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 smallest constant $M$, such that for any positive real numbers $x$, $y$, $z$, and $w$,\n\\[\\sqrt{\\frac{x}{y + z + w}} + \\sqrt{\\frac{y}{x + z + w}} + \\sqrt{\\frac{z}{x + y + w}} + \\sqrt{\\frac{w}{x + y + z}} < M.\\] Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{\\sqrt{3}}"
} | {
"split": "train",
"index": 23749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a balance scale, three different masses were put at random on each pan and the result is shown in the picture. The masses are 101, 102, 103, 104, 105, and 106 grams. What is the probability that the 106 gram mass stands on the heavier pan?\nA) 75%\nB) 80%\nC) 90%\nD) 95%\nE)... | math | {
"style": "rule",
"ground_truth": "80\\%"
} | {
"split": "train",
"index": 23750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that P and Q are points on the graphs of the functions $2x-y+6=0$ and $y=2\\ln x+2$ respectively, find the minimum value of the line segment |PQ|. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{6\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 23751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ moves on the ellipse $\\dfrac{x^2}{9} + \\dfrac{y^2}{25} = 1$, and points $A$ and $B$ move respectively on the circles $x^2 + (y-4)^2 = 16$ and $x^2 + (y+4)^2 = 4$. The maximum value of $PA + PB$ is \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final a... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the quadrilateral \\(ABCD\\), it is known that \\(\\angle ABD = \\angle ACD = 45^\\circ\\), \\(\\angle BAC = 30^\\circ\\), and \\(BC = 1\\). Find \\(AD\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 23753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all positive integers less than 500 that are fourth powers of even perfect squares? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "272"
} | {
"split": "train",
"index": 23754,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two diagonals of a regular decagon (a 10-sided polygon) are chosen. What is the probability that their intersection lies inside the decagon and forms a convex quadrilateral? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{42}{119}"
} | {
"split": "train",
"index": 23755,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $α∈\\left( \\frac{π}{2},π\\right) $, and $\\sin \\left(π-α\\right)+\\cos \\left(2π+α\\right)= \\frac{ \\sqrt{2}}{3} $. Find the values of:\n\n$(1)\\sin {α} -\\cos {α} .$\n\n$(2)\\tan {α} $. Let's think step by step and put only the final answer within \\boxed{}."
}... | math | {
"style": "rule",
"ground_truth": "- \\frac{9+4 \\sqrt{2}}{7}"
} | {
"split": "train",
"index": 23756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In solving the system of equations $y = 7$ and $x^2+ y^2= 100,$ what is the sum of the solutions for $x?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23757,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$, $y$, and $z$ are chosen independently and at random from the interval $[0, m]$ for some positive integer $m$. The probability that no two of $x$, $y$, and $z$ are within 2 units of each other is greater than $\\frac{1}{2}$. Determine the smallest possibl... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A computer software retailer has 1200 copies of a new software package to sell. Given that half of them will sell right away at the original price, two-thirds of the remainder will sell later when the price is reduced by 40%, and the remaining copies will sell in a clearance sa... | math | {
"style": "rule",
"ground_truth": "80.90"
} | {
"split": "train",
"index": 23759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length $1$, let $P$ be a moving point on the space diagonal $B C_{1}$ and $Q$ be a moving point on the base $A B C D$. Find the minimum value of $D_{1} P + P Q$. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "1 + \\frac{\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 23760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ricardo has $2020$ coins, some of which are pennies ($1$-cent coins) and the rest of which are nickels ($5$-cent coins). He has at least one penny and at least one nickel. What is the difference in cents between the greatest possible and least amounts of money that Ricardo can ... | math | {
"style": "rule",
"ground_truth": "8072"
} | {
"split": "train",
"index": 23761,
"solution": "1. **Define Variables:**\n Let $p$ be the number of pennies Ricardo has. Since the rest of the coins are nickels, he has $2020 - p$ nickels.\n\n2. **Constraints:**\n Ricardo has at least one penny and at least one nickel. Therefore, we have the constraints:\n ... |
deepscaler | [
{
"role": "user",
"content": "For what ratio of the bases of a trapezoid does there exist a line on which the six points of intersection with the diagonals, the lateral sides, and the extensions of the bases of the trapezoid form five equal segments? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "1:2"
} | {
"split": "train",
"index": 23762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three distinct integers, $x$, $y$, and $z$, are randomly chosen from the set $\\{1, 2, 3, \\dots, 12\\}$. What is the probability that $xyz - x - y - z$ is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{11}"
} | {
"split": "train",
"index": 23763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n$$\n\\int_{0}^{\\sqrt{2} / 2} \\frac{x^{4} \\cdot d x}{\\sqrt{\\left(1-x^{2}\\right)^{3}}}\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{4} - \\frac{3\\pi}{8}"
} | {
"split": "train",
"index": 23764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a certain group, the probability that each member uses mobile payment is $p$, and the payment methods of each member are independent of each other. Let $X$ be the number of members in the group of $10$ who use mobile payment, $D\\left(X\\right)=2.4$, $P\\left(X=4\\right) \\l... | math | {
"style": "rule",
"ground_truth": "0.6"
} | {
"split": "train",
"index": 23765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the smallest positive real number $d,$ such that for all nonnegative real numbers $x, y,$ and $z,$\n\\[\n\\sqrt{xyz} + d |x^2 - y^2 + z^2| \\ge \\frac{x + y + z}{3}.\n\\] 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": 23766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the area of the portion of the circle defined by \\(x^2 - 10x + y^2 = 9\\) that lies above the \\(x\\)-axis and to the left of the line \\(y = x-5\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4.25\\pi"
} | {
"split": "train",
"index": 23767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of \\(987 + 113 - 1000\\) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 23768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $102^{4} - 4 \\cdot 102^{3} + 6 \\cdot 102^2 - 4 \\cdot 102 + 1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100406401"
} | {
"split": "train",
"index": 23769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a triangle, two of the side lengths are 7 and 8, and the angle between them is $120^\\circ.$ Find the length of the third side. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(n)=1 \\times 3 \\times 5 \\times \\cdots \\times (2n-1)$ . Compute the remainder when $f(1)+f(2)+f(3)+\\cdots +f(2016)$ is divided by $100.$ *Proposed by James Lin* Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 23771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Four prime numbers are randomly selected without replacement from the first twelve prime numbers. What is the probability that the sum of the four selected numbers is even? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 23772,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the given figure, hexagon $ABCDEF$ is equiangular, $ABJI$ and $FEHG$ are squares with areas $18$ and $32$ respectively, $\\triangle JBK$ is equilateral and $FE=BC$. What is the area of $\\triangle KBC$? Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23773,
"solution": "1. **Determine the length of $\\overline{FE}$ and $\\overline{BC}$:**\n Since $FEHG$ is a square with area $32$, we have:\n \\[\n \\overline{FE}^2 = 32 \\implies \\overline{FE} = \\sqrt{32} = 4\\sqrt{2}.\n \\]\n Given that $\\overline{FE} = \\overline{BC}... |
deepscaler | [
{
"role": "user",
"content": "The diagonal of a regular 2006-gon \\(P\\) is called good if its ends divide the boundary of \\(P\\) into two parts, each containing an odd number of sides. The sides of \\(P\\) are also called good. Let \\(P\\) be divided into triangles by 2003 diagonals, none of which have co... | math | {
"style": "rule",
"ground_truth": "1003"
} | {
"split": "train",
"index": 23774,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $17^{-1} \\equiv 31 \\pmod{53}$, find $36^{-1} \\pmod{53}$ as a residue modulo 53 (i.e., a value between 0 and 52 inclusive). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 23775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of the positive odd divisors of $60$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 23776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the positive root of\n\\[x^3 - 3x^2 - x - \\sqrt{2} = 0.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 + \\sqrt{2}"
} | {
"split": "train",
"index": 23777,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the function $f(x)$ is an odd function defined on $\\mathbb{R}$ and $f(x+ \\frac{5}{2})=-\\frac{1}{f(x)}$, and when $x \\in [-\\frac{5}{2}, 0]$, $f(x)=x(x+ \\frac{5}{2})$, find $f(2016)=$ \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 23778,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fraction $\\frac {5x - 11}{2x^2 + x - 6}$ was obtained by adding the two fractions $\\frac {A}{x + 2}$ and $\\frac {B}{2x - 3}$. The values of $A$ and $B$ must be, respectively: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3, -1"
} | {
"split": "train",
"index": 23779,
"solution": "To find the values of $A$ and $B$ in the expression $\\frac{5x - 11}{2x^2 + x - 6}$, we first factorize the denominator:\n\n\\[ 2x^2 + x - 6 = (2x - 3)(x + 2) \\]\n\nGiven that the fraction can be expressed as the sum of two simpler fractions, we write:\n\n\\[ \\fr... |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\\triangle ABC$ to $\... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 23780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define $f\\left(n\\right)=\\textrm{LCM}\\left(1,2,\\ldots,n\\right)$ . Determine the smallest positive integer $a$ such that $f\\left(a\\right)=f\\left(a+2\\right)$ .\n\n*2017 CCA Math Bonanza Lightning Round #2.4* Let's think step by step and put only the final answer with... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 23781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\\frac{1}{3}$ and on region $B$ is $\\frac{1}{2}$. What is the probability of the arrow stopping on region $C$? Express your answer as a commo... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 23782,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many real \\( x \\) satisfy the equation \\( x = \\left\\lfloor \\frac{x}{2} \\right\\rfloor + \\left\\lfloor \\frac{x}{3} \\right\\rfloor + \\left\\lfloor \\frac{x}{5} \\right\\rfloor \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 23783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two opposite sides of a rectangle are each divided into $n$ congruent segments, and the endpoints of one segment are joined to the center to form triangle $A$. The other sides are each divided into $m$ congruent segments, and the endpoints of one of these segments are joined to... | math | {
"style": "rule",
"ground_truth": "\\frac{m}{n}"
} | {
"split": "train",
"index": 23784,
"solution": "1. **Positioning the Rectangle**: Place the rectangle on a coordinate grid with diagonal vertices at $(0,0)$ and $(x,y)$. This sets the rectangle's length along the x-axis as $x$ and its height along the y-axis as $y$.\n\n2. **Dividing the Sides**: \n - Each hori... |
deepscaler | [
{
"role": "user",
"content": "Each of the letters $W$, $X$, $Y$, and $Z$ represents a different integer in the set $\\{ 1,2,3,4\\}$, but not necessarily in that order. If $\\frac{W}{X} - \\frac{Y}{Z}=1$, then the sum of $W$ and $Y$ is Let's think step by step and put only the final answer within \\boxed{}.... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 23785,
"solution": "1. **Identify the possible values for fractions:** Given that $\\text{W}, \\text{X}, \\text{Y}, \\text{Z}$ are different integers from the set $\\{1, 2, 3, 4\\}$, we need to find pairs $(\\text{W}, \\text{X})$ and $(\\text{Y}, \\text{Z})$ such that $\\frac{\\text{W... |
deepscaler | [
{
"role": "user",
"content": "How many tetrahedrons can be formed using the vertices of a regular triangular prism? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 23786,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the parabola $C$: $y^{2}=2px(p > 0)$ with focus $F$ and passing through point $A(1,-2)$. \n$(1)$ Find the equation of the parabola $C$; \n$(2)$ Draw a line $l$ through $F$ at an angle of $45^{\\circ}$, intersecting the parabola $C$ at points $M$ and $N$, with $O$ being ... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 23787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In Rivertown, car plates each contain three symbols: two letters followed by a digit. The first letter is chosen from the set ${A, B, G, H, T}$, the second letter from ${E, I, O, U}$, and the digit from $0$ to $9$. \n\nTo accommodate an increase in the number of cars, Rivertown... | math | {
"style": "rule",
"ground_truth": "130"
} | {
"split": "train",
"index": 23788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among the non-empty subsets of the set \\( A = \\{1, 2, \\cdots, 10\\} \\), how many subsets have the sum of their elements being a multiple of 10? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 23789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Robot Petya displays three three-digit numbers every minute, which sum up to 2019. Robot Vasya swaps the first and last digits of each of these numbers and then sums the resulting numbers. What is the maximum sum that Vasya can obtain? Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "2118"
} | {
"split": "train",
"index": 23790,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The vertices of an equilateral triangle lie on the hyperbola \\( xy = 3 \\). The centroid of this triangle is at the origin, \\( (0,0) \\). What is the square of the area of the triangle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 23791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For the Shanghai World Expo, 20 volunteers were recruited, with each volunteer assigned a unique number from 1 to 20. If four individuals are to be selected randomly from this group and divided into two teams according to their numbers, with the smaller numbers in one team and ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\frac{1}{2^{1990}} \\sum_{n = 0}^{995} (-3)^n \\binom{1990}{2n}.\\] 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": 23793,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x, y)$ satisfying $1 \\le x \\le 5$ and $1 \\le y \\le 5$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2160"
} | {
"split": "train",
"index": 23794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A natural number \\( 1 \\leq n \\leq 221 \\) is called lucky if, when dividing 221 by \\( n \\), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "115"
} | {
"split": "train",
"index": 23795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integral value of $k$ such that\n$2x(kx-4)-x^2+6=0$ has no real roots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23796,
"solution": "1. **Expand and simplify the given quadratic equation**:\n \\[\n 2x(kx-4) - x^2 + 6 = 0\n \\]\n Expanding the terms:\n \\[\n 2kx^2 - 8x - x^2 + 6 = 0\n \\]\n Combine like terms:\n \\[\n (2k-1)x^2 - 8x + 6 = 0\n \\]\n\n2. **Condition for no rea... |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbf{u},$ $\\mathbf{v},$ and $\\mathbf{w}$ be vectors such that $\\|\\mathbf{u}\\| = 3,$ $\\|\\mathbf{v}\\| = 4,$ and $\\|\\mathbf{w}\\| = 5,$ and\n\\[\\mathbf{u} + \\mathbf{v} + \\mathbf{w} = \\mathbf{0}.\\]Compute $\\mathbf{u} \\cdot \\mathbf{v} + \\mathbf{u} \\cdot ... | math | {
"style": "rule",
"ground_truth": "-25"
} | {
"split": "train",
"index": 23797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $537_{8} + 246_{8}$. Express your answer in base $8$, and then convert your answer to base $16$, using alphabetic representation for numbers $10$ to $15$ (e.g., $A$ for $10$, $B$ for $11$, etc.). Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "205_{16}"
} | {
"split": "train",
"index": 23798,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of all numbers of the form $(2^{a^2}\\cdot 19^{b^2} \\cdot 53^{c^2} + 8)^{16} - 1$ where $a,b,c$ are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 23799,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $m$ and $n$ are positive integers such that $75m = n^{3}$. What is the minimum possible value of $m + n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 23800,
"solution": "1. **Prime Factorization of 75**: \n We start by factorizing 75 into its prime factors:\n \\[\n 75 = 5^2 \\cdot 3^1\n \\]\n\n2. **Form of \\(n^3\\)**: \n We know that \\(n^3\\) must be a cube of an integer, and \\(75m = n^3\\) implies that the product \... |
deepscaler | [
{
"role": "user",
"content": "Compute $\\left(\\sqrt{625681}\\right)^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "625681"
} | {
"split": "train",
"index": 23801,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence $\\{a_n\\}$ with $16$ terms, and $a_1=1, a_8=4$. Let the function related to $x$ be $f_n(x)=\\frac{x^3}{3}-a_nx^2+(a_n^2-1)x$, where $n\\in \\mathbf{N}^*$. If $x=a_{n+1}$ ($1\\leqslant n\\leqslant 15$) is the extremum point of the function $f_n(x)$, and the slo... | math | {
"style": "rule",
"ground_truth": "1176"
} | {
"split": "train",
"index": 23802,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x \\gt 0 \\gt y$, and $\\frac{1}{x+2}+\\frac{1}{1-y}=\\frac{1}{6}$, find the minimum value of $x-y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 23803,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A bag contains 7 red chips and 4 green chips. One chip is drawn from the bag, then placed back into the bag, and a second chip is drawn. What is the probability that the two selected chips are of different colors? Let's think step by step and put only the final answer within \\... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{121}"
} | {
"split": "train",
"index": 23804,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( (a_1, a_2, \\dots, a_{12}) \\) be a list of the first 12 positive integers such that for each \\( 2 \\le i \\le 12 \\), either \\( a_i+1 \\) or \\( a_i-1 \\) or both appear somewhere before \\( a_i \\) in the list. Determine the number of such lists. Let's think step by... | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"split": "train",
"index": 23805,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = |\\ln x|$, the solution set of the inequality $f(x) - f(x_0) \\geq c(x - x_0)$ is $(0, +\\infty)$, where $x_0 \\in (0, +\\infty)$, and $c$ is a constant. When $x_0 = 1$, the range of values for $c$ is \\_\\_\\_\\_\\_\\_; when $x_0 = \\frac{1}{2}$, the... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 23806,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the triangular pyramid P-ABC, PA is perpendicular to the base ABC, AB=2, AC=AP, BC is perpendicular to CA. If the surface area of the circumscribed sphere of the triangular pyramid P-ABC is $5\\pi$, find the value of BC. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 23807,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the curves $y=x^2-1$ and $y=1+x^3$ have perpendicular tangents at $x=x_0$, find the value of $x_0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{1}{\\sqrt[3]{6}}"
} | {
"split": "train",
"index": 23808,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "[asy]\ndraw((0,1)--(4,1)--(4,2)--(0,2)--cycle);\ndraw((2,0)--(3,0)--(3,3)--(2,3)--cycle);\ndraw((1,1)--(1,2));\nlabel(\"1\",(0.5,1.5));\nlabel(\"2\",(1.5,1.5));\nlabel(\"32\",(2.5,1.5));\nlabel(\"16\",(3.5,1.5));\nlabel(\"8\",(2.5,0.5));\nlabel(\"6\",(2.5,2.5));\n[/asy]\nThe im... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23809,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can we put 3 math books and 5 English books on a shelf if all the math books must stay together and all the English books must also stay together? (The math books are all different and so are the English books.) Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"split": "train",
"index": 23810,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square is contained in a cube when all of its points are in the faces or in the interior of the cube. Determine the biggest $\\ell > 0$ such that there exists a square of side $\\ell$ contained in a cube with edge $1$ . Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"split": "train",
"index": 23811,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the functions \\( f(\\alpha, x) \\) and \\( g(\\alpha) \\) be defined as\n\\[ f(\\alpha, x)=\\frac{\\left(\\frac{x}{2}\\right)^{\\alpha}}{x-1} \\]\n\\[ g(\\alpha)=\\left.\\frac{d^{4} f}{d x^{4}}\\right|_{x=2} \\]\n\nThen \\( g(\\alpha) \\) is a polynomial in \\( \\alpha \\)... | math | {
"style": "rule",
"ground_truth": "1/16"
} | {
"split": "train",
"index": 23812,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\begin{pmatrix} 2 & 0 \\\\ 5 & -3 \\end{pmatrix} \\begin{pmatrix} 8 & -2 \\\\ 1 & 1 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 16 & -4 \\\\ 37 & -13 \\end{pmatrix}"
} | {
"split": "train",
"index": 23813,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Erika, who is 14 years old, flips a fair coin whose sides are labeled 10 and 20, and then she adds the number on the top of the flipped coin to the number she rolls on a standard die. What is the probability that the sum equals her age in years? Express your answer as a common ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 23814,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $\\cos \\alpha = \\dfrac{1}{7}$, $\\cos (\\alpha - \\beta) = \\dfrac{13}{14}$, and $0 < \\beta < \\alpha < \\dfrac{\\pi}{2}$.\n\n1. Find the value of $\\tan 2\\alpha$.\n2. Find the value of $\\cos \\beta$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{2}"
} | {
"split": "train",
"index": 23815,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Expand $-(3-c)(c+2(3-c))$. What is the sum of the coefficients of the expanded form? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-10"
} | {
"split": "train",
"index": 23816,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Seven couples are at a social gathering. If each person shakes hands exactly once with everyone else except their spouse and one other person they choose not to shake hands with, how many handshakes were exchanged? Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"split": "train",
"index": 23817,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( a_{n} = 50 + n^{2} \\) for \\( n = 1, 2, \\cdots \\), find the maximum value of the greatest common divisor \\( d_{n} = \\gcd(a_{n}, a_{n+1}) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 23818,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $P$ is inside equilateral $\\triangle ABC$. Points $Q$, $R$, and $S$ are the feet of the perpendiculars from $P$ to $\\overline{AB}$, $\\overline{BC}$, and $\\overline{CA}$, respectively. Given that $PQ=1$, $PR=2$, and $PS=3$, what is $AB$ in terms of radicals? Let's thi... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{3}"
} | {
"split": "train",
"index": 23819,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "According to the chart shown, what was the average daily high temperature in Addington from September 15th, 2008 through September 19th, 2008, inclusive? Express your answer as a decimal to the nearest tenth. [asy]\nsize(300);\ndefaultpen(linewidth(.7pt)+fontsize(6pt));\n\nint[... | math | {
"style": "rule",
"ground_truth": "54.4"
} | {
"split": "train",
"index": 23820,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x) = 4x^4+12x^3-9x^2+x+3$ and $d(x) = x^2+3x-2$. If $f(x) = q(x)d(x) + r(x)$ for some polynomials $q(x)$ and $r(x)$ where $\\deg r < \\deg d$, calculate $q(1)+r(-1)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 23821,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange 3 male students and 4 female students in a row. Under the following different requirements, calculate the number of different arrangement methods: \n(1) Person A and Person B must stand at the two ends; \n(2) All male students must be grouped together; \n(3) Male stu... | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 23822,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a small pond there are eleven lily pads in a row labeled 0 through 10. A frog is sitting on pad 1. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\\frac{N}{10}$ and to pad $N+1$ with probability $1-\\frac{N}{10}$. Each jump is independent ... | math | {
"style": "rule",
"ground_truth": "\\frac{63}{146}"
} | {
"split": "train",
"index": 23823,
"solution": "Define \\( P(N) \\) as the probability that the frog survives starting from pad \\( N \\). We need to find \\( P(1) \\).\n\nThe recursive relationship given in the problem is:\n\\[ P(N) = \\frac{N}{10} P(N-1) + \\left(1 - \\frac{N}{10}\\right) P(N+1) \\]\nfor \\( 0... |
deepscaler | [
{
"role": "user",
"content": "Given the set $A=\\{x|x=a_0+a_1\\times3+a_2\\times3^2+a_3\\times3^3\\}$, where $a_k\\in\\{0,1,2\\}$ ($k=0,1,2,3$), and $a_3\\neq0$, calculate the sum of all elements in set $A$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2889"
} | {
"split": "train",
"index": 23824,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the monotonicity of the function $f(x) = \\frac{x}{x^2 + 1}$ on the interval $(1, +\\infty)$, and find the maximum and minimum values of the function when $x \\in [2, 3]$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{10}"
} | {
"split": "train",
"index": 23825,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many values of $\\theta$ in the interval $0 < \\theta \\leq 4\\pi$ satisfy the equation $2 + 4\\sin2\\theta - 3\\cos4\\theta + 2\\tan\\theta = 0$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 23826,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers $1,2,3, \\cdots, 2014$, select 315 different numbers (order does not matter) to form an arithmetic sequence. Among these, the number of ways to form an arithmetic sequence that includes the number 1 is ___. The total number of ways to form an arithmetic sequenc... | math | {
"style": "rule",
"ground_truth": "5490"
} | {
"split": "train",
"index": 23827,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[f(x) = \\frac{ax}{x + 1}.\\]Find the constant $a$ so that $f(f(x)) = x$ for all $x \\neq -1.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 23828,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A company offers its employees a salary increase, provided they increase their work productivity by 2% per week. If the company operates 5 days a week, by what percentage per day must employees increase their productivity to achieve the desired goal? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "0.4"
} | {
"split": "train",
"index": 23829,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-congruent triangles with perimeter 7 have integer side lengths? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 23830,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the remainder when $2^{2001}$ is divided by $2^{7}-1$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 23831,
"solution": "$2^{2001(\\bmod 7)}=2^{6}=64$."
} |
deepscaler | [
{
"role": "user",
"content": "A cube is constructed from $4$ white unit cubes and $4$ blue unit cubes. How many different ways are there to construct the $2 \\times 2 \\times 2$ cube using these smaller cubes? (Two constructions are considered the same if one can be rotated to match the other.) Let's think ... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"split": "train",
"index": 23832,
"solution": "To solve this problem, we use Burnside's Lemma, which is a powerful tool in combinatorics for counting distinct configurations under group actions, such as rotations in this case. The lemma states that the number of distinct configurations, up to symmetry, is the a... |
deepscaler | [
{
"role": "user",
"content": "Eight numbers \\( a_{1}, a_{2}, a_{3}, a_{4} \\) and \\( b_{1}, b_{2}, b_{3}, b_{4} \\) satisfy the following equations:\n\n$$\n\\left\\{\\begin{array}{c}\na_{1} b_{1}+a_{2} b_{3}=1 \\\\\na_{1} b_{2}+a_{2} b_{4}=0 \\\\\na_{3} b_{1}+a_{4} b_{3}=0 \\\\\na_{3} b_{2}+a_{4} b_{4}=1\... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 23833,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every positive real number $x$, let\n\\[g(x) = \\lim_{r \\to 0} ((x+1)^{r+1} - x^{r+1})^{\\frac{1}{r}}.\\]\nFind $\\lim_{x \\to \\infty} \\frac{g(x)}{x}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e"
} | {
"split": "train",
"index": 23834,
"solution": "The limit is $e$.\n\n\\textbf{First solution.}\nBy l'H\\^opital's Rule, we have\n\\begin{align*}\n&\\lim_{r\\to 0} \\frac{\\log((x+1)^{r+1}-x^{r+1})}{r} \\\\\n&\\quad = \\lim_{r\\to 0} \\frac{d}{dr} \\log((x+1)^{r+1}-x^{r+1}) \\\\\n&\\quad = \\lim_{r\\to 0} \\frac{... |
deepscaler | [
{
"role": "user",
"content": "How many positive three-digit integers less than 700 have at least two digits that are the same and none of the digits can be zero? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 23835,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $x$: $3^{2x} = \\sqrt{27}$. Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 23836,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Matt has somewhere between 1000 and 2000 pieces of paper he's trying to divide into piles of the same size (but not all in one pile or piles of one sheet each). He tries $2,3,4,5,6,7$, and 8 piles but ends up with one sheet left over each time. How many piles does he need? Let'... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 23837,
"solution": "The number of sheets will leave a remainder of 1 when divided by the least common multiple of $2,3,4,5,6,7$, and 8, which is $8 \\cdot 3 \\cdot 5 \\cdot 7=840$. Since the number of sheets is between 1000 and 2000, the only possibility is 1681. The number of piles m... |
deepscaler | [
{
"role": "user",
"content": "A particle moves so that it is at $(2t + 7, 4t - 13)$ at time $t.$ Find the speed of the particle, measured in unit of distance per unit of time. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\sqrt{5}"
} | {
"split": "train",
"index": 23838,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the distribution of the random variable $\\xi$ is $P(\\xi=x)= \\dfrac{xk}{15}$, where $x$ takes values $(1,2,3,4,5)$, find the value of $P\\left( \\left. \\dfrac{1}{2} < \\xi < \\dfrac{5}{2} \\right. \\right)$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{5}"
} | {
"split": "train",
"index": 23839,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( s = 1 + \\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\cdots + \\frac{1}{\\sqrt{10^6}} \\), what is the integer part of \\( s \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1998"
} | {
"split": "train",
"index": 23840,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many non-empty subsets of $\\{1,2,3,4,5,6,7,8\\}$ have exactly $k$ elements and do not contain the element $k$ for some $k=1,2, \\ldots, 8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 23841,
"solution": "Probably the easiest way to do this problem is to count how many non-empty subsets of $\\{1,2, \\ldots, n\\}$ have $k$ elements and do contain the element $k$ for some $k$. The element $k$ must have $k-1$ other elements with it to be in a subset of $k$ elements, so... |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\sqrt[3]{8+27} \\cdot \\sqrt[3]{8+\\sqrt{64}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt[3]{560}"
} | {
"split": "train",
"index": 23842,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fifth grade has 120 teachers and students going to visit the Natural History Museum. A transportation company offers two types of vehicles to choose from:\n(1) A bus with a capacity of 40 people, with a ticket price of 5 yuan per person. If the bus is full, the ticket price... | math | {
"style": "rule",
"ground_truth": "480"
} | {
"split": "train",
"index": 23843,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many perfect cubes are between 100 and 900? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 23844,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the imaginary number \\( z \\) satisfies \\( z^3 + 1 = 0 \\), \\( z \\neq -1 \\). Then \\( \\left( \\frac{z}{z-1} \\right)^{2018} + \\left( \\frac{1}{z-1} \\right)^{2018} = \\) . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 23845,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $10^{\\log_{10}9} = 8x + 5$ then $x$ equals: 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": 23846,
"solution": "1. **Understanding the Expression**: The expression given is $10^{\\log_{10}9}$. By the property of logarithms and exponents, $a^{\\log_a b} = b$, we can simplify this expression:\n \\[\n 10^{\\log_{10}9} = 9\n \\]\n\n2. **Setting Up the Equation**: We substi... |
deepscaler | [
{
"role": "user",
"content": "Is\n\\[f(x) = \\log (x + \\sqrt{1 + x^2})\\]an even function, odd function, or neither?\n\nEnter \"odd\", \"even\", or \"neither\". Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{odd}"
} | {
"split": "train",
"index": 23847,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "It is known that when 2008 is divided by certain natural numbers, the remainder is always 10. How many such natural numbers are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 23848,
"solution": ""
} |
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