data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "The function \\( f(x) = \\begin{cases} \nx + 2, & x \\leq -1, \\\\\nx^2, & -1 < x < 2, \\\\\n2x, & x \\geq 2 \n\\end{cases} \\), if \\( f(x) = 3 \\), then \\( x = \\) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"split": "train",
"index": 35449,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive four-digit integers of the form $\\_\\_45$ are divisible by 45? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 35450,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for the positive integer(s) \\( n \\) such that \\( \\phi\\left(n^{2}\\right) = 1000 \\phi(n) \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1000"
} | {
"split": "train",
"index": 35451,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We draw diagonals in some of the squares on a chessboard in such a way that no two diagonals intersect at a common point. What is the maximum number of diagonals that can be drawn this way? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 35452,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A prize fund is divided into first, second, and third prizes. The prize for each first prize is 3 times that of each second prize, and the prize for each second prize is 3 times that of each third prize. The total prize fund is 10,800 yuan. If the total prize money for the thir... | math | {
"style": "rule",
"ground_truth": "2700"
} | {
"split": "train",
"index": 35453,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $(2^0 - 1 + 5^2 - 0)^{-1} \\times 5?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{5}"
} | {
"split": "train",
"index": 35454,
"solution": "1. **Evaluate the expression inside the parentheses**: \n \\[\n 2^0 - 1 + 5^2 - 0\n \\]\n - \\(2^0 = 1\\) because any non-zero number raised to the power of 0 is 1.\n - \\(5^2 = 25\\) because squaring 5 gives 25.\n - Therefore, the expression simplifies... |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 35455,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $u_n$ be the $n^\\text{th}$ term of the sequence\n\\[1,\\,\\,\\,\\,\\,\\,2,\\,\\,\\,\\,\\,\\,5,\\,\\,\\,\\,\\,\\,6,\\,\\,\\,\\,\\,\\,9,\\,\\,\\,\\,\\,\\,12,\\,\\,\\,\\,\\,\\,13,\\,\\,\\,\\,\\,\\,16,\\,\\,\\,\\,\\,\\,19,\\,\\,\\,\\,\\,\\,22,\\,\\,\\,\\,\\,\\,23,\\ldots,\\]\n... | math | {
"style": "rule",
"ground_truth": "5898"
} | {
"split": "train",
"index": 35456,
"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, and it is given that $a\\cos B=(3c-b)\\cos A$. \n$(1)$ If $a\\sin B=2\\sqrt{2}$, find $b$; \n$(2)$ If $a=2\\sqrt{2}$ and the area of $\\triangle ABC$ is $\\sqrt{... | math | {
"style": "rule",
"ground_truth": "4+2\\sqrt{2}"
} | {
"split": "train",
"index": 35457,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ray of light originates from point $A$ and travels in a plane, being reflected $n$ times between lines $AD$ and $CD$ before striking a point $B$ (which may be on $AD$ or $CD$) perpendicularly and retracing its path back to $A$ (At each point of reflection the light makes two ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 35458,
"solution": "1. **Understanding the Reflections**: Each time the light is reflected, it forms an angle with the line equal to the angle of incidence. Given that $\\measuredangle CDA = 8^\\circ$, the light will reflect off line $CD$ and $AD$ such that the angle of incidence equa... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $AB = BC = 2$, $\\angle ABC = 120^\\circ$. A point $P$ is outside the plane of $\\triangle ABC$, and a point $D$ is on the line segment $AC$, such that $PD = DA$ and $PB = BA$. Find the maximum volume of the tetrahedron $PBCD$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 35459,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a,$ $b,$ $c$ be real numbers such that $0 \\leq a, b, c < 1$. Find the minimum value of\n\\[\n\\frac{1}{(2 - a)(2 - b)(2 - c)} + \\frac{1}{(2 + a)(2 + b)(2 + c)}.\n\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{8}"
} | {
"split": "train",
"index": 35460,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a geometric sequence $\\{a_n\\}$, where $a_1+a_2+a_3=4$ and $a_2+a_3+a_4=-2$, find the value of $a_3+a_4+a_5+a_6+a_7+a_8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{8}"
} | {
"split": "train",
"index": 35461,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given point $P(1, 2, 3)$, the symmetric point of $P$ about the $y$-axis is $P_1$, and the symmetric point of $P$ about the coordinate plane $xOz$ is $P_2$. Find the distance $|P_1P_2|$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{14}"
} | {
"split": "train",
"index": 35462,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money. Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "12.5\\%"
} | {
"split": "train",
"index": 35463,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a town of $351$ adults, every adult owns a car, motorcycle, or both. If $331$ adults own cars and $45$ adults own motorcycles, how many of the car owners do not own a motorcycle? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "306"
} | {
"split": "train",
"index": 35464,
"solution": "1. **Identify the total number of adults and their vehicle ownership**:\n - Total number of adults: $351$\n - Adults owning cars: $331$\n - Adults owning motorcycles: $45$\n\n2. **Apply the Principle of Inclusion-Exclusion (PIE)**:\n - The formula for PIE i... |
deepscaler | [
{
"role": "user",
"content": "Two concentric circles $\\omega, \\Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\\Omega$ and the tangent from $B$ to $\\omega$ touches $\\omega$ at $D$ . What is the length of $AD$ . Let's think step by step and put only the final answer within \\b... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 35465,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of eight-digit numbers for which the product of the digits equals 7000. The answer must be given as an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5600"
} | {
"split": "train",
"index": 35466,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The number $1000!$ has a long tail of zeroes. How many zeroes are there? (Reminder: The number $n!$ is the product of the integers from 1 to $n$. For example, $5!=5\\cdot 4\\cdot3\\cdot2\\cdot 1= 120$.) Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "249"
} | {
"split": "train",
"index": 35467,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $PQRS$ be a convex quadrilateral, and let $H_P,$ $H_Q,$ $H_R,$ $H_S$ denote the centroids of triangles $QRS,$ $PRS,$ $PQS,$ and $PQR,$ respectively. Calculate $\\frac{[H_P H_Q H_R H_S]}{[PQRS]}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 35468,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the sum $$ S =\\sum^{2021}_{j=1} \\left|\\sin \\frac{2\\pi j}{2021}\\right|. $$ The value of $S$ can be written as $\\tan \\left( \\frac{c\\pi}{d} \\right)$ for some relatively prime positive integers $c, d$ , satisfying $2c < d$ . Find the value of $c + d$ .... | math | {
"style": "rule",
"ground_truth": "3031"
} | {
"split": "train",
"index": 35469,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the product $\\dfrac{4}{3}\\cdot \\dfrac{5}{4}\\cdot \\dfrac{6}{5}\\cdot \\dfrac{7}{6}\\cdot \\ldots\\cdot \\dfrac{c}{d} = 12$, determine the sum of $c$ and $d$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "71"
} | {
"split": "train",
"index": 35470,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given non-zero vectors $\\overrightarrow{a}$ and $\\overrightarrow{b}$ satisfy $|\\overrightarrow{a}|=2|\\overrightarrow{b}|$, and $(\\overrightarrow{a}-\\overrightarrow{b})\\bot \\overrightarrow{b}$, find the angle between $\\overrightarrow{a}$ and $\\overrightarrow{b}$. Let's... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{3}"
} | {
"split": "train",
"index": 35471,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x^4 + 4x^3 + 6px^2 + 4qx + r$ is exactly divisible by $x^3 + 3x^2 + 9x + 3$, the value of $(p + q)r$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35472,
"solution": "1. **Define the polynomials and their roots:**\n Let $f(x) = x^3 + 3x^2 + 9x + 3$ and $g(x) = x^4 + 4x^3 + 6px^2 + 4qx + r$. Assume $f(x)$ has roots $r_1, r_2, r_3$. Since $f(x)$ divides $g(x)$, $g(x)$ must have roots $r_1, r_2, r_3$, and an additional root $r_4$... |
deepscaler | [
{
"role": "user",
"content": "Take a standard set of 28 dominoes and put back double 3, double 4, double 5, and double 6, as they will not be needed. Arrange the remaining dominoes to form 3 square frames, as shown in the image, so that the sum of the points along each side is equal. In the given example, t... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35473,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nCalculate the lengths of the arcs of the curves given by the parametric equations.\n\n$$\n\\begin{aligned}\n& \\left\\{\\begin{array}{l}\nx=\\left(t^{2}-2\\right) \\sin t+2 t \\cos t \\\\\ny=\\left(2-t^{2}\\right) \\cos t+2 t \\sin t\n\\end{array}\\right. \\\\\n& 0 \\leq t \\... | math | {
"style": "rule",
"ground_truth": "\\frac{8\\pi^3}{3}"
} | {
"split": "train",
"index": 35474,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), the three interior angles \\( \\angle A, \\angle B, \\angle C \\) satisfy \\( \\angle A = 3 \\angle B = 9 \\angle C \\). Find the value of \n\\[ \\cos A \\cdot \\cos B + \\cos B \\cdot \\cos C + \\cos C \\cdot \\cos A = \\quad . \\] Let's thi... | math | {
"style": "rule",
"ground_truth": "-1/4"
} | {
"split": "train",
"index": 35475,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an ellipse C: $$\\frac {x^{2}}{a^{2}}+ \\frac {y^{2}}{b^{2}}=1$$ ($a>b>0$) with its left and right foci being $F_1$ and $F_2$ respectively, and a point $$P(1, \\frac {3}{2})$$ on the ellipse such that the line connecting $P$ and the right focus of the ellipse is perpendic... | math | {
"style": "rule",
"ground_truth": "- \\frac { \\sqrt {3}}{12}"
} | {
"split": "train",
"index": 35476,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given real numbers $x$ and $y$ that satisfy the system of inequalities $\\begin{cases} x - 2y - 2 \\leqslant 0 \\\\ x + y - 2 \\leqslant 0 \\\\ 2x - y + 2 \\geqslant 0 \\end{cases}$, if the minimum value of the objective function $z = ax + by + 5 (a > 0, b > 0)$ is $2$, determi... | math | {
"style": "rule",
"ground_truth": "\\frac{10 + 4\\sqrt{6}}{3}"
} | {
"split": "train",
"index": 35477,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The mean of one set of seven numbers is 15, and the mean of a separate set of eight numbers is 22. What is the mean of the set of all fifteen numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18.73"
} | {
"split": "train",
"index": 35478,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of $x$ if $x$ is positive and $x\\cdot\\lfloor x\\rfloor=70$. Express your answer as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8.75"
} | {
"split": "train",
"index": 35479,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x)=\\sin \\omega x+\\cos \\omega x \\) where \\( \\omega > 0 \\) and \\( x \\in \\mathbb{R} \\), if the function \\( f(x) \\) is monotonically increasing on the interval \\( (-\\omega, \\omega) \\) and the graph of the function \\( y=f(x) \\) is symmetr... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{\\pi}}{2}"
} | {
"split": "train",
"index": 35480,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A shopping mall sells a batch of branded shirts, with an average daily sales volume of $20$ shirts, and a profit of $40$ yuan per shirt. In order to expand sales and increase profits, the mall decides to implement an appropriate price reduction strategy. After investigation, it... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35481,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a checkers tournament, students from 10th and 11th grades participated. Each player played against every other player exactly once. A win earned a player 2 points, a draw earned 1 point, and a loss earned 0 points. The number of 11th graders was 10 times the number of 10th g... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35482,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Brachycephalus frogs have three toes on each foot and two fingers on each hand. The common frog has five toes on each foot and four fingers on each hand. Some Brachycephalus and common frogs are in a bucket. Each frog has all its fingers and toes. Between them they have 122 toe... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35483,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given in $\\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$, and $\\sin A=\\sqrt{3}(1-\\cos A)$.\n\n$(1)$ Find $A$;\n\n$(2)$ If $a=7$ and $\\sin B+\\sin C=\\frac{13\\sqrt{3}}{14}$, find the area of $\\triangle ABC$. Let's think s... | math | {
"style": "rule",
"ground_truth": "10 \\sqrt{3}"
} | {
"split": "train",
"index": 35484,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[\\mathbf{M} = \\begin{pmatrix} 1 & 2 & 2 \\\\ 2 & 1 & -2 \\\\ a & 2 & b \\end{pmatrix}.\\]If $\\mathbf{M} \\mathbf{M}^T = 9 \\mathbf{I},$ then enter the ordered pair $(a,b).$\n\nNote: For a matrix $\\mathbf{A},$ $\\mathbf{A}^T$ is the transpose of $\\mathbf{A},$ which i... | math | {
"style": "rule",
"ground_truth": "(-2,-1)"
} | {
"split": "train",
"index": 35485,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain department store sells a batch of shirts. The cost price of each shirt is $80. On average, 30 shirts can be sold per day, with a profit of $50 per shirt. In order to increase sales and profits, the store decides to take appropriate price reduction measures. After inve... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 35486,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given in the polar coordinate system, the equation of curve Ω is $\\rho=6\\cos\\theta$. Taking the pole as the origin of the Cartesian coordinate system, with the polar axis as the positive half-axis of the x-axis, and using the same unit of length in both coordinate systems, e... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 35487,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two people, A and B, are collaborating to type a document. Initially, A types 100 characters per minute, and B types 200 characters per minute. When they reach half of the total amount, A's speed triples while B takes a 5-minute break and then continues at the original speed. B... | math | {
"style": "rule",
"ground_truth": "18000"
} | {
"split": "train",
"index": 35488,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two cards are chosen at random from a standard 52-card deck. What is the probability that the first card is a spade and the second card is a king? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{17}{884}"
} | {
"split": "train",
"index": 35489,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \\cdot B \\cdot C = 2023$. What is the largest possible value of the sum $A+B+C$? Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "297"
} | {
"split": "train",
"index": 35490,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sophia chooses a real number uniformly at random from the interval $[0, 3000]$. Independently, Oliver chooses a real number uniformly at random from the interval $[0, 4500]$. What is the probability that Oliver's number is at least twice as large as Sophia's number?\n**A) $\\fr... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 35491,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let an \"operation\" be randomly changing a given positive integer $n$ to a smaller nonnegative integer (each number has the same probability). What is the probability that after performing the operation on 2019 several times to obtain 0, the numbers 10, 100, and 1000 all appea... | math | {
"style": "rule",
"ground_truth": "1/2019000000"
} | {
"split": "train",
"index": 35492,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Granny Smith has $63. Elberta has $2 more than Anjou and Anjou has one-third as much as Granny Smith. How many dollars does Elberta have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 35493,
"solution": "1. **Identify the amount Granny Smith has**: Granny Smith has $63.\n\n2. **Calculate Anjou's amount**: \n - Anjou has one-third of what Granny Smith has. \n - Therefore, Anjou's amount is calculated as:\n \\[\n \\frac{1}{3} \\times 63 = 21\n \\]\n Anjou... |
deepscaler | [
{
"role": "user",
"content": "Given $|\\vec{a}|=1$, $|\\vec{b}|= \\sqrt{2}$, and $\\vec{a} \\perp (\\vec{a} - \\vec{b})$, find the angle between the vectors $\\vec{a}$ and $\\vec{b}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4}"
} | {
"split": "train",
"index": 35494,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The length of a rectangle is increased by $25\\%$, but the width of the rectangle is decreased to keep the area of the rectangle unchanged. By what percent was the rectangle's width decreased? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 35495,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a sequence of distinct positive integers $(i\\_1, i\\_2, \\ldots, i\\_n)$ where $n$ is a positive integer greater than or equal to 2, if $i\\_p > i\\_q$ when $p > q$, then $i\\_p$ and $i\\_q$ are called a \"good order\" of the sequence. The number of \"good orders\" in a ... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 35496,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A fair coin is tossed 4 times. What is the probability of at least two consecutive heads? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{8}"
} | {
"split": "train",
"index": 35497,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $A = (0,0)$ and $B = (b,2)$ be points on the coordinate plane. Let $ABCDEF$ be a convex equilateral hexagon such that $\\angle FAB = 120^\\circ,$ $\\overline{AB}\\parallel \\overline{DE},$ $\\overline{BC}\\parallel \\overline{EF,}$ $\\overline{CD}\\parallel \\overline{FA},$... | math | {
"style": "rule",
"ground_truth": "51"
} | {
"split": "train",
"index": 35498,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathbb R$ be the set of real numbers. Determine all functions $f:\\mathbb R\\to\\mathbb R$ that satisfy the equation\\[f(x+f(x+y))+f(xy)=x+f(x+y)+yf(x)\\]for all real numbers $x$ and $y$.\n\n[i] Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "f(x) = 2 - x \\text{ and } f(x) = x"
} | {
"split": "train",
"index": 35499,
"solution": "\nTo solve the functional equation:\n\n\\[\nf(x + f(x+y)) + f(xy) = x + f(x+y) + yf(x)\n\\]\n\nfor all \\( x, y \\in \\mathbb{R} \\), we start by considering particular values for \\( x \\) and \\( y \\) to simplify the equation and gain insight into the form of th... |
deepscaler | [
{
"role": "user",
"content": "The decimal representation of $m/n,$ where $m$ and $n$ are relatively prime positive integers and $m < n,$ contains the digits $2, 5$, and $1$ consecutively, and in that order. Find the smallest value of $n$ for which this is possible.\n Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "127"
} | {
"split": "train",
"index": 35500,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A regular octagon is inscribed in a circle and another regular octagon is circumscribed about the same circle. What is the ratio of the area of the larger octagon to the area of the smaller octagon? Express your answer as a common fraction. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "\\frac{4 - 2\\sqrt{2}}{2}"
} | {
"split": "train",
"index": 35501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box contains $28$ red balls, $20$ green balls, $19$ yellow balls, $13$ blue balls, $11$ white balls, and $9$ black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $15$ balls of a single color will be dr... | math | {
"style": "rule",
"ground_truth": "76"
} | {
"split": "train",
"index": 35502,
"solution": "To solve this problem, we need to determine the minimum number of balls that must be drawn to ensure that at least 15 balls of one color are drawn. We will consider the worst-case scenario where we draw the maximum number of balls possible without reaching 15 balls... |
deepscaler | [
{
"role": "user",
"content": "The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \\mathrm{km} ... | math | {
"style": "rule",
"ground_truth": "360"
} | {
"split": "train",
"index": 35503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $\\sin(3\\pi + \\alpha) = -\\frac{1}{2}$, find the value of $\\cos\\left(\\frac{7\\pi}{2} - \\alpha\\right)$. 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": 35504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For every dollar Ben spent on bagels, David spent $25$ cents less. Ben paid $\\$12.50$ more than David. How much did they spend in the bagel store together? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$87.50"
} | {
"split": "train",
"index": 35505,
"solution": "Let $B$ represent the amount Ben spent and $D$ represent the amount David spent at the bagel store. According to the problem, for every dollar Ben spent, David spent $25$ cents less. This can be expressed as:\n\\[ D = B - 0.25B = 0.75B \\]\nThis simplifies to:\n\\[... |
deepscaler | [
{
"role": "user",
"content": "Two circles are externally tangent to each other at point \\( A \\), and both are tangent to a third circle at points \\( B \\) and \\( C \\). The extension of chord \\( AB \\) of the first circle intersects the second circle at point \\( D \\), and the extension of chord \\( A... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 35506,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Bryan has some stamps of 3 cents, 4 cents, and 6 cents. What is the least number of stamps he can combine so the value of the stamps is 50 cents? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 35507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the sequence ${a_n}$ where $a_{1}= \\frac {3}{2}$, and $a_{n}=a_{n-1}+ \\frac {9}{2}(- \\frac {1}{2})^{n-1}$ (for $n\\geq2$).\n(I) Find the general term formula $a_n$ and the sum of the first $n$ terms $S_n$;\n(II) Let $T_{n}=S_{n}- \\frac {1}{S_{n}}$ ($n\\in\\mathbb{N}^*... | math | {
"style": "rule",
"ground_truth": "-\\frac{7}{12}"
} | {
"split": "train",
"index": 35508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If I choose four cards from a standard $52$-card deck, without replacement, what is the probability that I will end up with one card from each suit, in a sequential order (e.g., clubs, diamonds, hearts, spades)? Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "\\frac{2197}{499800}"
} | {
"split": "train",
"index": 35509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 8 identical rooks be placed on an $8 \\times 8$ chessboard symmetrically with respect to the diagonal that passes through the lower-left corner square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "139448"
} | {
"split": "train",
"index": 35510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that \\( f(x) \\) is a function defined for every real number \\( x \\) with \\( 0 \\leq x \\leq 1 \\) with the properties that\n\n- \\( f(1-x)=1-f(x) \\) for all real numbers \\( x \\) with \\( 0 \\leq x \\leq 1 \\),\n- \\( f\\left(\\frac{1}{3} x\\right)=\\frac{1}{2} f... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{4}"
} | {
"split": "train",
"index": 35511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Each of the numbers \\( m \\) and \\( n \\) is the square of an integer. The difference \\( m - n \\) is a prime number. Which of the following could be \\( n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "900"
} | {
"split": "train",
"index": 35512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, let $AB=6$, $BC=7$, $AC=4$, and $O$ be the incenter of $\\triangle ABC$. If $\\overrightarrow{AO} = p\\overrightarrow{AB} + q\\overrightarrow{AC}$, determine the value of $\\frac{p}{q}$. 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": 35513,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Julia invested a certain amount of money in two types of assets: real estate and mutual funds. The total amount she invested was $\\$200,000$. If she invested 6 times as much in real estate as she did in mutual funds, what was her total investment in real estate? Let's think st... | math | {
"style": "rule",
"ground_truth": "171,428.58"
} | {
"split": "train",
"index": 35514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that positive real numbers $x$ and $y$ satisfy $e^{x}=y\\ln x+y\\ln y$, then the minimum value of $\\frac{{e}^{x}}{x}-\\ln y$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "e-1"
} | {
"split": "train",
"index": 35515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Susie Q has $2000 to invest. She invests part of the money in Alpha Bank, which compounds annually at 4 percent, and the remainder in Beta Bank, which compounds annually at 6 percent. After three years, Susie's total amount is $\\$2436.29$. Determine how much Susie originally i... | math | {
"style": "rule",
"ground_truth": "820"
} | {
"split": "train",
"index": 35516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A net for hexagonal pyramid is constructed by placing a triangle with side lengths $x$ , $x$ , and $y$ on each side of a regular hexagon with side length $y$ . What is the maximum volume of the pyramid formed by the net if $x+y=20$ ? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "128\\sqrt{15}"
} | {
"split": "train",
"index": 35517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many of the four integers $222, 2222, 22222$, and $222222$ are multiples of 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 35518,
"solution": "We could use a calculator to divide each of the four given numbers by 3 to see which calculations give an integer answer. Alternatively, we could use the fact that a positive integer is divisible by 3 if and only if the sum of its digits is divisible by 3. The sums... |
deepscaler | [
{
"role": "user",
"content": "In parallelogram $EFGH$, point $Q$ is on $\\overline{EF}$ such that $\\frac{EQ}{EF} = \\frac{1}{8}$, and point $R$ is on $\\overline{EH}$ such that $\\frac{ER}{EH} = \\frac{1}{9}$. Let $S$ be the point of intersection of $\\overline{EG}$ and $\\overline{QR}$. Find the ratio $\\... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 35519,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find ... | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 35520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that α is an angle in the fourth quadrant, and $sin \\left( \\frac{\\pi}{2} + \\alpha \\right) = \\frac{4}{5}$, calculate the value of $tan(\\alpha)$. 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": 35521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate \\(14 \\cdot 31\\) and \\(\\left\\lfloor\\frac{2+\\sqrt{2}}{2}\\right\\rfloor + \\left\\lfloor\\frac{3+\\sqrt{3}}{3}\\right\\rfloor + \\left\\lfloor\\frac{4+\\sqrt{4}}{4}\\right\\rfloor + \\cdots + \\left\\lfloor\\frac{1989+\\sqrt{1989}}{1989}\\right\\rfloor + \\left\... | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"split": "train",
"index": 35522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The 200-digit number \\( M \\) is composed of 200 ones. What is the sum of the digits of the product \\( M \\times 2013 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1200"
} | {
"split": "train",
"index": 35523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the coefficient of $x^{8}$ in the expansion of \\\\((x^{3}+ \\frac{1}{2 \\sqrt {x}})^{5}\\\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{5}{2}"
} | {
"split": "train",
"index": 35524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For which maximal $N$ there exists an $N$-digit number with the following property: among any sequence of its consecutive decimal digits some digit is present once only?\n\t\n\tAlexey Glebov Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1023"
} | {
"split": "train",
"index": 35525,
"solution": "\nTo determine for which maximal \\( N \\) there exists an \\( N \\)-digit number satisfying the given property, we need to find an \\( N \\)-digit number such that in every sequence of consecutive decimal digits, there is at least one digit that appears only once.... |
deepscaler | [
{
"role": "user",
"content": "In an equilateral triangle $ABC$ with side length $6$, point $D$ is the midpoint of $BC$. Calculate $\\tan{\\angle BAD}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 35526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 2 teachers, 3 male students, and 4 female students taking a photo together. How many different standing arrangements are there under the following conditions? (Show the process, and represent the final result with numbers) \n(1) The male students must stand together;... | math | {
"style": "rule",
"ground_truth": "1728"
} | {
"split": "train",
"index": 35527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m \\ge 2$ be an integer and let $T = \\{2,3,4,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $T$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $a + b = c$. Let's think ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 35528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the product \\(x^{2} y^{2} z^{2} u\\) given the condition that \\(x, y, z, u \\geq 0\\) and:\n\n\\[ 2x + xy + z + yz u = 1 \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1/512"
} | {
"split": "train",
"index": 35529,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a bag contains 28 red balls, 20 green balls, 12 yellow balls, 20 blue balls, 10 white balls, and 10 black balls, determine the minimum number of balls that must be drawn to ensure that at least 15 balls of the same color are selected. Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "75"
} | {
"split": "train",
"index": 35530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a large circle with a radius of 11 and small circles with a radius of 1, find the maximum number of small circles that can be tangentially inscribed in the large circle without overlapping. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 35531,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\sqrt {3}\\sin x\\cos (x+ \\dfrac {π}{6})+\\cos x\\sin (x+ \\dfrac {π}{3})+ \\sqrt {3}\\cos ^{2}x- \\dfrac { \\sqrt {3}}{2}$.\n(I) Find the range of $f(x)$ when $x\\in(0, \\dfrac {π}{2})$;\n(II) Given $\\dfrac {π}{12} < α < \\dfrac {π}{3}$, $f(α)= \\dfrac {6}{5}$,... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{33}{65}"
} | {
"split": "train",
"index": 35532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The digits of 2021 can be rearranged to form other four-digit whole numbers between 1000 and 3000. Find the largest possible difference between two such four-digit whole numbers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1188"
} | {
"split": "train",
"index": 35533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ with a common difference $d > 0$, and $a\\_2$, $a\\_5-1$, $a\\_{10}$ form a geometric sequence. If $a\\_1=5$, and $S\\_n$ represents the sum of the first $n$ terms of the sequence, find the minimum value of $\\frac{2S\\_n+n+32}{a\\_n+1... | math | {
"style": "rule",
"ground_truth": "\\frac{20}{3}"
} | {
"split": "train",
"index": 35534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three boys \\( B_{1}, B_{2}, B_{3} \\) and three girls \\( G_{1}, G_{2}, G_{3} \\) are to be seated in a row according to the following rules:\n1) A boy will not sit next to another boy and a girl will not sit next to another girl,\n2) Boy \\( B_{1} \\) must sit next to girl \\... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 35535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the quadratic polynomial $p(x)$ such that $p(-2) = 13,$ $p(1) = -2,$ and $p(3) = 8.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2x^2 - 3x - 1"
} | {
"split": "train",
"index": 35536,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation $\\frac{1}{j} + \\frac{1}{k} = \\frac{1}{4}$, both $j$ and $k$ are positive integers. What is the sum of all possible values for $j+k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "59"
} | {
"split": "train",
"index": 35537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "550"
} | {
"split": "train",
"index": 35538,
"solution": "Recall that the arithmetic mean of all the $n$ digit palindromes is just the average of the largest and smallest $n$ digit palindromes, and in this case the $2$ palindromes are $101$ and $999$ and $\\frac{101+999}{2}=\\boxed{550},$ which is the final answer.\n ~ ma... |
deepscaler | [
{
"role": "user",
"content": "Find $x$ such that $\\lceil x \\rceil \\cdot x = 135$. Express $x$ as a decimal. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11.25"
} | {
"split": "train",
"index": 35539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The value of $$\\frac {1}{\\tan 20^\\circ} - \\frac {1}{\\cos 10^\\circ}$$ is equal to \\_\\_\\_\\_\\_\\_. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt {3}"
} | {
"split": "train",
"index": 35540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of $453_6$, $436_6$, and $42_6$ in base 6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1415_6"
} | {
"split": "train",
"index": 35541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $525^2 - 475^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50000"
} | {
"split": "train",
"index": 35542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are 17 people standing in a circle: each of them is either truthful (always tells the truth) or a liar (always lies). All of them said that both of their neighbors are liars. What is the maximum number of liars that can be in this circle? Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 35543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There were 100 cards lying face up with the white side on the table. Each card has one white side and one black side. Kostya flipped 50 cards, then Tanya flipped 60 cards, and after that Olya flipped 70 cards. As a result, all 100 cards ended up lying with the black side up. Ho... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 35544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\\hat{B}E = 90^\\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "10 + \\sqrt{2}"
} | {
"split": "train",
"index": 35545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Xiaoming's family raises chickens and pigs in a ratio of 26:5, and sheep to horses in a ratio of 25:9, while the ratio of pigs to horses is 10:3. Find the ratio of chickens, pigs, horses, and sheep. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "156:30:9:25"
} | {
"split": "train",
"index": 35546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The area of a trapezoid is 1. What is the minimum possible length of the longest diagonal of this trapezoid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 35547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alice cycled 240 miles in 4 hours, 30 minutes. Then, she cycled another 300 miles in 5 hours, 15 minutes. What was Alice's average speed in miles per hour for her entire journey? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55.38"
} | {
"split": "train",
"index": 35548,
"solution": ""
} |
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