data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Given 50 feet of fencing, where 5 feet is used for a gate that does not contribute to the enclosure area, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the remaining fencing? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "126.5625"
} | {
"split": "train",
"index": 20849,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $x=1$ is a root of the quadratic equation $\\left(m-2\\right)x^{2}+4x-m^{2}=0$, calculate the value(s) of $m$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20850,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $P$ is a point on the ellipse $\\frac{x^2}{25}+\\frac{y^2}{9}=1$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse, respectively. If $\\frac{{\\overrightarrow{PF_1} \\cdot \\overrightarrow{PF_2}}}{{|\\overrightarrow{PF_1}| \\cdot |\\overrightarrow{PF_2}... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 20851,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\([x]\\) represent the integral part of the decimal number \\(x\\). Given that \\([3.126] + \\left[3.126 + \\frac{1}{8}\\right] + \\left[3.126 + \\frac{2}{8}\\right] + \\ldots + \\left[3.126 + \\frac{7}{8}\\right] = P\\), find the value of \\(P\\). Let's think step by step... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 20852,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n\n\\[\n\\int_{0}^{\\frac{\\pi}{4}} \\frac{2 \\cos x + 3 \\sin x}{(2 \\sin x - 3 \\cos x)^{3}} \\, dx\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{17}{18}"
} | {
"split": "train",
"index": 20853,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sultan gathered 300 court sages and proposed a trial. There are 25 different colors of hats, known in advance to the sages. The sultan informed them that each sage would be given one of these hats, and if they wrote down the number of hats for each color, all these numbers ... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 20854,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three vertices of parallelogram $PQRS$ are $P(-3,-2)$, $Q(1,-5)$, $R(9,1)$ with $P$ and $R$ diagonally opposite. The sum of the coordinates of vertex $S$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 20855,
"solution": "To find the coordinates of vertex $S$ of parallelogram $PQRS$, where $P$ and $R$ are diagonally opposite, we can use the property that the diagonals of a parallelogram bisect each other. This means that the midpoint of diagonal $PR$ is also the midpoint of diagonal... |
deepscaler | [
{
"role": "user",
"content": "The distance from the point of intersection of a circle's diameter with a chord of length 18 cm to the center of the circle is 7 cm. This point divides the chord in the ratio 2:1. Find the radius.\n\n$$\nAB = 18, EO = 7, AE = 2BE, R = ?\n$$ Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 20856,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six standard six-sided dice are rolled. We are told there is a pair and a three-of-a-kind, but no four-of-a-kind initially. The pair and the three-of-a-kind are set aside, and the remaining die is re-rolled. What is the probability that after re-rolling this die, at least four ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 20857,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three identical rods each have a piece broken off at a random point. What is the probability that the three resulting pieces can form a triangle?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 20858,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the lines $x-y-1=0$ and $x-y-5=0$ both intersect circle $C$ creating chords of length 10, find the area of circle $C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27\\pi"
} | {
"split": "train",
"index": 20859,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Beginning with a $3 \\mathrm{~cm}$ by $3 \\mathrm{~cm}$ by $3 \\mathrm{~cm}$ cube, a $1 \\mathrm{~cm}$ by $1 \\mathrm{~cm}$ by $1 \\mathrm{~cm}$ cube is cut from one corner and a $2 \\mathrm{~cm}$ by $2 \\mathrm{~cm}$ by $2 \\mathrm{~cm}$ cube is cut from the opposite corner. I... | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 20860,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Nine integers from 1 to 5 are written on a board. It is known that seven of them are at least 2, six are greater than 2, three are at least 4, and one is at least 5. Find the sum of all the numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 20861,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A store has equal amounts of candies priced at 2 rubles per kilogram and candies priced at 3 rubles per kilogram. At what price should the mixture of these candies be sold? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2.4"
} | {
"split": "train",
"index": 20862,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The concept of negative numbers first appeared in the ancient Chinese mathematical book \"Nine Chapters on the Mathematical Art.\" If income of $5$ yuan is denoted as $+5$ yuan, then expenses of $5$ yuan are denoted as $-5$ yuan. Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 20863,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Perpendiculars \\( B E \\) and \\( D F \\), dropped from the vertices \\( B \\) and \\( D \\) of parallelogram \\( A B C D \\) onto sides \\( A D \\) and \\( B C \\) respectively, divide the parallelogram into three parts of equal area. On the extension of diagonal \\( B D \\) ... | math | {
"style": "rule",
"ground_truth": "1:1"
} | {
"split": "train",
"index": 20864,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$ ) with four colors. A coloring is called multichromatic if whenever $a, b$ and $\\operatorname{gcd}(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"split": "train",
"index": 20865,
"solution": "First, we show that $N$ cannot have three distinct prime divisors. For the sake of contradiction, suppose $p q r \\mid N$ for three distinct primes $p, q, r$. Then by the problem statement, $(p, q, 1),(p, r, 1)$, and $(q, r, 1)$ have three distinct colors, so $(p, ... |
deepscaler | [
{
"role": "user",
"content": "Determine the maximum integer $ n $ such that for each positive integer $ k \\le \\frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 20866,
"solution": "\nWe need to determine the maximum integer \\( n \\) such that for each positive integer \\( k \\leq \\frac{n}{2} \\), there are two positive divisors of \\( n \\) with difference \\( k \\).\n\nTo solve this, we begin by considering the divisors of \\( n \\). Let t... |
deepscaler | [
{
"role": "user",
"content": "Given positive numbers \\(a,b\\) satisfying \\(2ab=\\dfrac{2a-b}{2a+3b},\\) then the maximum value of \\(b\\) is \\_\\_\\_\\_\\_ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 20867,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $C$ be a point not on line $AE$. Points $D$ and $F$ lie on line $AE$ such that $CD \\perp AE$ and $CF \\perp AE$. Meanwhile, $B$ is a point on line $CE$ such that $AB \\perp CE$. Given $AB = 3$, $CD = 10$, and $AE = 6$, find the lengths of $CE$ and $CF$. Let's think step by... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 20868,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Every bag of Dummies (a chocolate candy) contains the same number of pieces.\n\nThe Dummies in one bag can't be divided equally among $9$ kids, because after each kid gets the same (whole) number of pieces, $7$ pieces are left over.\n\nIf the Dummies in three bags are divided e... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 20869,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the parametric equations of curve $C_{1}$ are $\\left\\{{\\begin{array}{l}{x=-\\sqrt{3}t}\\\\{y=t}\\end{array}}\\right.$ ($t$ is the parameter), and the parametric equations of curve $C_{2}$ are $\\left\\{{\\begin{array}{l}{x=4\\cos\\theta}\\... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 20870,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nTwo candidates participated in an election with \\(p+q\\) voters. Candidate \\(A\\) received \\(p\\) votes and candidate \\(B\\) received \\(q\\) votes, with \\(p > q\\). During the vote counting, only one vote is registered at a time on a board. Let \\(r\\) be the probabilit... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2019}"
} | {
"split": "train",
"index": 20871,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $3250$ and $3500$ have four distinct digits arranged in increasing order? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20872,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Angle $ABC$ of $\\triangle ABC$ is a right angle. The sides of $\\triangle ABC$ are the diameters of semicircles as shown. The area of the semicircle on $\\overline{AB}$ equals $8\\pi$, and the arc of the semicircle on $\\overline{AC}$ has length $8.5\\pi$. What is the radius o... | math | {
"style": "rule",
"ground_truth": "7.5"
} | {
"split": "train",
"index": 20873,
"solution": "1. **Identify the given information and the relationship between the elements:**\n - $\\triangle ABC$ is a right triangle with $\\angle ABC = 90^\\circ$.\n - The sides of $\\triangle ABC$ are the diameters of semicircles.\n - The area of the semicircle on $\\... |
deepscaler | [
{
"role": "user",
"content": "The area of a circle is \\( 64\\pi \\) square units. Calculate both the radius and the circumference of the circle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16\\pi"
} | {
"split": "train",
"index": 20874,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Grammar club has 20 members: 10 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{295}{323}"
} | {
"split": "train",
"index": 20875,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A metal bar with a temperature of $20{ }^{\\circ} \\mathrm{C}$ is placed into water that is initially at $80{ }^{\\circ} \\mathrm{C}$. After thermal equilibrium is reached, the temperature is $60{ }^{\\circ} \\mathrm{C}$. Without removing the first bar from the water, another m... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 20876,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence $\\{a_n\\}$ is an arithmetic sequence, and the sequence $\\{b_n\\}$ satisfies $b_n = a_na_{n+1}a_{n+2} (n \\in \\mathbb{N}^*)$. Let $S_n$ be the sum of the first $n$ terms of $\\{b_n\\}$. If $a_{12} = \\frac{3}{8}a_5 > 0$, then when $S_n$ reaches its maximum value,... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 20877,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\tan x+\\tan y=4$ and $\\cot x+\\cot y=5$, compute $\\tan (x+y)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 20878,
"solution": "We have $\\cot x+\\cot y=\\frac{\\tan x+\\tan y}{\\tan x \\tan y}$, so $\\tan x \\tan y=\\frac{4}{5}$. Thus, by the $\\tan$ sum formula, $\\tan (x+y)=\\frac{\\tan x+\\tan y}{1-\\tan x \\tan y}=20$."
} |
deepscaler | [
{
"role": "user",
"content": ".4 + .02 + .006 = Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".426"
} | {
"split": "train",
"index": 20879,
"solution": "To solve the problem, we add the decimal numbers together. It's helpful to align the numbers by their decimal places to ensure accurate addition:\n\n1. Write each number with the same number of decimal places:\n \\[\n 0.4 = 0.400 \\\\\n 0.02 = 0.020 \\\\\n ... |
deepscaler | [
{
"role": "user",
"content": "Find all real numbers $a$ such that the roots of the polynomial\n$$x^3 - 6x^2 + 21x + a$$form an arithmetic progression and are not all real. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-26"
} | {
"split": "train",
"index": 20880,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the greatest common divisor of $8!$ and $(6!)^2.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5760"
} | {
"split": "train",
"index": 20881,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2$, $|\\overrightarrow{b}|=\\sqrt{2}$, and $\\overrightarrow{a}\\perp(\\overrightarrow{a}+2\\overrightarrow{b})$, calculate the projection of $\\overrightarrow{b}$ in the direction of $... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 20882,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure, it is given that angle $C = 90^{\\circ}$, $\\overline{AD} = \\overline{DB}$, $DE \\perp AB$, $\\overline{AB} = 20$, and $\\overline{AC} = 12$. The area of quadrilateral $ADEC$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "58\\frac{1}{2}"
} | {
"split": "train",
"index": 20883,
"solution": "1. **Identify the given information and the goal:**\n - $\\angle C = 90^\\circ$ indicates $\\triangle ABC$ is a right triangle.\n - $\\overline{AD} = \\overline{DB}$ implies $D$ is the midpoint of $\\overline{AB}$.\n - $DE \\perp AB$ means $\\triangle BDE$ is... |
deepscaler | [
{
"role": "user",
"content": "Given $x_1$ and $x_2$ are the two real roots of the quadratic equation in $x$: $x^2 - 2(m+2)x + m^2 = 0$.\n\n(1) When $m=0$, find the roots of the equation; \n(2) If $(x_1 - 2)(x_2 - 2) = 41$, find the value of $m$; \n(3) Given an isosceles triangle $ABC$ with one side length... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 20884,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be a rectangle such that $\\overline{AB}=\\overline{CD}=30$, $\\overline{BC}=\\overline{DA}=50$ and point $E$ lies on line $AB$, 20 units from $A$. Find the area of triangle $BEC$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 20885,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $EFGH$ is a parallelogram. A line through point $G$ makes a $30^\\circ$ angle with side $GH$. Determine the degree measure of angle $E$. \n[asy]\nsize(100);\ndraw((0,0)--(5,2)--(6,7)--(1,5)--cycle);\ndraw((5,2)--(7.5,3)); // transversal line\ndraw(Arc((5,2),1,-60,... | math | {
"style": "rule",
"ground_truth": "150"
} | {
"split": "train",
"index": 20886,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are two ways of choosing six different numbers from the list \\( 1,2,3,4,5,6,7,8,9 \\) so that the product of the six numbers is a perfect square. Suppose that these two perfect squares are \\( m^{2} \\) and \\( n^{2} \\), with \\( m \\) and \\( n \\) positive integers an... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 20887,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For what real values of $K$ does $x = K^2 (x-1)(x-2)$ have real roots? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "all"
} | {
"split": "train",
"index": 20888,
"solution": "1. **Rewrite the given equation**: Start by rewriting the equation $x = K^2 (x-1)(x-2)$ in a standard quadratic form:\n \\[\n x = K^2x^2 - 3K^2x + 2K^2.\n \\]\n Rearranging terms, we get:\n \\[\n K^2x^2 - (3K^2 + 1)x + 2K^2 = 0.\n \\]\n\n2. **Identify... |
deepscaler | [
{
"role": "user",
"content": "Ten families have an average of 2 children per family. If exactly two of these families are childless, what is the average number of children in the families with children? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "2.5"
} | {
"split": "train",
"index": 20889,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three faces of a right rectangular prism have areas of 54, 56, and 60 square units. Calculate the volume of the prism in cubic units and round it to the nearest whole number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "426"
} | {
"split": "train",
"index": 20890,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \\in \\mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$,... | math | {
"style": "rule",
"ground_truth": "{r=\\frac{a+1}a,a\\leq1010}"
} | {
"split": "train",
"index": 20891,
"solution": "\nConsider the setup of Alice's solitaire game on the number line. Initially, there is a red bead at position \\( 0 \\) and a blue bead at position \\( 1 \\). During each move, Alice chooses an integer \\( k \\) and a bead to move. If the red bead is at position \\... |
deepscaler | [
{
"role": "user",
"content": "The average of 12 numbers is 90. If the numbers 80, 85, and 92 are removed from the set of numbers, what is the average of the remaining numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{823}{9}"
} | {
"split": "train",
"index": 20892,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 20893,
"solution": "\nWe are given that there are 7 children and 19 wheels. Each child is riding either a bicycle (which has 2 wheels) or a tricycle (which has 3 wheels). We need to determine the number of tricycles.\n\n#### Solution 1: Using a substitution method\n1. Let $b$ be the n... |
deepscaler | [
{
"role": "user",
"content": "Given a decreasing arithmetic sequence $\\{a_n\\}$, where $a_3 = -1$, and $a_1$, $a_4$, $-a_6$ form a geometric sequence. Find the value of $S_7$, where $S_n$ represents the sum of the first $n$ terms of $\\{a_n\\}$. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "-14"
} | {
"split": "train",
"index": 20894,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \\equal{} x^2 \\plus{} ax \\plus{} b$ passing through the point $ (1,\\ \\minus{} 1)$ and the $ x$ axis about the $ x$ axis Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "\\frac{16\\pi}{15}"
} | {
"split": "train",
"index": 20895,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Green Valley School has 120 students enrolled, consisting of 70 boys and 50 girls. If $\\frac{1}{7}$ of the boys and $\\frac{1}{5}$ of the girls are absent on a particular day, what percent of the total student population is absent? Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "16.67\\%"
} | {
"split": "train",
"index": 20896,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the cosine of the angle between the non-intersecting diagonals of two adjacent lateral faces of a regular triangular prism, where the lateral edge is equal to the side of the base. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 20897,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ed has five identical green marbles, and a large supply of identical red marbles. He arranges the green marbles and some of the red ones in a row and finds that the number of marbles whose right hand neighbor is the same color as themselves is equal to the number of marbles who... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 20898,
"solution": "We are limited by the number of marbles whose right hand neighbor is not the same color as the marble. By surrounding every green marble with red marbles - RGRGRGRGRGR. That's 10 \"not the same colors\" and 0 \"same colors.\" Now, for every red marble we add, we wi... |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the four-digit positive integers that end in 0? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4945500"
} | {
"split": "train",
"index": 20899,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of $8$ -tuples $(\\epsilon_1, \\epsilon_2,...,\\epsilon_8)$ such that $\\epsilon_1, \\epsilon_2, ..., 8 \\in \\{1,-1\\}$ and $\\epsilon_1 + 2\\epsilon_2 + 3\\epsilon_3 +...+ 8\\epsilon_8$ is a multiple of $3$ . Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "88"
} | {
"split": "train",
"index": 20900,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Ten women sit in $10$ seats in a line. All of the $10$ get up and then reseat themselves using all $10$ seats, each sitting in the seat she was in before or a seat next to the one she occupied before. In how many ways can the women be reseated? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 20901,
"solution": "To solve this problem, we will use a recursive approach to determine the number of ways the women can be reseated under the given constraints. We define $S_n$ as the number of ways $n$ women can be reseated in $n$ seats such that each woman sits in her original sea... |
deepscaler | [
{
"role": "user",
"content": "If $\\vec{e}_1$ and $\\vec{e}_2$ are unit vectors with an angle of $\\frac{\\pi}{3}$ between them, and $\\vec{a}=2\\vec{e}_1+ \\vec{e}_2$, $\\vec{b}=-3\\vec{e}_1+2\\vec{e}_2$, calculate the value of $\\vec{a}\\cdot \\vec{b}$. Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{2}"
} | {
"split": "train",
"index": 20902,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Altitudes $\\overline{AP}$ and $\\overline{BQ}$ of an acute triangle $\\triangle ABC$ intersect at point $H$. If $HP=3$ and $HQ=4$, then calculate $(BP)(PC)-(AQ)(QC)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-7"
} | {
"split": "train",
"index": 20903,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\\frac{1}{9}+\\frac{1}{18}=\\frac{1}{\\square}$, what is the number that replaces the $\\square$ to make the equation true? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 20904,
"solution": "We simplify the left side and express it as a fraction with numerator 1: $\\frac{1}{9}+\\frac{1}{18}=\\frac{2}{18}+\\frac{1}{18}=\\frac{3}{18}=\\frac{1}{6}$. Therefore, the number that replaces the $\\square$ is 6."
} |
deepscaler | [
{
"role": "user",
"content": "For $a>0$ , denote by $S(a)$ the area of the part bounded by the parabolas $y=\\frac 12x^2-3a$ and $y=-\\frac 12x^2+2ax-a^3-a^2$ .\nFind the maximum area of $S(a)$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{8\\sqrt{2}}{3}"
} | {
"split": "train",
"index": 20905,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\\( P \\) is the midpoint of the height \\( V O \\) of the regular quadrilateral pyramid \\( V-ABCD \\). If the distance from point \\( P \\) to the side faces is 3, and the distance to the base is 5, what is the volume of the regular quadrilateral pyramid? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "750"
} | {
"split": "train",
"index": 20906,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system $xOy$, it is known that the line $l_1$ is defined by the parametric equations $\\begin{cases}x=t\\cos \\alpha\\\\y=t\\sin \\alpha\\end{cases}$ (where $t$ is the parameter), and the line $l_2$ by $\\begin{cases}x=t\\cos(\\alpha + \\frac{\\pi}{4... | math | {
"style": "rule",
"ground_truth": "2\\sqrt{2}"
} | {
"split": "train",
"index": 20907,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the diagram which is the same as the previous one. If $AB = 36$ units and $CD = 72$ units, what is the area of square $BCFE$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2592"
} | {
"split": "train",
"index": 20908,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A traffic light cycles as follows: green for 45 seconds, then yellow for 5 seconds, and then red for 50 seconds. Mark chooses a random five-second interval to observe the light. What is the probability that the color changes during his observation? Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{20}"
} | {
"split": "train",
"index": 20909,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In recent years, live streaming e-commerce has gradually become an emerging marketing model, bringing new growth points to the e-commerce industry. At the beginning of the first year, a certain live streaming platform had an initial capital of 5 million yuan. Due to the partici... | math | {
"style": "rule",
"ground_truth": "46.8"
} | {
"split": "train",
"index": 20910,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Distribute 7 students into two dormitories, A and B, with each dormitory having at least 2 students. How many different distribution plans are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 20911,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A point $(x,y)$ is randomly picked from inside the rectangle with vertices $(0,0)$, $(3,0)$, $(3,2)$, and $(0,2)$. What is the probability that $x < y$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{1}{3}"
} | {
"split": "train",
"index": 20912,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the largest real number $c$ such that for any 2017 real numbers $x_{1}, x_{2}, \\ldots, x_{2017}$, the inequality $$\\sum_{i=1}^{2016} x_{i}\\left(x_{i}+x_{i+1}\\right) \\geq c \\cdot x_{2017}^{2}$$ holds. Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "-\\frac{1008}{2017}"
} | {
"split": "train",
"index": 20913,
"solution": "Let $n=2016$. Define a sequence of real numbers \\left\\{p_{k}\\right\\} by $p_{1}=0$, and for all $k \\geq 1$, $$p_{k+1}=\\frac{1}{4\\left(1-p_{k}\\right)}$$ Note that, for every $i \\geq 1$, $$\\left(1-p_{i}\\right) \\cdot x_{i}^{2}+x_{i} x_{i+1}+p_{i+1} x_{i+1}^... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, the lengths of the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a = 3, cosC = $- \\frac{1}{15}$, and 5sin(B + C) = 3sin(A + C).\n(1) Find the length of side c.\n(2) Find the value of sin(B - $\\frac{\\pi}{3}$). Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{14} - 5\\sqrt{3}}{18}"
} | {
"split": "train",
"index": 20914,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $C_{n}^{4}$, $C_{n}^{5}$, and $C_{n}^{6}$ form an arithmetic sequence, find the value of $C_{n}^{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 20915,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the natural number that is divisible by 9 and 5 and has 14 distinct divisors. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3645"
} | {
"split": "train",
"index": 20916,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three points $(3,-5)$, $(-a + 2, 3)$, and $(2a+3,2)$ lie on the same line. What is $a$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-7}{23}"
} | {
"split": "train",
"index": 20917,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $f(x)=\\dfrac{x+1}{3x-4}$, what is the value of $f(5)$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\dfrac{6}{11}"
} | {
"split": "train",
"index": 20918,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the largest integer that divides $m^{5}-5 m^{3}+4 m$ for all $m \\geq 5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 20919,
"solution": "5!=120."
} |
deepscaler | [
{
"role": "user",
"content": "Bully Vasya loves to run on the escalator in the subway, and he runs down twice as fast as up. If the escalator is not working, it takes Vasya 6 minutes to run up and down. If the escalator is moving down, it takes Vasya 13.5 minutes to run up and down. How many seconds will it... | math | {
"style": "rule",
"ground_truth": "324"
} | {
"split": "train",
"index": 20920,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two numbers need to be inserted between $4$ and $16$ such that the first three numbers are in arithmetic progression and the last three numbers are in geometric progression. What is the sum of those two numbers?\nA) $6\\sqrt{3} + 8$\nB) $10\\sqrt{3} + 6$\nC) $6 + 10\\sqrt{3}$\n... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{3} + 8"
} | {
"split": "train",
"index": 20921,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $D$ be the circle with the equation $x^2 + 8x + 20y + 89 = -y^2 - 6x$. Find the value of $c + d + s$ where $(c, d)$ is the center of $D$ and $s$ is its radius. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-17 + 2\\sqrt{15}"
} | {
"split": "train",
"index": 20922,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The inclination angle $\\alpha$ of the line $l: \\sqrt{3}x+3y+1=0$ is $\\tan^{-1}\\left( -\\frac{\\sqrt{3}}{3} \\right)$. Calculate the value of the angle $\\alpha$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5\\pi}{6}"
} | {
"split": "train",
"index": 20923,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Under normal circumstances, for people aged between 18 and 38 years old, the regression equation for weight $y$ (kg) based on height $x$ (cm) is $y=0.72x-58.5$. Zhang Honghong, who is neither fat nor thin, has a height of 1.78 meters. His weight should be around \\_\\_\\_\\_\\_... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 20924,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, sides a, b, and c are opposite to angles A, B, and C, respectively. Given $\\vec{m} = (a-b, c)$ and $\\vec{n} = (a-c, a+b)$, and that $\\vec{m}$ and $\\vec{n}$ are collinear, find the value of $2\\sin(\\pi+B) - 4\\cos(-B)$. Let's think step by step... | math | {
"style": "rule",
"ground_truth": "-\\sqrt{3} - 2"
} | {
"split": "train",
"index": 20925,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A B C \\) be a triangle. The midpoints of the sides \\( B C, A C \\), and \\( A B \\) are denoted by \\( D, E \\), and \\( F \\) respectively.\n\nThe two medians \\( A D \\) and \\( B E \\) are perpendicular to each other and have lengths \\( \\overline{A D} = 18 \\) an... | math | {
"style": "rule",
"ground_truth": "22.5"
} | {
"split": "train",
"index": 20926,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "(1) Given $\\sin \\left( \\frac{\\pi }{3}-\\alpha \\right)=\\frac{1}{2}$, find the value of $\\cos \\left( \\frac{\\pi }{6}+\\alpha \\right)$;\n\n(2) Given $\\cos \\left( \\frac{5\\pi }{12}+\\alpha \\right)=\\frac{1}{3}$ and $-\\pi < \\alpha < -\\frac{\\pi }{2}$, find the value... | math | {
"style": "rule",
"ground_truth": "- \\frac{1}{3}+ \\frac{2 \\sqrt{2}}{3}"
} | {
"split": "train",
"index": 20927,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n\\[\\frac{5}{3^2 \\cdot 7^2} + \\frac{9}{7^2 \\cdot 11^2} + \\frac{13}{11^2 \\cdot 15^2} + \\dotsb.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{72}"
} | {
"split": "train",
"index": 20928,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Krystyna has some raisins. After giving some away and eating some, she has 16 left. How many did she start with? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "54"
} | {
"split": "train",
"index": 20929,
"solution": "Working backwards, Krystyna had 36 raisins before eating 4, and 54 raisins initially."
} |
deepscaler | [
{
"role": "user",
"content": "A hospital's internal medicine ward has 15 nurses, who work in pairs, rotating shifts every 8 hours. After two specific nurses work the same shift together, calculate the maximum number of days required for them to work the same shift again. Let's think step by step and put onl... | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 20930,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $M(\\sqrt{6}, \\sqrt{2})$ on the ellipse $G$: $\\frac{x^{2}}{a^{2}} + \\frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ with an eccentricity of $\\frac{\\sqrt{6}}{3}$.\n\n1. Find the equation of ellipse $G$.\n2. If the line $l$ with a slope of $1$ intersects ellipse $G$ at poin... | math | {
"style": "rule",
"ground_truth": "\\frac{9}{2}"
} | {
"split": "train",
"index": 20931,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the natural numbers $1, 2, 3, \\ldots, 377$ is painted either red or blue (both colors are present). It is known that the number of red numbers is equal to the smallest red number, and the number of blue numbers is equal to the largest blue number. What is the smallest ... | math | {
"style": "rule",
"ground_truth": "189"
} | {
"split": "train",
"index": 20932,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On a line \\( l \\) in space, points \\( A \\), \\( B \\), and \\( C \\) are sequentially located such that \\( AB = 18 \\) and \\( BC = 14 \\). Find the distance between lines \\( l \\) and \\( m \\) if the distances from points \\( A \\), \\( B \\), and \\( C \\) to line \\( ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 20933,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\binom{18}{11}=31824$, $\\binom{18}{12}=18564$, and $\\binom{20}{13}=77520$, find the value of $\\binom{19}{13}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "27132"
} | {
"split": "train",
"index": 20934,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( I \\) is the incenter of \\( \\triangle ABC \\), and\n\\[ 9 \\overrightarrow{CI} = 4 \\overrightarrow{CA} + 3 \\overrightarrow{CB}. \\]\n\nLet \\( R \\) and \\( r \\) be the circumradius and inradius of \\( \\triangle ABC \\), respectively. Find \\(\\frac{r}{R} =... | math | {
"style": "rule",
"ground_truth": "5/16"
} | {
"split": "train",
"index": 20935,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Quadrilateral $ABCD$ is inscribed in circle $O$ and has side lengths $AB=3, BC=2, CD=6$, and $DA=8$. Let $X$ and $Y$ be points on $\\overline{BD}$ such that $\\frac{DX}{BD} = \\frac{1}{4}$ and $\\frac{BY}{BD} = \\frac{11}{36}$.\nLet $E$ be the intersection of line $AX$ and the ... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 20936,
"solution": "1. **Identify Similar Triangles and Use Parallel Lines**:\n - Since $AC \\parallel EF$, triangles $\\triangle ACX$ and $\\triangle EFX$ are similar. Therefore, $\\frac{XF}{XC} = \\frac{XE}{XA}$.\n - Since $YE \\parallel AD$, triangles $\\triangle EYX$ and $\\tr... |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a right triangle with $\\angle A=90^{\\circ}$. Let $D$ be the midpoint of $AB$ and let $E$ be a point on segment $AC$ such that $AD=AE$. Let $BE$ meet $CD$ at $F$. If $\\angle BFC=135^{\\circ}$, determine $BC/AB$. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{13}}{2}"
} | {
"split": "train",
"index": 20937,
"solution": "Let $\\alpha=\\angle ADC$ and $\\beta=\\angle ABE$. By exterior angle theorem, $\\alpha=\\angle BFD+\\beta=$ $45^{\\circ}+\\beta$. Also, note that $\\tan \\beta=AE/AB=AD/AB=1/2$. Thus, $$1=\\tan 45^{\\circ}=\\tan (\\alpha-\\beta)=\\frac{\\tan \\alpha-\\tan \\beta}{... |
deepscaler | [
{
"role": "user",
"content": "Let \\( p, q, r, s \\) be distinct real numbers such that the roots of \\( x^2 - 12px - 13q = 0 \\) are \\( r \\) and \\( s \\), and the roots of \\( x^2 - 12rx - 13s = 0 \\) are \\( p \\) and \\( q \\). Find the value of \\( p + q + r + s \\). Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "2028"
} | {
"split": "train",
"index": 20938,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, it is known that the internal angle $A= \\frac{\\pi}{3}$, side $BC=2\\sqrt{3}$. Let internal angle $B=x$, and the area be $y$.\n(1) If $x=\\frac{\\pi}{4}$, find the length of side $AC$;\n(2) Find the maximum value of $y$. Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 20939,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $c$ be a constant. The simultaneous equations\n\\begin{align*}x-y = &\\ 2 \\\\\ncx+y = &\\ 3 \\\\\n\\end{align*}have a solution $(x, y)$ inside Quadrant I if and only if Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1<c<\\frac{3}{2}"
} | {
"split": "train",
"index": 20940,
"solution": "1. **Start by solving the system of equations**:\n Given the equations:\n \\[\n \\begin{align*}\n x - y &= 2, \\\\\n cx + y &= 3.\n \\end{align*}\n \\]\n We can solve for $y$ from the first equation:\n \\[\n y = x - 2.\n \\]\n Substitute $y$... |
deepscaler | [
{
"role": "user",
"content": "For $\\mathbf{v} = \\begin{pmatrix} -10 \\\\ 6 \\end{pmatrix}$ and $\\mathbf{w} = \\begin{pmatrix} 15 \\\\ -9 \\end{pmatrix}$, compute $\\text{proj}_{\\mathbf{w}} \\mathbf{v}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -10 \\\\ 6 \\end{pmatrix}"
} | {
"split": "train",
"index": 20941,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of four-digit numbers in which all digits are different, the first digit is divisible by 2, and the sum of the first and last digits is divisible by 3. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "672"
} | {
"split": "train",
"index": 20942,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function $f$ is defined by $f(z) = i\\overline{z}$, where $i^2 = -1$ and $\\overline{z}$ is the complex conjugate of $z$. How many values of $z$ satisfy both $|z| = 5$ and $f(z) = z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 20943,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a permutation $\\pi$ of the set $\\{1,2, \\ldots, 10\\}$, define a rotated cycle as a set of three integers $i, j, k$ such that $i<j<k$ and $\\pi(j)<\\pi(k)<\\pi(i)$. What is the total number of rotated cycles over all permutations $\\pi$ of the set $\\{1,2, \\ldots, 10\\... | math | {
"style": "rule",
"ground_truth": "72576000"
} | {
"split": "train",
"index": 20944,
"solution": "Let us consider a triple $(i, j, k)$ with $i<j<k$ and determine how many permutations rotate it. There are $\\binom{10}{3}$ choices for the values of $\\pi(i), \\pi(j), \\pi(k)$ and the choice of this set of three determines the values of $\\pi(i), \\pi(j), \\pi(k)... |
deepscaler | [
{
"role": "user",
"content": "If $\\lfloor{\\sqrt{x}}\\rfloor=6$, how many possible integer values of $x$ are there? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 20945,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the arithmetic sequence $\\{a\\_n\\}$, $S=10$, $S\\_9=45$, find the value of $a\\_{10}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 20946,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three balls of the same size but different colors in a pocket. One ball is drawn each time, the color is recorded, and then it is put back. The drawing stops when all three colors of balls have been drawn. If it stops after exactly 5 draws, the number of different way... | math | {
"style": "rule",
"ground_truth": "42"
} | {
"split": "train",
"index": 20947,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The root(s) of $\\frac {15}{x^2 - 4} - \\frac {2}{x - 2} = 1$ is (are): Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3 \\text{ and } 5"
} | {
"split": "train",
"index": 20948,
"solution": "1. **Identify the common denominator and simplify the equation:**\n Given the equation:\n \\[\n \\frac{15}{x^2 - 4} - \\frac{2}{x - 2} = 1\n \\]\n We recognize that $x^2 - 4 = (x-2)(x+2)$. Thus, the common denominator for the fractions is $x^2 - 4$. Rewri... |
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