data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "\\( A, B, C \\) are positive integers. It is known that \\( A \\) has 7 divisors, \\( B \\) has 6 divisors, \\( C \\) has 3 divisors, \\( A \\times B \\) has 24 divisors, and \\( B \\times C \\) has 10 divisors. What is the minimum value of \\( A + B + C \\)? Let's think step b... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 24249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $$Q(x) = \\left(1 + x + x^2 + \\ldots + x^{20}\\right)^2 - x^{20},$$ find the sum $$\\beta_1 + \\beta_2 + \\beta_6$$ where the complex zeros of $Q(x)$ are written in the form, $\\beta_k=r_k[\\cos(2\\pi\\beta_k)+i\\sin(2\\pi\\beta_k)]$, with $0<\\beta_1\\le\... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{7}"
} | {
"split": "train",
"index": 24250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A room is 24 feet long and 14 feet wide. Find the ratio of the length to its perimeter and the ratio of the width to its perimeter. Express each ratio in the form $a:b$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7:38"
} | {
"split": "train",
"index": 24251,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve for $n$: $\\frac{2-n}{n+1} + \\frac{2n-4}{2-n} = 1$. 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": 24252,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From a container filled to the brim with $100\\%$ juice, fifth-grader Masha drank 1 liter of juice in a day and in the evening refilled the container with 1 liter of water. The next day, after thoroughly mixing the contents, she drank 1 liter of the mixture and in the evening r... | math | {
"style": "rule",
"ground_truth": "1.75"
} | {
"split": "train",
"index": 24253,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=e^{x}$, and $f(x)=g(x)-h(x)$, where $g(x)$ is an even function, and $h(x)$ is an odd function. If there exists a real number $m$ such that the inequality $mg(x)+h(x)\\geqslant 0$ holds for $x\\in [-1,1]$, determine the minimum value of $m$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "\\dfrac{e^{2}-1}{e^{2}+1}"
} | {
"split": "train",
"index": 24254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all triangles $ABC,$ find the maximum value of $\\sin A + \\sin B \\sin C.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1 + \\sqrt{5}}{2}"
} | {
"split": "train",
"index": 24255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that quadrilateral \\(ABCD\\) is an isosceles trapezoid with \\(AB \\parallel CD\\), \\(AB = 6\\), and \\(CD = 16\\). Triangle \\(ACE\\) is a right-angled triangle with \\(\\angle AEC = 90^\\circ\\), and \\(CE = BC = AD\\). Find the length of \\(AE\\). Let's think step by... | math | {
"style": "rule",
"ground_truth": "4\\sqrt{6}"
} | {
"split": "train",
"index": 24256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be a positive integer such that $12n^2+12n+11$ is a $4$ -digit number with all $4$ digits equal. Determine the value of $n$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 24257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of $V_3$ in Horner's method (also known as Qin Jiushao algorithm) for finding the value of the polynomial $f(x) = 4x^6 + 3x^5 + 4x^4 + 2x^3 + 5x^2 - 7x + 9$ when $x = 4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 24258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "During the \"Cool Summer Happy Shopping\" promotion held in a certain shopping mall, Xiao Yang bought $m$ items of type A goods priced at $5$ yuan each, and $n$ items of type B goods priced at $17 yuan each, spending a total of $203$ yuan. Then the maximum value of $m+n$ is ___... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 24259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Observe the following equations: \n1=1 \n1-4=-（1+2）=-3 \n1-4+9=1+2+3=6 \n1-4+9-16=-（1+2+3+4）=-10 \nThen, the 5th equation is \nThe value of the 20th equation is \nThese equations reflect a certain pattern among integers. Let $n$ represent a positive integer, try ... | math | {
"style": "rule",
"ground_truth": "-210"
} | {
"split": "train",
"index": 24260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the number of ordered pairs of positive integers $(a, b)$ such that $a < b$ and the harmonic mean of $a$ and $b$ is equal to $12^4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"split": "train",
"index": 24261,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In 1980, the per capita income in our country was $255; by 2000, the standard of living had reached a moderately prosperous level, meaning the per capita income had reached $817. What was the annual average growth rate? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "6\\%"
} | {
"split": "train",
"index": 24262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a+a_{1}(x+2)+a_{2}(x+2)^{2}+\\ldots+a_{12}(x+12)^{12}=(x^{2}-2x-2)^{6}$, where $a_{i}$ are constants. Find the value of $2a_{2}+6a_{3}+12a_{4}+20a_{5}+\\ldots+132a_{12}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "492"
} | {
"split": "train",
"index": 24263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A belt is installed on two pulleys with radii of 14 inches and 4 inches respectively. The belt is taut and does not intersect itself. If the distance between the points where the belt touches the two pulleys is 24 inches, find the distance between the centers of the two pulleys... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 24264,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{P}$ be the parabola in the plane determined by the equation $y = 2x^2$. Suppose a circle $\\mathcal{C}$ intersects $\\mathcal{P}$ at four distinct points. If three of these points are $(-7, 98)$, $(-1, 2)$, and $(6, 72)$, find the sum of the distances from the ve... | math | {
"style": "rule",
"ground_truth": "\\sqrt{9653} + \\sqrt{5} + \\sqrt{5220} + \\sqrt{68}"
} | {
"split": "train",
"index": 24265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Among all the factors of $11!$ (where $11! = 11 \\times 10 \\times \\cdots \\times 1$), the largest factor that can be expressed in the form $6k + 1$ (where $k$ is a natural number) is $\\qquad$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "385"
} | {
"split": "train",
"index": 24266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Oleg drew an empty $50 \\times 50$ table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, with 50 of them being rational and the remaining 50 irrational. Then, in each cell of the table, he wrote the sum... | math | {
"style": "rule",
"ground_truth": "1250"
} | {
"split": "train",
"index": 24267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a recipe that prepares $8$ servings of fruit punch requires $3$ oranges, $2$ liters of juice, and $1$ liter of soda, and Kim has $10$ oranges, $12$ liters of juice, and $5$ liters of soda, determine the greatest number of servings of fruit punch that she can prepare by ma... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 24268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two lines are perpendicular and intersect at point $O$. Points $A$ and $B$ move along these two lines at a constant speed. When $A$ is at point $O$, $B$ is 500 yards away from point $O$. After 2 minutes, both points $A$ and $B$ are equidistant from $O$. After another 8 minutes,... | math | {
"style": "rule",
"ground_truth": "2: 3"
} | {
"split": "train",
"index": 24269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ant starts from vertex \\( A \\) of rectangular prism \\( ABCD-A_1B_1C_1D_1 \\) and travels along the surface to reach vertex \\( C_1 \\) with the shortest distance being 6. What is the maximum volume of the rectangular prism? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "12\\sqrt{3}"
} | {
"split": "train",
"index": 24270,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base 10 with no leading zeros. A 6-digit palindrome $n$ is chosen uniformly at random. What is the probability that $\\frac{n}{11}$ is also a palindrome? Let's think step by ... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{30}"
} | {
"split": "train",
"index": 24271,
"solution": "To solve this problem, we first need to understand the structure of a 6-digit palindrome and then determine how many of these palindromes are divisible by 11 and also form a palindrome when divided by 11.\n\n1. **Structure of a 6-digit palindrome**: \n A 6-digit ... |
deepscaler | [
{
"role": "user",
"content": "The function $g(x)$ satisfies\n\\[ 3g(x) + 4g(1 - x) = 6x^2 \\] for all real numbers \\(x\\). Find $g(5)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{-66}{7}"
} | {
"split": "train",
"index": 24272,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$ . She noticed that the pages formed only by even digits have a blue mark, and that every three pages since page two have a red mark. How many pages of the encyclopedia have both colors? Let's thi... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 24273,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of $\\left(81\\right)^{0.25} \\cdot \\left(81\\right)^{0.2}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\cdot \\sqrt[5]{3^4}"
} | {
"split": "train",
"index": 24274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The Benton Youth Soccer Team has 20 players on the team, including reserves. Of these, three are goalies. Today, the team is having a contest to see which goalie can block the most number of penalty kicks. For each penalty kick, a goalie stands in the net while the rest of the ... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 24275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Circles $A$ and $B$ each have a radius of 1 and are tangent to each other. Circle $C$ has a radius of 2 and is tangent to the midpoint of $\\overline{AB}.$ What is the area inside circle $C$ but outside circle $A$ and circle $B?$\nA) $1.16$\nB) $3 \\pi - 2.456$\nC) $4 \\pi - 4.... | math | {
"style": "rule",
"ground_truth": "4 \\pi - 4.912"
} | {
"split": "train",
"index": 24276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sequence \\(\\left\\{a_{n}\\right\\}\\) satisfies \\(a_{1}=1, \\sqrt{\\frac{1}{a_{n}^{2}}+4}=\\frac{1}{a_{n+1}}\\). Let \\(S_{n}=\\sum_{i=1}^{n} a_{i}^{2}\\). If \\(S_{2 n+1}-S_{n} \\leqslant \\frac{t}{30}\\) holds for any \\(n \\in \\mathbf{N}^{*}\\), what is the smallest ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 24277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of the expression:\n\\[ 2(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4(1 + 4))))))))) \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "699050"
} | {
"split": "train",
"index": 24278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many numbers between 10 and 13000, when read from left to right, are formed by consecutive digits in ascending order? For example, 456 is one of these numbers, but 7890 is not. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 24279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The orthogonal projections of the triangle \\(ABC\\) onto two mutually perpendicular planes are equilateral triangles with sides of length 1. Find the perimeter of triangle \\(ABC\\), given that \\(AB = \\frac{\\sqrt{5}}{2}\\). Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2} + \\sqrt{5}"
} | {
"split": "train",
"index": 24280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Carrie has a rectangular garden that measures $6$ feet by $8$ feet. She plants the entire garden with strawberry plants. Carrie is able to plant $4$ strawberry plants per square foot, and she harvests an average of $10$ strawberries per plant. How many strawberries can she expe... | math | {
"style": "rule",
"ground_truth": "1920"
} | {
"split": "train",
"index": 24281,
"solution": "1. **Calculate the area of the garden**: \n The garden is a rectangle with dimensions $6$ feet by $8$ feet. The area \\( A \\) of a rectangle is given by the formula:\n \\[\n A = \\text{length} \\times \\text{width}\n \\]\n Substituting the given dimensi... |
deepscaler | [
{
"role": "user",
"content": "Expand the product ${(x+5)(x+7)}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^2 + 12x + 35"
} | {
"split": "train",
"index": 24282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f$ defined on the non-negative reals and taking non-negative real values such that: $f(2)=0,f(x)\\ne0$ for $0\\le x<2$, and $f(xf(y))f(y)=f(x+y)$ for all $x,y$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = \\begin{cases}\n\\frac{2}{2 - x}, & 0 \\leq x < 2, \\\\\n0, & x \\geq 2.\n\\end{cases}"
} | {
"split": "train",
"index": 24283,
"solution": "\nWe need to find all functions \\( f: [0, \\infty) \\to [0, \\infty) \\) that satisfy the following conditions:\n1. \\( f(2) = 0 \\).\n2. \\( f(x) \\neq 0 \\) for \\( 0 \\leq x < 2 \\).\n3. \\( f(xf(y))f(y) = f(x+y) \\) for all \\( x, y \\geq 0 \\).\n\nLet's begin... |
deepscaler | [
{
"role": "user",
"content": "The force needed to loosen a bolt varies inversely with the length of the handle of the wrench used. A wrench with a handle length of 9 inches requires 375 pounds of force to loosen a certain bolt. A wrench of 15 inches will require how many pounds of force to loosen the same b... | math | {
"style": "rule",
"ground_truth": "225"
} | {
"split": "train",
"index": 24284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all natural numbers which are divisible by $30$ and which have exactly $30$ different divisors. \n\n(M Levin) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11250, 4050, 7500, 1620, 1200, 720"
} | {
"split": "train",
"index": 24285,
"solution": "\nTo find all natural numbers divisible by 30 with exactly 30 different divisors, we first explore the properties of divisors and the structure of such numbers.\n\nA natural number \\( n \\) has exactly 30 divisors if its prime factorization can be expressed to fit... |
deepscaler | [
{
"role": "user",
"content": "A right circular cone sits on a table, pointing up. The cross-section triangle, perpendicular to the base, has a vertex angle of 60 degrees. The diameter of the cone's base is $12\\sqrt{3}$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the con... | math | {
"style": "rule",
"ground_truth": "288\\pi"
} | {
"split": "train",
"index": 24286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the independent college admissions process, a high school has obtained 5 recommendation spots, with 2 for Tsinghua University, 2 for Peking University, and 1 for Fudan University. Both Peking University and Tsinghua University require the participation of male students. The ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 24287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of rooks that can be placed in an $8 \\times 8 \\times 8$ cube so that no two rooks attack each other? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 24288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular garden measures $12$ meters in width and $20$ meters in length. It is paved with tiles that are $2$ meters by $2$ meters each. A cat runs from one corner of the rectangular garden to the opposite corner but must leap over a small pond that exactly covers one tile ... | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 24289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $A B C D$ with area 1, point $M$ is selected on $\\overline{A B}$ and points $X, Y$ are selected on $\\overline{C D}$ such that $A X<A Y$. Suppose that $A M=B M$. Given that the area of triangle $M X Y$ is $\\frac{1}{2014}$, compute the area of trapezoid $A X Y B$.... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}+\\frac{1}{2014} \\text{ OR } \\frac{504}{1007}"
} | {
"split": "train",
"index": 24290,
"solution": "Notice that $[A M X]+[B Y M]=\\frac{1}{2}[A B C D]=\\frac{1}{2}$. Thus, $$[A X Y B]=[A M X]+[B Y M]+[M X Y]=\\frac{1}{2}+\\frac{1}{2014}=\\frac{504}{1007}$$"
} |
deepscaler | [
{
"role": "user",
"content": "There are 54 students in a class, and there are 4 tickets for the Shanghai World Expo. Now, according to the students' ID numbers, the tickets are distributed to 4 students through systematic sampling. If it is known that students with ID numbers 3, 29, and 42 have been selecte... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 24291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the cubic polynomial $p(x)$ such that $p(1) = -7,$ $p(2) = -9,$ $p(3) = -15,$ and $p(4) = -31.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-x^3 + 4x^2 - 7x - 3"
} | {
"split": "train",
"index": 24292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The positive integers $x$ and $y$ are the two smallest positive integers for which the product of $360$ and $x$ is a square and the product of $360$ and $y$ is a cube. What is the sum of $x$ and $y$? Let's think step by step and put only the final answer within \\boxed{}."
}
... | math | {
"style": "rule",
"ground_truth": "85"
} | {
"split": "train",
"index": 24293,
"solution": "To solve this problem, we need to find the smallest integers $x$ and $y$ such that $360x$ is a perfect square and $360y$ is a perfect cube. We start by factoring $360$:\n$$360 = 2^3 \\cdot 3^2 \\cdot 5.$$\n\n#### Finding $x$ such that $360x$ is a perfect square:\nF... |
deepscaler | [
{
"role": "user",
"content": "Let $A = \\{x \\mid x^2 - ax + a^2 - 19 = 0\\}$, $B = \\{x \\mid x^2 - 5x + 6 = 0\\}$, and $C = \\{x \\mid x^2 + 2x - 8 = 0\\}$. \n(1) If $A = B$, find the value of $a$; \n(2) If $B \\cap A \\neq \\emptyset$ and $C \\cap A = \\emptyset$, find the value of $a$. Let's think ste... | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 24294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Moving along a particular line in the Cartesian plane, when the $x$-value increases by 3 units, the $y$-value increases by 7 units. When the $x$-value increases by 9 units, by how many units will the $y$-value increase? Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 24295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular piece of paper $A B C D$ is folded and flattened as shown in the diagram, so that triangle $D C F$ falls onto triangle $D E F$, with vertex $E$ precisely landing on side $A B$. Given $\\angle 1 = 22^\\circ$, find the measure of $\\angle 2$. Let's think step by ste... | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 24296,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the given data \"$20$ nanoseconds\" in scientific notation. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2 \\times 10^{-8}"
} | {
"split": "train",
"index": 24297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The integer 636405 may be written as the product of three 2-digit positive integers. What is the sum of these three integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "259"
} | {
"split": "train",
"index": 24298,
"solution": "We begin by factoring the given integer into prime factors. Since 636405 ends in a 5, it is divisible by 5, so $636405=5 \\times 127281$. Since the sum of the digits of 127281 is a multiple of 3, then it is a multiple of 3, so $636405=5 \\times 3 \\times 42427$. Th... |
deepscaler | [
{
"role": "user",
"content": "In the diagram, \\( B, C \\) and \\( D \\) lie on a straight line, with \\(\\angle ACD=100^{\\circ}\\), \\(\\angle ADB=x^{\\circ}\\), \\(\\angle ABD=2x^{\\circ}\\), and \\(\\angle DAC=\\angle BAC=y^{\\circ}\\). The value of \\( x \\) is: Let's think step by step and put only th... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 24299,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circle with a radius of 2 is inscribed in triangle \\(ABC\\) and touches side \\(BC\\) at point \\(D\\). Another circle with a radius of 4 touches the extensions of sides \\(AB\\) and \\(AC\\), as well as side \\(BC\\) at point \\(E\\). Find the area of triangle \\(ABC\\) if ... | math | {
"style": "rule",
"ground_truth": "\\frac{56}{\\sqrt{3}}"
} | {
"split": "train",
"index": 24300,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) divides the side \\( BC \\) of the parallelogram \\( ABCD \\) in the ratio \\( BM:MC = 1:2 \\). The line \\( AM \\) intersects the diagonal \\( BD \\) at point \\( K \\). Find the area of the quadrilateral \\( CMKD \\) if the area of the parallelogram \\( ABCD \... | math | {
"style": "rule",
"ground_truth": "\\frac{11}{24}"
} | {
"split": "train",
"index": 24301,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the equation $w^3+x^3+y^3=z^3$, $w^3$, $x^3$, $y^3$, and $z^3$ are distinct, consecutive positive perfect cubes listed in ascending order. What is the smallest possible value of $z$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 24302,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose that $n$ is a positive integer such that in base $7$, then $n$ can be expressed as $\\overline{ABC}_7$, and in base $11$, then $n$ can be expressed as $\\overline{CBA}_{11}$. Find the largest possible value of $n$ in base $10$. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "247"
} | {
"split": "train",
"index": 24303,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f : \\mathbb{R} \\to \\mathbb{R}$ be a function such that\n\\[f(xf(y) + x) = xy + f(x)\\]for all $x,$ $y.$\n\nLet $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \\times s.$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "0"
} | {
"split": "train",
"index": 24304,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a strip of paper with three types of scale lines that divide the strip into 6 parts, 10 parts, and 12 parts along its length. If the strip is cut along all the scale lines, into how many parts is the strip divided? Let's think step by step and put only the final answer... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 24305,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many four-digit numbers are divisible by 17? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "530"
} | {
"split": "train",
"index": 24306,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x\\%$ of four-digit numbers have a repeated digit (the repeated digits do not need to be adjacent), then what is $x$? Express your answer as a decimal to the nearest tenth. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "49.6"
} | {
"split": "train",
"index": 24307,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AX = XY = YB = BC$ and the measure of angle $ABC$ is 120 degrees. What is the number of degrees in the measure of angle $BAC$?\n\n[asy]\npair A,X,Y,B,C;\nX = A + dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(-30);\ndraw(B--Y--X--B--C--A--X);\nlabel(\... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24308,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a_1,a_2,\\ldots$ be a sequence determined by the rule $a_n= \\frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n=3a_{n-1}+1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \\le 2008$ is it true that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$? Let's think ste... | math | {
"style": "rule",
"ground_truth": "502"
} | {
"split": "train",
"index": 24309,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An icosahedron is a regular polyhedron with twenty faces, all of which are equilateral triangles. If an icosahedron is rotated by $\\theta$ degrees around an axis that passes through two opposite vertices so that it occupies exactly the same region of space as before, what is t... | math | {
"style": "rule",
"ground_truth": "72^{\\circ}"
} | {
"split": "train",
"index": 24310,
"solution": "Because this polyhedron is regular, all vertices must look the same. Let's consider just one vertex. Each triangle has a vertex angle of $60^{\\circ}$, so we must have fewer than 6 triangles; if we had 6 , there would be $360^{\\circ}$ at each vertex and you wouldn... |
deepscaler | [
{
"role": "user",
"content": "Given the origin of the rectangular coordinate system xOy as the pole and the positive semi-axis of the x-axis as the polar axis, establish a polar coordinate system with the same unit length. The parametric equation of the line l is $$\\begin{cases} \\overset{x=2+t}{y=1+t}\\en... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 24311,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is GCF(LCM(16, 21), LCM(14, 18))? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 24312,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The greatest prime number that is a divisor of $16,385$ can be deduced similarly, find the sum of the digits of this greatest prime number. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 24313,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mr. Green measures his rectangular garden by walking two of the sides and finds that it is $15$ steps by $20$ steps. Each of Mr. Green's steps is $2$ feet long. Mr. Green expects a half a pound of potatoes per square foot from his garden. How many pounds of potatoes does Mr. Gr... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 24314,
"solution": "1. **Convert steps to feet**: \n Mr. Green's garden measures $15$ steps by $20$ steps. Given that each step is $2$ feet long, we convert the dimensions from steps to feet:\n \\[\n 15 \\text{ steps} \\times 2 \\text{ feet/step} = 30 \\text{ feet}\n \\]\n \... |
deepscaler | [
{
"role": "user",
"content": "In a speech contest held in a class, there are a total of $5$ contestants, including $3$ females (among which is female A) and $2$ males. If the two males cannot appear consecutively, and female A cannot be the first to appear, what is the number of ways for the appearance orde... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 24315,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arithmetic sequences $\\left(a_n\\right)$ and $\\left(b_n\\right)$ have integer terms with $a_1=b_1=1<a_2 \\le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"split": "train",
"index": 24316,
"solution": "\n1. **Identify the form of the sequences**: Given that $\\left(a_n\\right)$ and $\\left(b_n\\right)$ are arithmetic sequences with integer terms and $a_1 = b_1 = 1$, we can express the $n$-th terms of these sequences as:\n \\[\n a_n = 1 + (n-1)x \\quad \\text{... |
deepscaler | [
{
"role": "user",
"content": "What is the smallest positive integer $n$ such that $5n \\equiv 2024 \\pmod{26}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 24317,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equation $(x^2 + 2x + 1)^2+(x^2 + 3x + 2)^2+(x^2 + 4x +3)^2+...+(x^2 + 1996x + 1995)^2= 0$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 24318,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Six rhombi of side length 1 are arranged as shown. What is the perimeter of this figure? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"split": "train",
"index": 24319,
"solution": "The first rhombus and the last rhombus each have three edges that form part of the exterior of the figure, and so they each contribute 3 to the perimeter. The inner four rhombi each have two edges that form part of the exterior of the figure, and so they each contr... |
deepscaler | [
{
"role": "user",
"content": "Given that odd prime numbers \\( x, y, z \\) satisfy\n\\[ x \\mid (y^5 + 1), \\quad y \\mid (z^5 + 1), \\quad z \\mid (x^5 + 1). \\]\n\nFind the minimum value of the product \\( xyz \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2013"
} | {
"split": "train",
"index": 24320,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are exactly 120 ways to color five cells in a \\( 5 \\times 5 \\) grid such that each row and each column contains exactly one colored cell.\n\nThere are exactly 96 ways to color five cells in a \\( 5 \\times 5 \\) grid without the corner cell such that each row and each ... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 24321,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a grid where the dimensions are 7 steps in width and 6 steps in height, how many paths are there from the bottom left corner $C$ to the top right corner $D$, considering that each step must either move right or move up? Let's think step by step and put only the final answer ... | math | {
"style": "rule",
"ground_truth": "1716"
} | {
"split": "train",
"index": 24322,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose the estimated $20$ billion dollar cost to send a person to the planet Mars is shared equally by the $250$ million people in the U.S. Then each person's share is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 24323,
"solution": "1. **Identify the total cost and the number of people sharing the cost:** \n The total cost to send a person to Mars is estimated to be $20$ billion dollars. This cost is shared equally among $250$ million people in the U.S.\n\n2. **Convert the total cost into a... |
deepscaler | [
{
"role": "user",
"content": "Calculate the sum of 5739204.742 and -176817.835, and round the result to the nearest integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5562387"
} | {
"split": "train",
"index": 24324,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a right triangle $\\triangle PQR$, we know that $\\tan Q = 0.5$ and the length of $QP = 16$. What is the length of $QR$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8 \\sqrt{5}"
} | {
"split": "train",
"index": 24325,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, points $A$, $B$, $C$, $D$, $E$, and $F$ lie on a straight line with $AB=BC=CD=DE=EF=3$. Semicircles with diameters $AF$, $AB$, $BC$, $CD$, $DE$, and $EF$ create a shape as depicted. What is the area of the shaded region underneath the largest semicircle that exc... | math | {
"style": "rule",
"ground_truth": "\\frac{45}{2}\\pi"
} | {
"split": "train",
"index": 24326,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain rectangle had its dimensions expressed in whole numbers of decimeters. Then, it changed its dimensions three times. First, one of its dimensions was doubled and the other was adjusted so that the area remained the same. Then, one dimension was increased by $1 \\mathrm... | math | {
"style": "rule",
"ground_truth": "4:1"
} | {
"split": "train",
"index": 24327,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the area of the smallest square that can contain a circle with a radius of 6. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 24328,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rectangle $DEFA$ below is a $3 \\times 4$ rectangle with $DC=CB=BA=1$. The area of the \"bat wings\" (shaded area) is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\frac{1}{2}"
} | {
"split": "train",
"index": 24329,
"solution": "1. **Assign Coordinates to Points**:\n - Let $E = (0, 0)$, $F = (3, 0)$, $A = (3, 4)$, $D = (0, 4)$.\n - Given $DC = CB = BA = 1$, we find coordinates for $C$ and $B$:\n - $C = (1, 4)$ (since $DC = 1$ and $D = (0, 4)$)\n - $B = (3, 3)$ (since $BA = 1$ a... |
deepscaler | [
{
"role": "user",
"content": "The real numbers $a$ and $b$ satisfy\n\\[\\begin{pmatrix} 2 \\\\ a \\\\ -7 \\end{pmatrix} \\times \\begin{pmatrix} 5 \\\\ 4 \\\\ b \\end{pmatrix} = \\mathbf{0}.\\]Enter the ordered pair $(a,b).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\left( \\frac{8}{5}, -\\frac{35}{2} \\right)"
} | {
"split": "train",
"index": 24330,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A telephone pole is supported by a steel cable which extends from the top of the pole to a point on the ground 3 meters from its base. When Leah walks 2.5 meters from the base of the pole toward the point where the cable is attached to the ground, her head just touches the cabl... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 24331,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( A = (2, 0) \\) and \\( B = (8, 6) \\). Let \\( P \\) be a point on the circle \\( x^2 + y^2 = 8x \\). Find the smallest possible value of \\( AP + BP \\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2}"
} | {
"split": "train",
"index": 24332,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let\n\\[x^8 - 98x^4 + 1 = p(x) q(x),\\]where $p(x)$ and $q(x)$ are monic, non-constant polynomials with integer coefficients. Find $p(1) + q(1).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 24333,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Fifty students are standing in a line facing the teacher. The teacher first asks everyone to count off from left to right as $1, 2, \\cdots, 50$; then asks the students whose numbers are multiples of 3 to turn around, and then asks the students whose numbers are multiples of 7 ... | math | {
"style": "rule",
"ground_truth": "31"
} | {
"split": "train",
"index": 24334,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose $a$, $b$, $c$ and $d$ are integers satisfying: $a-b+c=5$, $b-c+d=6$, $c-d+a=3$, and $d-a+b=2$. What is the value of $a+b+c+d$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 24335,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that \\( a \\) is a positive real number and \\( b \\) is an integer between \\( 2 \\) and \\( 500 \\), inclusive, find the number of ordered pairs \\( (a,b) \\) that satisfy the equation \\( (\\log_b a)^{1001}=\\log_b(a^{1001}) \\). Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "1497"
} | {
"split": "train",
"index": 24336,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Convex quadrilateral $ABCD$ has $AB=3$, $BC=4$, $CD=13$, $AD=12$, and $\\angle ABC=90^{\\circ}$, as shown. What is the area of the quadrilateral? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 24337,
"solution": "1. **Identify the right triangle**: Given $\\angle ABC = 90^\\circ$ and the sides $AB = 3$ and $BC = 4$, we can apply the Pythagorean theorem to find $AC$:\n \\[\n AC = \\sqrt{AB^2 + BC^2} = \\sqrt{3^2 + 4^2} = \\sqrt{9 + 16} = \\sqrt{25} = 5.\n \\]\n\n2. **C... |
deepscaler | [
{
"role": "user",
"content": "Let $r$ be the speed in miles per hour at which a wheel, $15$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\\frac{1}{3}$ of a second, the speed $r$ is increased by $4$ miles per hour. Determine the original speed $r$.\nA) 9\n... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 24338,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagram shows the miles traveled by bikers Alberto and Bjorn. After four hours, about how many more miles has Alberto biked than Bjorn? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 24339,
"solution": "1. **Identify the distance traveled by each biker after 4 hours**: According to the problem, Bjorn biked 45 miles and Alberto biked 60 miles in the same time period.\n\n2. **Calculate the difference in miles traveled**: To find out how many more miles Alberto biked... |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f$ defined on all real numbers and taking real values such that \\[f(f(y)) + f(x - y) = f(xf(y) - x),\\] for all real numbers $x, y.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = 0"
} | {
"split": "train",
"index": 24340,
"solution": " \nTo solve the given functional equation, we need to find all functions \\( f : \\mathbb{R} \\rightarrow \\mathbb{R} \\) such that:\n\\[\nf(f(y)) + f(x - y) = f(xf(y) - x)\n\\]\nholds for all real numbers \\( x \\) and \\( y... |
deepscaler | [
{
"role": "user",
"content": "Given $f(x) = x^5 + 2x^3 + 3x^2 + x + 1$, calculate the value of $v_3$ when evaluating the value of $f(3)$ using Horner's method. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 24341,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $x^{2}+\\left(m-1\\right)x+9$ is a perfect square trinomial, then the value of $m$ is ____. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-5"
} | {
"split": "train",
"index": 24342,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A convex polyhedron \\(Q\\) has \\(30\\) vertices, \\(70\\) edges, and \\(40\\) faces, of which \\(30\\) are triangular and \\(10\\) are pentagonal. Compute the number of space diagonals in polyhedron \\(Q\\). Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "315"
} | {
"split": "train",
"index": 24343,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cookie Monster now finds a bigger cookie with the boundary described by the equation $x^2 + y^2 - 8 = 2x + 4y$. He wants to know both the radius and the area of this cookie to determine if it's enough for his dessert. Let's think step by step and put only the final answer withi... | math | {
"style": "rule",
"ground_truth": "13\\pi"
} | {
"split": "train",
"index": 24344,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many times during a day does the angle between the hour and minute hands measure exactly $17^{\\circ}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "44"
} | {
"split": "train",
"index": 24345,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the diagram, triangles $ABC$ and $CDE$ are isosceles. The perimeter of $\\triangle CDE$ is $22,$ the perimeter of $\\triangle ABC$ is $24,$ and the length of $CE$ is $9.$ What is the length of $AB?$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 24346,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many different combinations of 4 marbles can be made from 5 indistinguishable red marbles, 4 indistinguishable blue marbles, and 2 indistinguishable black marbles? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 24347,
"solution": "$5+4+3=12$."
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the sum \\[\\frac{1}{2^1} + \\frac{2}{2^2} + \\frac{3}{2^3} + \\cdots + \\frac{k}{2^k} + \\cdots \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 24348,
"solution": ""
} |
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