $AD=10$, $BE=6$, $CF=24$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicula... | math | {
"style": "rule",
"ground_truth": "\\frac{40}{3}"
} | {
"split": "train",
"index": 35254,
"solution": "1. **Identify the Coordinates of Points $A$, $B$, and $C$ Relative to Line $RS$:**\n Given that segments $AD=10$, $BE=6$, and $CF=24$ are perpendicular to line $RS$, we can consider $RS$ as a horizontal line (like the x-axis in a coordinate system). The lengths o... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\\sin 2B= \\sqrt{3}b\\sin A$.\n\n1. Find $B$;\n2. If $\\cos A= \\dfrac{1}{3}$, find the value of $\\sin C$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\dfrac{2\\sqrt{6}+1}{6}"
} | {
"split": "train",
"index": 35255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(\\frac{y}{x-z}=\\frac{x+y}{z}=\\frac{x}{y}\\) for three positive numbers \\(x,y\\) and \\(z\\), all different, then \\(\\frac{x}{y}=\\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35256,
"solution": "Given the equation:\n\\[\n\\frac{y}{x-z} = \\frac{x+y}{z} = \\frac{x}{y}\n\\]\nWe can use the property that if \\(\\frac{a_1}{b_1} = \\frac{a_2}{b_2} = \\ldots = \\frac{a_n}{b_n} = k\\), then:\n\\[\n\\frac{a_1 + a_2 + \\ldots + a_n}{b_1 + b_2 + \\ldots + b_n} = k\n... |
deepscaler | [
{
"role": "user",
"content": "The sum of 20 consecutive integers is a triangular number. What is the smallest such sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "190"
} | {
"split": "train",
"index": 35257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_{n}\\}$, $\\{b_{n}\\}$, with their sums of the first $n$ terms being $S_{n}$ and $T_{n}$ respectively, and satisfying $\\frac{S_n}{T_n}=\\frac{n+3}{2n-1}$, find $\\frac{a_5}{T_9}$. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{51}"
} | {
"split": "train",
"index": 35258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the definite integral:\n\n$$\n\\int_{\\frac{\\pi}{2}}^{2 \\operatorname{arctg} 2} \\frac{d x}{\\sin ^{2} x(1-\\cos x)}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55/96"
} | {
"split": "train",
"index": 35259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has side lengths $AB=4$, $BC=5$, and $CA=6$. Points $D$ and $E$ are on ray $AB$ with $AB<AD<AE$. The point $F \\neq C$ is a point of intersection of the circumcircles of $\\triangle ACD$ and $\\triangle EBC$ satisfying $DF=2$ and $EF=7$. Then $BE$ can be expresse... | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 35260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A five-character license plate is composed of English letters and digits. The first four positions must contain exactly two English letters (letters $I$ and $O$ cannot be used). The last position must be a digit. Xiao Li likes the number 18, so he hopes that his license plate c... | math | {
"style": "rule",
"ground_truth": "23040"
} | {
"split": "train",
"index": 35261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the equation in terms of \\( x \\)\n$$\nx^{4}-16 x^{3}+(81-2a) x^{2}+(16a-142) x + a^{2} - 21a + 68 = 0\n$$\nwhere all roots are integers, find the value of \\( a \\) and solve the equation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 35262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\\\(f(x) = x^2 + 2ax + 4\\\\) and the interval \\\\([-3,5]\\\\), calculate the probability that the function has no real roots. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 35263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A$ be the set of positive integers that have no prime factors other than $2$, $3$, or $5$. The infinite sum \\[\\frac{1}{1} + \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\frac{1}{5} + \\frac{1}{6} + \\frac{1}{8} + \\frac{1}{9} + \\frac{1}{10} + \\frac{1}{12} + \\frac{1}{... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 35264,
"solution": "To solve the problem, we need to calculate the sum of the reciprocals of all positive integers that have no prime factors other than $2$, $3$, or $5$. These integers can be expressed in the form $2^a 3^b 5^c$ where $a$, $b$, and $c$ are non-negative integers.\n\n1.... |
deepscaler | [
{
"role": "user",
"content": "There are 22 black and 3 blue balls in a bag. Ahmet chooses an integer $ n$ in between 1 and 25. Betül draws $ n$ balls from the bag one by one such that no ball is put back to the bag after it is drawn. If exactly 2 of the $ n$ balls are blue and the second blue ball is ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 35265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\sin(α + \\frac{π}{6}) - \\cos α = \\frac{1}{3}$, find the value of $2 \\sin α \\cos(α + \\frac{π}{6})$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 35266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Person A and person B start simultaneously from points A and B, respectively, and move towards each other. When person A reaches the midpoint C of A and B, person B is still 240 meters away from point C. When person B reaches point C, person A has already moved 360 meters past ... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 35267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $10\\times10\\times10$ grid of points consists of all points in space of the form $(i,j,k)$, where $i$, $j$, and $k$ are integers between $1$ and $10$, inclusive. Find the number of different lines that contain exactly $8$ of these points. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 35268,
"solution": "Considered the cases $(1, 2, ..., 8), (2, 3, ...,9), (3, 4, ..., 10)$ and reverse. Also, consider the constant subsequences of length 8 $(1, 1, ..., 1), (2, 2, ..., 2), ..., (10, 10, ..., 10)$. Of all the triplets that work they cannot be extended to form another p... |
deepscaler | [
{
"role": "user",
"content": "In the drawing, there is a grid composed of 25 small equilateral triangles.\n\nHow many rhombuses can be formed from two adjacent small triangles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ is a right triangle with legs $AB$ and $AC$. Points $X$ and $Y$ lie on legs $AB$ and $AC$, respectively, so that $AX:XB = AY:YC = 1:2$. If $BY = 16$ units, and $CX = 28$ units, what is the length of hypotenuse $BC$? Express your answer in simplest radical form. L... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{26}"
} | {
"split": "train",
"index": 35270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A non-increasing sequence of 100 non-negative reals has the sum of the first two terms at most 100 and the sum of the remaining terms at most 100. What is the largest possible value for the sum of the squares of the terms? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "10000"
} | {
"split": "train",
"index": 35271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a square \\(ABCD\\), let \\(P\\) be a point on the side \\(BC\\) such that \\(BP = 3PC\\) and \\(Q\\) be the midpoint of \\(CD\\). If the area of the triangle \\(PCQ\\) is 5, what is the area of triangle \\(QDA\\)? Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a rectangular grid where grid lines are spaced $1$ unit apart, the acronym XYZ is depicted below. The X is formed by two diagonal lines crossing, the Y is represented with a 'V' shape starting from a bottom point going up to join two endpoints with horizontal lines, the Z is... | math | {
"style": "rule",
"ground_truth": "4 + 5\\sqrt{2}"
} | {
"split": "train",
"index": 35273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of real solutions to the equation:\n\\[\n\\frac{1^2}{x - 1} + \\frac{2^2}{x - 2} + \\frac{3^2}{x - 3} + \\dots + \\frac{120^2}{x - 120} = x.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"split": "train",
"index": 35274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 4~.$$ Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"split": "train",
"index": 35275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\theta \\in (0^\\circ, 360^\\circ)$ and the terminal side of angle $\\theta$ is symmetric to the terminal side of the $660^\\circ$ angle with respect to the x-axis, and point $P(x, y)$ is on the terminal side of angle $\\theta$ (not the origin), find the value of $$\\frac ... | math | {
"style": "rule",
"ground_truth": "\\frac {\\sqrt {3}}{4}"
} | {
"split": "train",
"index": 35276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the height of Jack's house, in feet, if the house casts a shadow 56 feet long at the same time a 21-foot tree casts a shadow that is 24 feet long? Express your answer to the nearest whole number. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "49"
} | {
"split": "train",
"index": 35277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its sec... | math | {
"style": "rule",
"ground_truth": "1609"
} | {
"split": "train",
"index": 35278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A woman wants freshly baked cookies delivered exactly at 18:00 for an event. Delivery trucks, upon finishing baking, travel with varying speeds due to potential traffic conditions:\n- If there is moderate traffic, the trucks travel at an average speed of 60 km/h and would arriv... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is inscribed in the ellipse\n\\[\\frac{x^2}{3} + \\frac{y^2}{6} = 1,\\]so that its sides are parallel to the coordinate axes. Find the area of the square.\n\n[asy]\nunitsize(1 cm);\n\ndraw(xscale(sqrt(3))*yscale(sqrt(6))*Circle((0,0),1));\ndraw((sqrt(2),sqrt(2))--(-sq... | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 35280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that in triangle PQR, side PR = 6 cm and side PQ = 10 cm, point S is the midpoint of QR, and the length of the altitude from P to QR is 4 cm, calculate the length of QR. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{5}"
} | {
"split": "train",
"index": 35281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Increase Grisha's yield by 40% and Vasya's yield by 20%.\n\nGrisha, the most astute among them, calculated that in the first case their total yield would increase by 1 kg; in the second case, it would decrease by 0.5 kg; in the third case, it would increase by 4 kg. What was th... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A toy factory has a total of 450 labor hours and 400 units of raw materials for production. Producing a bear requires 15 labor hours and 20 units of raw materials, with a selling price of 80 yuan; producing a cat requires 10 labor hours and 5 units of raw materials, with a sell... | math | {
"style": "rule",
"ground_truth": "2200"
} | {
"split": "train",
"index": 35283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given numbers \\( x_{1}, \\cdots, x_{1991} \\) satisfy the condition\n$$\n\\left|x_{1}-x_{2}\\right|+\\cdots+\\left|x_{1990}-x_{1991}\\right|=1991 ,\n$$\n\nwhere \\( y_{k}=\\frac{1}{k}\\left(x_{1}+\\cdots+x_{k}\\right) \\) for \\( k = 1, \\cdots, 1991 \\). Find the maximum poss... | math | {
"style": "rule",
"ground_truth": "1990"
} | {
"split": "train",
"index": 35284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{500}{1001}"
} | {
"split": "train",
"index": 35285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define an ordered quadruple of integers $(a, b, c, d)$ as captivating if $1 \\le a < b < c < d \\le 15$, and $a+d > 2(b+c)$. How many captivating ordered quadruples are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 35286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate $\\int_{0}^{1} \\frac{\\sin x}{x} \\, dx$ with an accuracy of 0.01. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.94"
} | {
"split": "train",
"index": 35287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One base of a trapezoid is $100$ units longer than the other base. The segment that joins the midpoints of the legs divides the trapezoid into two regions whose areas are in the ratio $2: 3$. Let $x$ be the length of the segment joining the legs of the trapezoid that is paralle... | math | {
"style": "rule",
"ground_truth": "181"
} | {
"split": "train",
"index": 35288,
"solution": "Let the shorter base have length $b$ (so the longer has length $b+100$), and let the height be $h$. The length of the midline of the trapezoid is the average of its bases, which is $\\frac{b+b+100}{2} = b+50$. The two regions which the midline divides the trapezoid... |
deepscaler | [
{
"role": "user",
"content": "Let \\\\((x^{2}+1)(2x+1)^{9}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\\ldots+a_{n}(x+2)^{n}\\\\), then \\\\(a_{0}+a_{1}+a_{2}+\\ldots+a_{n}=\\\\) \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 35289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the domain of function $y=f(x)$ is $D$. If for any $x_{1}, x_{2} \\in D$, when $x_{1} + x_{2} = 2a$, it always holds that $f(x_{1}) + f(x_{2}) = 2b$, then the point $(a,b)$ is called the center of symmetry of the graph of the function $y=f(x)$. Investigate a center of s... | math | {
"style": "rule",
"ground_truth": "-4035"
} | {
"split": "train",
"index": 35290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures $3$ by $4$ by $5$ units. Given that the volume of this set is $\\frac{m + n\\pi}{p},$ where $m, n,$ and $p$ are positive integers, and $n$ and $p$ are relatively pr... | math | {
"style": "rule",
"ground_truth": "505"
} | {
"split": "train",
"index": 35291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve in positive integers the following equation:\n\\[{1\\over n^2}-{3\\over 2n^3}={1\\over m^2}\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(m, n) = (4, 2)"
} | {
"split": "train",
"index": 35292,
"solution": "\nTo solve the equation in positive integers:\n\n\\[\n\\frac{1}{n^2} - \\frac{3}{2n^3} = \\frac{1}{m^2},\n\\]\n\nwe start by simplifying the left-hand side of the equation. Begin by finding a common denominator:\n\n\\[\n\\frac{1}{n^2} - \\frac{3}{2n^3} = \\frac{2}{... |
deepscaler | [
{
"role": "user",
"content": "The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 35293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be the least common multiple of all integers from $10$ to $40$, inclusive. Let $Q$ be the least common multiple of $P$ and the integers $41, 42, 43, 44, 45, 46, 47, 48, 49,$ and $50$. What is the value of $\\frac{Q}{P}?$\nA) 1763\nB) 82861\nC) 9261\nD) 14756 Let's think... | math | {
"style": "rule",
"ground_truth": "82861"
} | {
"split": "train",
"index": 35294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the community center has 8 cans of soup and 2 loaves of bread, with each can of soup feeding 4 adults or 7 children and each loaf of bread feeding 3 adults or 4 children, and the center needs to feed 24 children, calculate the number of adults that can be fed with the remain... | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 35295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people, Jia, Yi, and Bing, participated in a competition and they took the top 3 places (with no ties). Jia said: \"I am first\", Yi said: \"I am not first\", and Bing said: \"I am not third\". Only one of them is telling the truth. If the rankings of Jia, Yi, and Bing ar... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 35296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{1}$ and $F_{2}$ are respectively the left and right foci of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\ (a > b > 0)$, a circle with the origin as its center and the semi-focal distance as its radius intersects the right branch of the hyperbola at po... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} + 1"
} | {
"split": "train",
"index": 35297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three people jointly start a business with a total investment of 143 million yuan. The ratio of the highest investment to the lowest investment is 5:3. What is the maximum and minimum amount the third person could invest in millions of yuan? Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 35298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point selected uniformly at random in the cube $[0,1]^{3}$. The plane parallel to $x+y+z=0$ passing through $P$ intersects the cube in a two-dimensional region $\\mathcal{R}$. Let $t$ be the expected value of the perimeter of $\\mathcal{R}$. If $t^{2}$ can be writt... | math | {
"style": "rule",
"ground_truth": "12108"
} | {
"split": "train",
"index": 35299,
"solution": "We can divide the cube into 3 regions based on the value of $x+y+z$ which defines the plane: $x+y+z<1,1 \\leq x+y+z \\leq 2$, and $x+y+z>2$. The two regions on the ends create tetrahedra, each of which has volume $1 / 6$. The middle region is a triangular antiprism... |
deepscaler | [
{
"role": "user",
"content": "Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{35}{12}"
} | {
"split": "train",
"index": 35300,
"solution": "1. **Identify the Geometry of the Problem:**\n - Triangle $ABC$ is a right triangle with $AB$ as the hypotenuse, so $AB = 10$, $BC = 6$, and $AC = 8$.\n - Circle centered at $O$ is tangent to $BC$ at $B$ and passes through $A$.\n - Circle centered at $P$ is t... |
deepscaler | [
{
"role": "user",
"content": "Triangle $PQR$ is a right triangle with legs $PQ$ and $PR$. Points $U$ and $V$ are on legs $PQ$ and $PR$, respectively so that $PU:UQ = PV:VR = 1:3$. If $QU = 18$ units, and $RV = 45$ units, what is the length of hypotenuse $PQ$? Express your answer in simplest radical form. Le... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{29}"
} | {
"split": "train",
"index": 35301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One hundred number cards are laid out in a row in ascending order: \\(00, 01, 02, 03, \\ldots, 99\\). Then, the cards are rearranged so that each subsequent card is obtained from the previous one by increasing or decreasing exactly one of the digits by 1 (for example, after 29 ... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 35302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{5}"
} | {
"split": "train",
"index": 35303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Usain runs one lap around the school stadium at a constant speed, and photographers Arina and Marina are positioned near the track. After the start, for 4 seconds, Usain was closer to Arina, then for 21 seconds he was closer to Marina, and then until the finish, he was again cl... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 35304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x,$ $y,$ and $z$ be nonnegative real numbers such that $x + y + z = 1.$ Find the maximum value of $x + y^2 + z^3.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 35305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. If $a=2$, $b= \\sqrt {3}$, $B= \\frac {\\pi}{3}$, then $A=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{2}"
} | {
"split": "train",
"index": 35306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P=\\{1,2,\\ldots,6\\}$, and let $A$ and $B$ be two non-empty subsets of $P$. Find the number of pairs of sets $(A,B)$ such that the maximum number in $A$ is less than the minimum number in $B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "129"
} | {
"split": "train",
"index": 35307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the real numbers $m$, $n$, $s$, $t$ are all distinct and satisfy $mn=st$, then $m$, $n$, $s$, $t$ are said to have the property of \"quasi-geometric progression.\" Now, randomly select $4$ different numbers from the $7$ numbers $2$, $4$, $8$, $16$, $32$, $64$, $128$. The pro... | math | {
"style": "rule",
"ground_truth": "\\frac{13}{35}"
} | {
"split": "train",
"index": 35308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $ 4\\times 4$ table is divided into $ 16$ white unit square cells. Two cells are called neighbors if they share a common side. A [i]move[/i] consists in choosing a cell and the colors of neighbors from white to black or from black to white. After exactly $ n$ moves all the $ ... | math | {
"style": "rule",
"ground_truth": "6, 8, 10, 12, 14, 16, \\ldots"
} | {
"split": "train",
"index": 35309,
"solution": "\nTo solve this problem, we must determine the number of moves, \\( n \\), necessary to change all 16 cells of a \\( 4 \\times 4 \\) grid from white to black. The transformation involves a series of operations, each toggling the color (from white to black or black ... |
deepscaler | [
{
"role": "user",
"content": "Richard starts with the string HHMMMMTT. A move consists of replacing an instance of HM with MH , replacing an instance of MT with TM, or replacing an instance of TH with HT. Compute the number of possible strings he can end up with after performing zero or more moves. Let's th... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 35310,
"solution": "The key claim is that the positions of the Ms fully determines the end configuration. Indeed, since all Hs are initially left of all Ts, the only successful swaps that can occur will involve Ms. So, picking $\\binom{8}{4}=70$ spots for Ms and then filling in the re... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos(\\frac{\\pi}{6} - \\alpha) = \\frac{\\sqrt{3}}{3}$, find the value of $\\sin(\\frac{5\\pi}{6} - 2\\alpha)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{3}"
} | {
"split": "train",
"index": 35311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the numbers \\(1, 2, 3, \\ldots, 1000\\). Find the largest number \\(m\\) with the following property: no matter which \\(m\\) of these numbers are removed, among the remaining \\(1000-m\\) numbers, there will be two such that one of them divides the other. Let's think st... | math | {
"style": "rule",
"ground_truth": "499"
} | {
"split": "train",
"index": 35312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=|x-p|+|x-15|+|x-p-15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \\leq x\\leq15$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35313,
"solution": "It is best to get rid of the absolute values first.\n Under the given circumstances, we notice that $|x-p|=x-p$, $|x-15|=15-x$, and $|x-p-15|=15+p-x$.\n Adding these together, we find that the sum is equal to $30-x$, which attains its minimum value (on the given in... |
deepscaler | [
{
"role": "user",
"content": "The function $f(x)=\\log_{a}(1-x)+\\log_{a}(x+3)$ where $0 < a < 1$. \n$(1)$ Find the zeros of the function $f(x)$. \n$(2)$ If the minimum value of the function $f(x)$ is $-2$, find the value of $a$. 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": 35314,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Palindromes are numbers that read the same backwards and forwards, like 5665. What is the least possible positive four-digit palindrome that is divisible by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1221"
} | {
"split": "train",
"index": 35315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, $ABCD$ is a trapezoid with bases $AB$ and $CD$ such that $AB$ is parallel to $CD$ and $CD$ is three times the length of $AB$. The area of $ABCD$ is $27$. Find the area of $\\triangle ABC$.\n\n[asy]\ndraw((0,0)--(3,6)--(9,6)--(12,0)--cycle);\ndraw((3,6)--(0,0));\... | math | {
"style": "rule",
"ground_truth": "6.75"
} | {
"split": "train",
"index": 35316,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a small reserve, a biologist counted a total of 300 heads comprising of two-legged birds, four-legged mammals, and six-legged insects. The total number of legs counted was 980. Calculate the number of two-legged birds. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 35317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below.\n\\[\\begin{array}{c@{\\hspace{8em}} c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{6pt}}c@{\\hspace{4pt}}c@{\\hspace{2pt}} c@{\\hspace{2pt}}c@{\\hspace{2... | math | {
"style": "rule",
"ground_truth": "62"
} | {
"split": "train",
"index": 35318,
"solution": "Call the row x, and the number from the leftmost side t. Call the first term in the ratio $N$. $N = \\dbinom{x}{t}$. The next term is $N * \\frac{x-t}{t+1}$, and the final term is $N * \\frac{(x-t)*(x-t-1)}{(t+1)*(t+2)}$. Because we have the ratio, $N : N * \\frac{... |
deepscaler | [
{
"role": "user",
"content": "If $x-y>x$ and $x+y<y$, then Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x<0,y<0"
} | {
"split": "train",
"index": 35319,
"solution": "We are given two inequalities:\n1. \\(x - y > x\\)\n2. \\(x + y < y\\)\n\n#### Analyzing the first inequality:\nStarting from the first inequality:\n\\[ x - y > x \\]\nSubtract \\(x\\) from both sides:\n\\[ -y > 0 \\]\nThis implies:\n\\[ y < 0 \\]\nThus, \\(y\\) is... |
deepscaler | [
{
"role": "user",
"content": "An $m\\times n\\times p$ rectangular box has half the volume of an $(m + 2)\\times(n + 2)\\times(p + 2)$ rectangular box, where $m, n,$ and $p$ are integers, and $m\\le n\\le p.$ What is the largest possible value of $p$? Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 35320,
"solution": "Similarly as above, we solve for $p,$ but we express the denominator differently:\n\\[p=\\dfrac{2(m+2)(n+2)}{(m+2)(n+2)-4(m+n+2)} \\implies \\dfrac{1}{p}=\\dfrac{1}{2}-\\dfrac{2(m+n+2)}{(m+2)(n+2)}.\\] Hence, it suffices to maximize $\\dfrac{m+n+2}{(m+2)(n+2)},$ un... |
deepscaler | [
{
"role": "user",
"content": "Three male students and three female students, a total of six students, stand in a row. If male student A does not stand at either end, and exactly two of the three female students stand next to each other, then the number of different arrangements is ______. Let's think step b... | math | {
"style": "rule",
"ground_truth": "288"
} | {
"split": "train",
"index": 35321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 35322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the set $S = \\{P_1, P_2, \\dots, P_{12}\\}$ consist of the twelve vertices of a regular $12$-gon. A subset $Q$ of $S$ is called \"communal\" if there is a circle such that all points of $Q$ are inside the circle, and all points of $S$ not in $Q$ are outside of the circle. ... | math | {
"style": "rule",
"ground_truth": "134"
} | {
"split": "train",
"index": 35323,
"solution": "By looking at the problem and drawing a few pictures, it quickly becomes obvious that one cannot draw a circle that covers $2$ disjoint areas of the $12$-gon without including all the vertices in between those areas. In other words, in order for a subset to be comm... |
deepscaler | [
{
"role": "user",
"content": "What is $4\\cdot 6+8\\cdot 3-28\\div 2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 35324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\cos 75^\\circ.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6} - \\sqrt{2}}{4}"
} | {
"split": "train",
"index": 35325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given circle M: $(x+1)^2+y^2=1$, and circle N: $(x-1)^2+y^2=9$, a moving circle P is externally tangent to circle M and internally tangent to circle N. The trajectory of the center of circle P is curve C. \n(1) Find the equation of C: \n(2) Let $l$ be a line that is tangent t... | math | {
"style": "rule",
"ground_truth": "\\frac{18}{7}"
} | {
"split": "train",
"index": 35326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean smaller than the largest of these numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.8"
} | {
"split": "train",
"index": 35327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $x$ be a positive integer, and define the integers $n=x^2+2x+17$ and $d=2x+5$. When dividing $n$ by $d$, the quotient is $x$, and the remainder is $7$. Find $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a quadratic function $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \\geq 2ax + b$ is $\\mathbb{R}$ (the set of all real numbers), then the maximum value of $\\frac{b^2}{a^2 + c^2}$ is __________. Let's think st... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2} - 2"
} | {
"split": "train",
"index": 35329,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Call a three-digit number $\\overline{ABC}$ $\\textit{spicy}$ if it satisfies $\\overline{ABC}=A^3+B^3+C^3$ . Compute the unique $n$ for which both $n$ and $n+1$ are $\\textit{spicy}$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "370"
} | {
"split": "train",
"index": 35330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos^{4}x-2\\sin x\\cos x-\\sin^{4}x$.\n\n(1) Find the smallest positive period of the function $f(x)$;\n\n(2) When $x\\in\\left[0,\\frac{\\pi}{2}\\right]$, find the minimum value of $f(x)$ and the set of $x$ values where the minimum value is obtained.... | math | {
"style": "rule",
"ground_truth": "\\left\\{\\frac{3\\pi}{8}\\right\\}"
} | {
"split": "train",
"index": 35331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of $(ax + b)^{2000},$ where $a$ and $b$ are relatively prime positive integers, the coefficients of $x^{2}$ and $x^{3}$ are equal. Find $a + b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "667"
} | {
"split": "train",
"index": 35332,
"solution": "Using the binomial theorem, $\\binom{2000}{1998} b^{1998}a^2 = \\binom{2000}{1997}b^{1997}a^3 \\Longrightarrow b=666a$.\n Since $a$ and $b$ are positive relatively prime integers, $a=1$ and $b=666$, and $a+b=\\boxed{667}$."
} |
deepscaler | [
{
"role": "user",
"content": "$3^n = 3 \\cdot 9^3 \\cdot 81^2$. What is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle DEF$ and $\\triangle D'E'F'$ are in the coordinate plane with vertices $D(2,2)$, $E(2,14)$, $F(18,2)$, $D'(32,26)$, $E'(44,26)$, $F'(32,10)$. A rotation of $n$ degrees clockwise around the point $(u,v)$ where $0<n<180$, will transform $\\triangle DEF$ to $... | math | {
"style": "rule",
"ground_truth": "124"
} | {
"split": "train",
"index": 35334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c, x, y, z \\) be nonzero complex numbers such that\n\\[ a = \\frac{b+c}{x-3}, \\quad b = \\frac{a+c}{y-3}, \\quad c = \\frac{a+b}{z-3}, \\]\nand \\( xy + xz + yz = 10 \\) and \\( x + y + z = 6 \\), find \\( xyz \\). Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $F_{2}$ is the right focus of the ellipse $mx^{2}+y^{2}=4m\\left(0 \\lt m \\lt 1\\right)$, point $A\\left(0,2\\right)$, and point $P$ is any point on the ellipse, and the minimum value of $|PA|-|PF_{2}|$ is $-\\frac{4}{3}$, then $m=$____. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 35336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the only integer whose square is less than its double? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 35337,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At the 2013 Winnebago County Fair a vendor is offering a \"fair special\" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the \"fair special\" t... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35338,
"solution": "1. **Calculate the total cost without any discounts:** \n The regular price for one pair of sandals is $50. Therefore, the cost for three pairs of sandals at the regular price is:\n \\[\n 3 \\times 50 = 150 \\text{ dollars}\n \\]\n\n2. **Calculate the tota... |
deepscaler | [
{
"role": "user",
"content": "A section of a book fell out. The first page of this section is numbered 143, and the number of the last page consists of the same digits but in a different order. How many pages fell out of the book? Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "172"
} | {
"split": "train",
"index": 35339,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Yesterday, Sasha cooked soup and added too little salt, so he had to add more salt later. Today he doubled the amount of salt, but still had to add more salt, though only half as much as yesterday. By what factor does Sasha need to increase today's portion of salt so that tomor... | math | {
"style": "rule",
"ground_truth": "1.5"
} | {
"split": "train",
"index": 35340,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let points $A(x_1, y_1)$ and $B(x_2, y_2)$ be on the graph of $f(x) = x^2$. The points $C$ and $D$ trisect the segment $\\overline{AB}$ with $AC < CB$. A horizontal line drawn through $C$ intersects the curve at another point $E(x_3, y_3)$. Find $x_3$ if $x_1 = 1$ and $x_2 = 4$... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 35341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 1985 sets, each consisting of 45 elements, and the union of any two sets contains exactly 89 elements. How many elements are in the union of all these 1985 sets? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "87341"
} | {
"split": "train",
"index": 35342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nOlga Ivanovna, the class teacher of Grade 5B, is organizing a \"Mathematical Ballet.\" She wants to arrange boys and girls so that at a distance of 5 meters from each girl there are exactly 2 boys. What is the maximum number of girls that can participate in the ballet, given ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Pigsy picked a bag of wild fruits from the mountain to bring back to Tang Monk. As he walked, he got hungry and took out half of them, hesitated, put back two, and then ate the rest. This happened four times: taking out half, putting back two, and then eating. By the time he re... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( AB = 8 \\), \\( BC = 11 \\), and \\( AC = 6 \\). The points \\( P \\) and \\( Q \\) are on \\( BC \\) such that \\( \\triangle PBA \\) and \\( \\triangle QAC \\) are each similar to \\( \\triangle ABC \\). What is the length of \\( PQ \\)... | math | {
"style": "rule",
"ground_truth": "\\frac{21}{11}"
} | {
"split": "train",
"index": 35345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "12 students are standing in two rows, with 4 in the front row and 8 in the back row. Now, 2 students from the back row are to be selected to stand in the front row. If the relative order of the other students remains unchanged, the number of different rearrangement methods is _... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 35346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\( AB = AC \\), \\( \\angle A = 80^\\circ \\), and point \\( D \\) is inside the triangle such that \\( \\angle DAB = \\angle DBA = 10^\\circ \\). Find the measure of \\( \\angle ACD \\). Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 35347,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that\n$\\frac{((3!)!)!}{3!} = k \\cdot n!,$\n where $k$ and $n$ are positive integers and $n$ is as large as possible, find $k + n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "839"
} | {
"split": "train",
"index": 35348,
"solution": "Note that\\[{{\\left((3!)!\\right)!}\\over{3!}}= {{(6!)!}\\over{6}}={{720!}\\over6}={{720\\cdot719!}\\over6}=120\\cdot719!.\\]Because $120\\cdot719!<720!$, conclude that $n$ must be less than 720, so the maximum value of $n$ is 719. The requested value of $k+n$ is ... |
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