data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of the following function $f(x) $ defined at $0<x<\\frac{\\pi}{2}$ .\r\n\r\n\\[f(x)=\\int_0^x \\frac{d\\theta}{\\cos \\theta}+\\int_x^{\\frac{\\pi}{2}} \\frac{d\\theta}{\\sin \\theta}\\] Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "\\ln(3 + 2\\sqrt{2})"
} | {
"split": "train",
"index": 3200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maria ordered a certain number of televisions for the stock of a large store, paying R\\$ 1994.00 per television. She noticed that in the total amount to be paid, the digits 0, 7, 8, and 9 do not appear. What is the smallest number of televisions she could have ordered? Let's t... | math | {
"style": "rule",
"ground_truth": "56"
} | {
"split": "train",
"index": 3201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that 2(tanA + tanB) = $\\frac{\\text{tanA}}{\\text{cosB}} + \\frac{\\text{tanB}}{\\text{cosA}}$.\n(1) Find the value of $\\frac{a+b}{c}$;\n(2) If c = 2 and C = $\\frac{\\pi}{3}$, find ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 3202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "With about six hours left on the van ride home from vacation, Wendy looks for something to do. She starts working on a project for the math team.\n\nThere are sixteen students, including Wendy, who are about to be sophomores on the math team. Elected as a math team officer, o... | math | {
"style": "rule",
"ground_truth": "1881"
} | {
"split": "train",
"index": 3203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $m=(\\sin x,-1)$ and $n=\\left( \\sqrt{3}\\cos x,-\\frac{1}{2}\\right)$, and the function $f(x)=(m+n)\\cdot m$.\n\n1. Find the interval where the function $f(x)$ is monotonically decreasing.\n2. Given $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, an... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 3204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Walking is a form of exercise that falls between walking and racewalking. It is a simple and safe aerobic exercise that can enhance lung capacity and promote heart health. A sports physiologist conducted a large number of surveys on the body fat percentage ($X$) of people engag... | math | {
"style": "rule",
"ground_truth": "0.03"
} | {
"split": "train",
"index": 3205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive three-digit integers less than 500 have at least two digits that are the same? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 3206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convert the point $(1,-\\sqrt{3})$ in rectangular coordinates to polar coordinates. Enter your answer in the form $(r,\\theta),$ where $r > 0$ and $0 \\le \\theta < 2 \\pi.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( 2, \\frac{5 \\pi}{3} \\right)"
} | {
"split": "train",
"index": 3207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Mrs. Marsh's algebra class, 30 of the 36 students took the Chapter 4 test, and their mean score was $72 \\%$. The next day the other six students took the test and their mean score was $78 \\%$. What is the new class mean? Express your answer as a percent. Let's think step b... | math | {
"style": "rule",
"ground_truth": "73\\%"
} | {
"split": "train",
"index": 3208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nThe medians \\( A M \\) and \\( B E \\) of triangle \\( A B C \\) intersect at point \\( O \\). Points \\( O, M, E, C \\) lie on the same circle. Find \\( A B \\) if \\( B E = A M = 3 \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{3}"
} | {
"split": "train",
"index": 3209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence elements \\( a_{n} \\) such that \\( a_{1}=1337 \\) and \\( a_{2n+1}=a_{2n}=n-a_{n} \\) for all positive integers \\( n \\). Determine the value of \\( a_{2004} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"split": "train",
"index": 3210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Forty teams play a tournament in which every team plays every other team exactly once. No ties occur, and each team has a $50 \\%$ chance of winning any game it plays. The probability that no two teams win the same number of games is $\\frac mn,$ where $m$ and $n$ are relativel... | math | {
"style": "rule",
"ground_truth": "742"
} | {
"split": "train",
"index": 3211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three marbles are randomly selected, without replacement, from a bag containing two red, two blue and two green marbles. What is the probability that one marble of each color is selected? Express your answer as a common fraction. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 3212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\frac{\\sqrt{2}}{\\sqrt{3}} \\cdot \\frac{\\sqrt{4}}{\\sqrt{5}} \\cdot \\frac{\\sqrt{6}}{\\sqrt{7}}$ and rationalize the denominator of the resulting fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{35}}{35}"
} | {
"split": "train",
"index": 3213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $X_1, X_2, \\ldots, X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$ . Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$ , that is, $X_i\\cap X_{i+1}=\\emptyset$ and $X_i\\cup X_{i+1}\\neq S$ , for all $i\\in\\{1... | math | {
"style": "rule",
"ground_truth": "|S| \\ge 8"
} | {
"split": "train",
"index": 3214,
"solution": "The answer is that $|S| \\ge 8$ .\nFirst, we provide a inductive construction for $S = \\left\\{ 1, \\dots, 8 \\right\\}$ . Actually, for $n \\ge 4$ we will provide a construction for $S = \\left\\{ 1, \\dots, n \\right\\}$ which has $2^{n-1} + 1$ elements in a line... |
deepscaler | [
{
"role": "user",
"content": "Find the number of solutions to\n\\[\\sin x = \\left( \\frac{1}{3} \\right)^x\\] on the interval $(0,100\\pi).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 3215,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $D$, $E$, and $F$ be constants such that the equation \\[\\frac{(x+E)(Dx+36)}{(x+F)(x+9)} = 3\\] has infinitely many solutions for $x$. For these values of $D$, $E$, and $F$, it turns out that there are only finitely many values of $x$ which are not solutions to the equatio... | math | {
"style": "rule",
"ground_truth": "-21"
} | {
"split": "train",
"index": 3216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers \\( a, b, c \\) satisfy\n\\[\na^{2}+b^{2}-4a \\leqslant 1, \\quad b^{2}+c^{2}-8b \\leqslant -3, \\quad c^{2}+a^{2}-12c \\leqslant -26,\n\\]\nwhat is the value of \\( (a+b)^{c} \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "27"
} | {
"split": "train",
"index": 3217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that, for all positive integers $k$,\n$1^2+2^2+3^2+\\ldots+k^{2}=\\frac{k(k+1)(2k+1)}6$.\n Find the smallest positive integer $k$ such that $1^2+2^2+3^2+\\ldots+k^2$ is a multiple of $200$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 3218,
"solution": "$\\frac{k(k+1)(2k+1)}{6}$ is a multiple of $200$ if $k(k+1)(2k+1)$ is a multiple of $1200 = 2^4 \\cdot 3 \\cdot 5^2$. So $16,3,25|k(k+1)(2k+1)$.\n Since $2k+1$ is always odd, and only one of $k$ and $k+1$ is even, either $k, k+1 \\equiv 0 \\pmod{16}$.\n Thus, $k \\e... |
deepscaler | [
{
"role": "user",
"content": "The simple (i.e., non-intersecting) quadrilateral \\(ABCD\\) has sides \\(AB\\), \\(BC\\), and \\(CD\\) with lengths 4, 5, and 20, respectively. If the angles \\(B\\) and \\(C\\) are obtuse, and \\(\\sin C = -\\cos B = \\frac{3}{5}\\), then what is the length of the side \\(AD\... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 3219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the random variable $X \\sim N(1, \\sigma^{2})$, if $P(0 < x < 3)=0.5$, $P(0 < X < 1)=0.2$, then $P(X < 3)=$\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_\\_ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.8"
} | {
"split": "train",
"index": 3220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a scalene triangle with integer side lengths $a, b, c$, the following relation holds. What is the smallest height of the triangle?\n\n$$\n\\frac{a^{2}}{c}-(a-c)^{2}=\\frac{b^{2}}{c}-(b-c)^{2}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 3221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( f(n) \\) be the number of 0's in the decimal representation of the positive integer \\( n \\). For example, \\( f(10001123) = 3 \\) and \\( f(1234567) = 0 \\). Find the value of\n\\[ f(1) + f(2) + f(3) + \\ldots + f(99999) \\] Let's think step by step and put only the f... | math | {
"style": "rule",
"ground_truth": "38889"
} | {
"split": "train",
"index": 3222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, $AB= \\sqrt {3}$, $BC=1$, and $\\sin C= \\sqrt {3}\\cos C$, the area of $\\triangle ABC$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {3}}{2}"
} | {
"split": "train",
"index": 3223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Kermit the frog enjoys hopping around the infinite square grid in his backyard. It takes him 1 Joule of energy to hop one step north or one step south, and 1 Joule of energy to hop one step east or one step west. He wakes up one morning on the grid with 100 Joules of energy, an... | math | {
"style": "rule",
"ground_truth": "10201"
} | {
"split": "train",
"index": 3224,
"solution": "It is easy to see that the coordinates of the frog's final position must have the same parity. Suppose that the frog went to sleep at $(x, y)$. Then, we have that $-100 \\leq y \\leq 100$ and $|x| \\leq 100-|y|$, so $x$ can take on the values $-100+|y|,-98+|y|, \\ld... |
deepscaler | [
{
"role": "user",
"content": "One of the sides of a triangle is divided into segments of $6$ and $8$ units by the point of tangency of the inscribed circle. If the radius of the circle is $4$, then the length of the shortest side is Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3225,
"solution": "1. **Identify the segments created by the inscribed circle**: Let's label the triangle as $\\triangle ABC$ with the inscribed circle touching $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$ respectively. Given that one side is divided into segments of $6$ and $8$ u... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\begin{cases} x+2 & (x\\leqslant -1) \\\\ x^{2} & (-1< x < 2) \\\\ 2x & (x\\geqslant 2) \\end{cases}$ \n$(1)$ Find $f(2)$, $f\\left(\\dfrac{1}{2}\\right)$, $f[f(-1)]$; \n$(2)$ If $f(a)=3$, find the value of $a$. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 3226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( n \\geq 1 \\) be a positive integer. We say that an integer \\( k \\) is a fan of \\( n \\) if \\( 0 \\leq k \\leq n-1 \\) and there exist integers \\( x, y, z \\in \\mathbb{Z} \\) such that\n\\[ \n\\begin{aligned}\nx^2 + y^2 + z^2 &\\equiv 0 \\pmod{n}; \\\\\nxyz &\\equ... | math | {
"style": "rule",
"ground_truth": "101"
} | {
"split": "train",
"index": 3227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\\triangle ABC$, respectively, and it satisfies $(2b-a) \\cdot \\cos C = c \\cdot \\cos A$.\n$(1)$ Find the size of angle $C$;\n$(2)$ Let $y = -4\\sqrt{3}\\sin^2\\frac{A}{2} + 2\\sin(C-B)$, find the max... | math | {
"style": "rule",
"ground_truth": "2-2 \\sqrt {3}"
} | {
"split": "train",
"index": 3228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arrangement of letters and numerals below, by how many different paths can one spell AMC8? Beginning at the A in the middle, a path only allows moves from one letter to an adjacent (above, below, left, or right, but not diagonal) letter. One example of such a path is tra... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3229,
"solution": "1. **Identify the starting point and possible moves**: The problem specifies that we start at the central 'A' and can move to adjacent letters (up, down, left, right, but not diagonal). \n\n2. **Count the number of 'M's adjacent to 'A'**: From the central 'A', there... |
deepscaler | [
{
"role": "user",
"content": "Given that \\( P \\) is any point on the arc \\( \\overparen{AD} \\) of the circumscribed circle of square \\( ABCD \\), find the value of \\( \\frac{PA + PC}{PB} \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 3230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gretchen has ten socks, two of each color: red, blue, green, yellow, and purple. She randomly draws five socks. What is the probability that she has exactly two pairs of socks with the same color? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{42}"
} | {
"split": "train",
"index": 3231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The cubic polynomial $p(x)$ satisfies $p(2) = 1,$ $p(7) = 19,$ $p(15) = 11,$ and $p(20) = 29.$ Find\n\\[p(1) + p(2) + p(3) + \\dots + p(21).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 3232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A 1-liter carton of milk used to cost 80 rubles. Recently, in an effort to cut costs, the manufacturer reduced the carton size to 0.9 liters and increased the price to 99 rubles. By what percentage did the manufacturer's revenue increase? Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "37.5"
} | {
"split": "train",
"index": 3233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular box measures $a \\times b \\times c$, where $a$, $b$, and $c$ are integers and $1\\leq a \\leq b \\leq c$. The volume and the surface area of the box are numerically equal. How many ordered triples $(a,b,c)$ are possible? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3234,
"solution": "1. **Equating Volume and Surface Area**: Given a rectangular box with dimensions $a \\times b \\times c$, the volume is $abc$ and the surface area is $2(ab + bc + ca)$. Setting these equal gives:\n \\[\n 2(ab + bc + ca) = abc.\n \\]\n Dividing both sides by ... |
deepscaler | [
{
"role": "user",
"content": "Given a family of sets \\(\\{A_{1}, A_{2}, \\ldots, A_{n}\\}\\) that satisfies the following conditions:\n(1) Each set \\(A_{i}\\) contains exactly 30 elements;\n(2) For any \\(1 \\leq i < j \\leq n\\), the intersection \\(A_{i} \\cap A_{j}\\) contains exactly 1 element;\n(3) T... | math | {
"style": "rule",
"ground_truth": "871"
} | {
"split": "train",
"index": 3235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many distinct right triangles exist with one leg equal to \\( \\sqrt{2016} \\), and the other leg and hypotenuse expressed as natural numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $2011 \\cdot 2012 \\cdot 2013 \\cdot 2014$ modulo 5. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 3237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the 4040 integers ranging from -2020 to 2019, three numbers are randomly chosen and multiplied together. Let the smallest possible product be $m$ and the largest possible product be $n$. What is the value of $\\frac{m}{n}$? Provide the answer in simplest fraction form. Let... | math | {
"style": "rule",
"ground_truth": "-\\frac{2020}{2017}"
} | {
"split": "train",
"index": 3238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3x^2+4"
} | {
"split": "train",
"index": 3239,
"solution": "1. **Use the Remainder Theorem**: Given that $3x^3 - 9x^2 + kx - 12$ is divisible by $x - 3$, we apply the Remainder Theorem which states that if a polynomial $f(x)$ is divisible by $x - a$, then $f(a) = 0$. Here, $a = 3$.\n\n2. **Substitute $x = 3$ into the polynom... |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos(\\alpha+\\pi)= \\frac {3}{5}$ and $\\pi \\leq \\alpha < 2\\pi$, calculate the value of $\\sin(-\\alpha-2\\pi)$. 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": 3240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate:<br/>$(1)-1^{2023}+8×(-\\frac{1}{2})^{3}+|-3|$;<br/>$(2)(-25)×\\frac{3}{2}-(-25)×\\frac{5}{8}+(-25)÷8($simplified calculation). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 3241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A positive integer \\( n \\) is said to be good if \\( 3n \\) is a re-ordering of the digits of \\( n \\) when they are expressed in decimal notation. Find a four-digit good integer which is divisible by 11. Let's think step by step and put only the final answer within \\boxed{... | math | {
"style": "rule",
"ground_truth": "2475"
} | {
"split": "train",
"index": 3242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $m$ is a prime number between 30 and 50. If you divide $m$ by 12, the remainder is 7. What is the value of $m$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 3243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Real numbers $x,$ $y,$ and $z$ satisfy the following equality:\n\\[4(x + y + z) = x^2 + y^2 + z^2.\\]Let $M$ be the maximum value of $xy + xz + yz,$ and let $m$ be the minimum value of $xy + xz + yz.$ Find $M + 10m.$ Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3244,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\\circ$ angle with $AB$. At the ... | math | {
"style": "rule",
"ground_truth": "160"
} | {
"split": "train",
"index": 3245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a football championship with 16 teams, each team played with every other team exactly once. A win was awarded 3 points, a draw 1 point, and a loss 0 points. A team is considered successful if it scored at least half of the maximum possible points. What is the maximum number ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 3246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a unit cube \\(ABCDA_1B_1C_1D_1\\), eight planes \\(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\\) and \\(D_1AB_1\\) intersect the cube. What is the volume of the part that contains the center of the cube? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "1/6"
} | {
"split": "train",
"index": 3247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 0, 1, 2, 3, and 4, select three different digits to form a three-digit number and calculate the total number of such numbers that are odd. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 3248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a, b, c$ are real numbers such that $a^2 + 2b = 7$, $b^2 + 4c = -7$, and $c^2 + 6a = -14$, find $a^2 + b^2 + c^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 3249,
"solution": "1. **Given Equations**: We start with the given equations:\n - \\(a^2 + 2b = 7\\)\n - \\(b^2 + 4c = -7\\)\n - \\(c^2 + 6a = -14\\)\n\n2. **Adding Equations**: Add all three equations together:\n \\[\n (a^2 + 2b) + (b^2 + 4c) + (c^2 + 6a) = 7 - 7 - 14\n \... |
deepscaler | [
{
"role": "user",
"content": "Expand $(x-2)(x+2)(x^2+4)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^4-16"
} | {
"split": "train",
"index": 3250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. Given that $a > b$, $a=5$, $c=6$, and $\\sin B= \\frac{3}{5}$.\n\n(Ⅰ) Find the values of $b$ and $\\sin A$;\n\n(Ⅱ) Find the value of $\\sin \\left(2A+ \\frac{\\pi}... | math | {
"style": "rule",
"ground_truth": "\\frac{7\\sqrt{2}}{26}"
} | {
"split": "train",
"index": 3251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $2022$ has the following property: it is a multiple of $6$ and the sum of its digits is $6$. Such positive integers are called \"auspicious numbers.\" Among all three-digit positive integers, the number of \"auspicious numbers\" is ____. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ , $b$ , $c$ be positive integers such that $abc + bc + c = 2014$ . Find the minimum possible value of $a + b + c$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 3253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 9 people sit around a round table? (Two seatings are considered the same if one is a rotation of the other.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40,\\!320"
} | {
"split": "train",
"index": 3254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a parallelogram with vertices $A(0, 0)$, $B(0, 5)$, $C(x, 5)$, and $D(x, 0)$ where $x > 0$. The parallelogram is titled such that $AD$ and $BC$ make an angle of $30^\\circ$ with the horizontal axis. If the area of the parallelogram is 35 square units, find the val... | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 3255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. and Mrs. Lopez have two children. When they get into their family car, two people sit in the front, and the other two sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 3256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive number $\\lambda$ such that for any triangle with side lengths $a, b, c$, if $a \\geqslant \\frac{b+c}{3}$, then the following inequality holds:\n$$\nac + bc - c^2 \\leqslant \\lambda \\left( a^2 + b^2 + 3c^2 + 2ab - 4bc \\right).\n$$ Let's think step... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{2} + 1}{7}"
} | {
"split": "train",
"index": 3257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two circles that share the same center have radii $10$ meters and $20$ meters. An aardvark runs along the path shown, starting at $A$ and ending at $K$. How many meters does the aardvark run? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20\\pi + 40"
} | {
"split": "train",
"index": 3258,
"solution": "To solve this problem, we need to calculate the total distance the aardvark runs along different segments of the circles and straight lines. We assume the path is symmetric and consists of arcs and radial segments.\n\n1. **Arc of the larger circle (radius = 20 meter... |
deepscaler | [
{
"role": "user",
"content": "Given points \\(A(4,5)\\), \\(B(4,0)\\) and \\(C(0,5)\\), compute the line integral of the second kind \\(\\int_{L}(4 x+8 y+5) d x+(9 x+8) d y\\) where \\(L:\\)\na) the line segment \\(OA\\);\nb) the broken line \\(OCA\\);\nc) the parabola \\(y=k x^{2}\\) passing through the po... | math | {
"style": "rule",
"ground_truth": "\\frac{796}{3}"
} | {
"split": "train",
"index": 3259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $ABCD$ has center $O,\\ AB=900,\\ E$ and $F$ are on $AB$ with $AE<BF$ and $E$ between $A$ and $F, m\\angle EOF =45^\\circ,$ and $EF=400.$ Given that $BF=p+q\\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p... | math | {
"style": "rule",
"ground_truth": "307"
} | {
"split": "train",
"index": 3260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six students stand in a row for a photo. Among them, student A and student B are next to each other, student C is not next to either student A or student B. The number of different ways the students can stand is ______ (express the result in numbers). Let's think step by step a... | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 3261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{m}=(\\sin A, \\frac {1}{2})$ and $\\overrightarrow{n}=(3,\\sin A+ \\sqrt {3}\\cos A)$ are collinear, where $A$ is an internal angle of $\\triangle ABC$. \n$(1)$ Find the size of angle $A$; \n$(2)$ If $BC=2$, find the maximum value of the area $... | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 3262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the product of all constants $t$ such that the quadratic $x^2 + tx + 6$ can be factored in the form $(x+a)(x+b)$, where $a$ and $b$ are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1225"
} | {
"split": "train",
"index": 3263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any finite set $S$, let $|S|$ denote the number of elements in $S$. Find the number of ordered pairs $(A,B)$ such that $A$ and $B$ are (not necessarily distinct) subsets of $\\{1,2,3,4,5\\}$ that satisfy \\[|A| \\cdot |B| = |A \\cap B| \\cdot |A \\cup B|\\] Let's think step... | math | {
"style": "rule",
"ground_truth": "454"
} | {
"split": "train",
"index": 3264,
"solution": "The answer is \\begin{align*} \\sum_{k=0}^{5}\\left[2\\binom{5}{k}2^{5-k}-\\binom{5}{k}\\right] &= 2\\sum_{k=0}^{5}\\binom{5}{k}2^{5-k}-\\sum_{k=0}^{5}\\binom{5}{k} \\\\ &=2(2+1)^5-(1+1)^5 \\\\ &=2(243)-32 \\\\ &=\\boxed{454}. \\end{align*} ~MRENTHUSIASM"
} |
deepscaler | [
{
"role": "user",
"content": "Given the quadratic function \\( y = ax^2 + bx + c \\) with its graph intersecting the \\( x \\)-axis at points \\( A \\) and \\( B \\), and its vertex at point \\( C \\):\n\n(1) If \\( \\triangle ABC \\) is a right-angled triangle, find the value of \\( b^2 - 4ac \\).\n\n(2) C... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 3265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)$ be the product of functions made by taking four functions from three functions $x,\\ \\sin x,\\ \\cos x$ repeatedly. Find the minimum value of $\\int_{0}^{\\frac{\\pi}{2}}f(x)\\ dx.$ Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi^5}{160}"
} | {
"split": "train",
"index": 3266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In an arithmetic sequence \\(\\{a_{n}\\}\\), if \\(\\frac{a_{11}}{a_{10}} < -1\\) and the sum of the first \\(n\\) terms \\(S_{n}\\) has a maximum value, then the value of \\(n\\) when \\(S_{n}\\) attains its smallest positive value is \\(\\qquad\\). Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 3267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $x^3 - ax^2 + bx - 2310$ has three positive integer roots. What is the smallest possible value of $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 3268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. On a semicircle with AB as the diameter, besides points A and B, there are 6 other points. Since there are also 4 other points on AB, making a total of 12 points, how many quadrilaterals can be formed with these 12 points as vertices?\n2. On one side of angle A, there are fi... | math | {
"style": "rule",
"ground_truth": "200"
} | {
"split": "train",
"index": 3269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\( ACE \\) with a point \\( B \\) on segment \\( AC \\) and a point \\( D \\) on segment \\( CE \\) such that \\( BD \\) is parallel to \\( AE \\). A point \\( Y \\) is chosen on segment \\( AE \\), and segment \\( CY \\) is drawn, intersecting \\( BD \\) at p... | math | {
"style": "rule",
"ground_truth": "39/25"
} | {
"split": "train",
"index": 3270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the circumference of a sector is $20\\,cm$ and its area is $9\\,cm^2$, find the radian measure of the central angle of the sector. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{9}"
} | {
"split": "train",
"index": 3271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The square of $5-\\sqrt{y^2-25}$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "y^2-10\\sqrt{y^2-25}"
} | {
"split": "train",
"index": 3272,
"solution": "We are given the expression $(5-\\sqrt{y^2-25})^2$ and need to simplify it. We will use the formula for the square of a binomial, which is $(a-b)^2 = a^2 - 2ab + b^2$.\n\n1. **Identify $a$ and $b$:**\n - Here, $a = 5$ and $b = \\sqrt{y^2-25}$.\n\n2. **Apply the bi... |
deepscaler | [
{
"role": "user",
"content": "Given three lines $l_1$: $4x+y-4=0$, $l_2$: $mx+y=0$, $l_3$: $x-my-4=0$ that do not intersect at the same point:\n\n(1) When these three lines cannot form a triangle, find the value of the real number $m$.\n\n(2) When $l_3$ is perpendicular to both $l_1$ and $l_2$, find the dis... | math | {
"style": "rule",
"ground_truth": "\\frac{4\\sqrt{17}}{17}"
} | {
"split": "train",
"index": 3273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $ABC$ with equal legs $AC$ and $BC$, a circle is constructed with $AC$ as its diameter, intersecting side $AB$ at point $M$. Find the distance from vertex $B$ to the center of this circle if $BM = \\sqrt{2}$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 3274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Keisha's basketball team must decide on a new uniform. The seventh-graders will pick the color of the shorts (black or gold) and the eighth-graders will pick the color of the jersey (black, white, or gold), but the two groups of players will not confer together. If, for both ... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 3275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The base of the pyramid is an isosceles right triangle, where each leg measures 8. Each of the pyramid's lateral edges is 9. Find the volume of the pyramid. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "224/3"
} | {
"split": "train",
"index": 3276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John's flight departed from Chicago at 3:15 PM and landed in Denver at 4:50 PM. Considering that Denver is one hour behind Chicago, if his flight took $h$ hours and $m$ minutes, with $0 < m < 60$, calculate the value of $h + m$. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 3277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, if $\\frac {\\tan A}{\\tan B}+ \\frac {\\tan A}{\\tan C}=3$, then the maximum value of $\\sin A$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac { \\sqrt {21}}{5}"
} | {
"split": "train",
"index": 3278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $8 \\tan \\theta = 3 \\cos \\theta$ and $0 < \\theta < \\pi,$ then determine the value of $\\sin \\theta.$ 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": 3279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$ , while $BD$ and $CE$ meet at $Q$ . Find the area of $APQD$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 3280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of an infinite geometric series is $64$ times the series that results if the first four terms of the original series are removed. What is the value of the series' common ratio? 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": 3281,
"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": "499"
} | {
"split": "train",
"index": 3282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is a parallelogram with an area of $50$ square units. Points $P$ and $Q$ are located on sides $AB$ and $CD$ respectively, such that $AP = \\frac{1}{3}AB$ and $CQ = \\frac{2}{3}CD$. What is the area of triangle $APD$? Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "16.67"
} | {
"split": "train",
"index": 3283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The amount of heat \\( Q \\) received by a certain substance when heated from 0 to \\( T \\) is determined by the formula \\( Q = 0.1054t + 0.000002t^2 \\) (\\( Q \\) is in joules, \\( t \\) is in kelvins). Find the heat capacity of this substance at \\( 100 \\) K. Let's think ... | math | {
"style": "rule",
"ground_truth": "0.1058"
} | {
"split": "train",
"index": 3284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A furniture store received a batch of office chairs that were identical except for their colors: 15 chairs were black and 18 were brown. The chairs were in demand and were being bought in a random order. At some point, a customer on the store's website discovered that only two ... | math | {
"style": "rule",
"ground_truth": "0.489"
} | {
"split": "train",
"index": 3285,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some constants $x$ and $a$, the third, fourth, and fifth terms in the expansion of $(x + a)^n$ are 84, 280, and 560, respectively. Find $n.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 3286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle of radius $3$ is cut into six congruent arcs. These arcs are then rearranged symmetrically to form a hexagonal star as illustrated below. Determine the ratio of the area of the hexagonal star to the area of the original circle.\nA) $\\frac{4.5}{\\pi}$\nB) $\\frac{4.5\\... | math | {
"style": "rule",
"ground_truth": "\\frac{4.5\\sqrt{3}}{\\pi}"
} | {
"split": "train",
"index": 3287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two semicircles of diameters 2 and 4 are placed such that the smaller is inside the larger one and their diameters are along the same line. A square of side 1 is inscribed in the larger semicircle such that one of its sides is along the diameter of the semicircles. Calculate th... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}\\pi - 1"
} | {
"split": "train",
"index": 3288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ satisfies $a_1 = 1$ and $5^{a_{n + 1} - a_n} - 1 = \\frac {1}{n + \\frac {2}{3}}$ for $n \\geq 1$. Find the least integer $k$ greater than $1$ for which $a_k$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 3289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number $x$, $\\lfloor x \\rfloor$ represents the largest integer not exceeding $x$, for example: $\\lfloor 2 \\rfloor = 2$, $\\lfloor 3.2 \\rfloor = 3$. Then $\\lfloor \\log_{2}1 \\rfloor + \\lfloor \\log_{2}2 \\rfloor + \\lfloor \\log_{2}3 \\rfloor + \\ldots + \\l... | math | {
"style": "rule",
"ground_truth": "264"
} | {
"split": "train",
"index": 3290,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer. A [i]Nordic[/i] square is an $n \\times n$ board containing all the integers from $1$ to $n^2$ so that each cell contains exactly one number. Two different cells are considered adjacent if they share a common side. Every cell that is adjacent only... | math | {
"style": "rule",
"ground_truth": "2n(n - 1) + 1"
} | {
"split": "train",
"index": 3291,
"solution": "\nTo solve the problem of finding the smallest possible total number of uphill paths in a Nordic square, we begin by understanding the structure and constraints involved:\n\nFirstly, consider an \\( n \\times n \\) board containing all integers from \\( 1 \\) to \\(... |
deepscaler | [
{
"role": "user",
"content": "Let $q(x)=q^{1}(x)=2x^{2}+2x-1$, and let $q^{n}(x)=q(q^{n-1}(x))$ for $n>1$. How many negative real roots does $q^{2016}(x)$ have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2017+1}{3}"
} | {
"split": "train",
"index": 3292,
"solution": "Define $g(x)=2x^{2}-1$, so that $q(x)=-\\frac{1}{2}+g(x+\\frac{1}{2})$. Thus $q^{N}(x)=0 \\Longleftrightarrow \\frac{1}{2}=g^{N}(x+\\frac{1}{2})$ where $N=2016$. But, viewed as function $g:[-1,1] \\rightarrow[-1,1]$ we have that $g(x)=\\cos(2 \\arccos(x))$. Thus, th... |
deepscaler | [
{
"role": "user",
"content": "A triple of positive integers $(a, b, c)$ is tasty if $\\operatorname{lcm}(a, b, c) \\mid a+b+c-1$ and $a<b<c$. Find the sum of $a+b+c$ across all tasty triples. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 3293,
"solution": "The condition implies $c \\mid b+a-1$. WLOG assume $c>b>a$; since $b+a-1<2 c$ we must have $b+a-1=c$. Substituting into $b \\mid a+c-1$ and $a \\mid c+b-1$ gives $$\\begin{aligned} & b \\mid 2 a-2 \\\\ & a \\mid 2 b-2 \\end{aligned}$$ Since $2 a-2<2 b$ we must eithe... |
deepscaler | [
{
"role": "user",
"content": "Given a triangle \\(A B C\\) with \\(A B = A C\\) and \\(\\angle A = 110^{\\circ}\\). Inside the triangle, a point \\(M\\) is chosen such that \\(\\angle M B C = 30^{\\circ}\\) and \\(\\angle M C B = 25^{\\circ}\\). Find \\(\\angle A M C\\). Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 3294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A biased coin with the probability of landing heads as 1/3 is flipped 12 times. What is the probability of getting exactly 9 heads in the 12 flips? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1760}{531441}"
} | {
"split": "train",
"index": 3295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store increased the price of a certain Super VCD by 40% and then advertised a \"10% discount and a free 50 yuan taxi fare\" promotion. As a result, each Super VCD still made a profit of 340 yuan. What was the cost price of each Super VCD? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "1500"
} | {
"split": "train",
"index": 3296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $2n$ students in a school $(n \\in \\mathbb{N}, n \\geq 2)$. Each week $n$ students go on a trip. After several trips the following condition was fulfilled: every two students were together on at least one trip. What is the minimum number of trips needed for this to h... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3297,
"solution": "We prove that for any $n \\geq 2$ the answer is 6. First we show that less than 6 trips is not sufficient. In that case the total quantity of students in all trips would not exceed $5n$. A student meets $n-1$ other students in each trip, so he or she takes part on a... |
deepscaler | [
{
"role": "user",
"content": "The values of $x$ and $y$ are always positive, and $x^2$ and $y$ vary inversely. If $y$ is 10 when $x$ is 2, then find $x$ when $y$ is 4000. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{10}"
} | {
"split": "train",
"index": 3298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 17 people at a party, and each has a reputation that is either $1,2,3,4$, or 5. Some of them split into pairs under the condition that within each pair, the two people's reputations differ by at most 1. Compute the largest value of $k$ such that no matter what the rep... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 3299,
"solution": "First, note that $k=8$ fails when there are $15,0,1,0,1$ people of reputation 1, 2, 3, 4, 5, respectively. This is because the two people with reputation 3 and 5 cannot pair with anyone, and there can only be at maximum $\\left\\lfloor\\frac{15}{2}\\right\\rfloor=7$... |
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