data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "If $a=\\frac{1}{2}$ and $(a+1)(b+1)=2$ then the radian measure of $\\arctan a + \\arctan b$ equals 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": 3500,
"solution": "1. **Given Values and Equation Setup:**\n - We are given $a = \\frac{1}{2}$.\n - We also have the equation $(a+1)(b+1) = 2$.\n\n2. **Solving for $b$:**\n - Substitute $a = \\frac{1}{2}$ into the equation:\n \\[\n \\left(\\frac{1}{2} + 1\\right)(b+1) = ... |
deepscaler | [
{
"role": "user",
"content": "In right triangle $ABC$, $\\sin A = \\frac{8}{17}$ and $\\sin B = 1$. Find $\\sin C$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{17}"
} | {
"split": "train",
"index": 3501,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One angle of a triangle is twice another, and the sides opposite these angles have lengths 12 and 18. Compute the length of the third side of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 3502,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\( \\triangle ABC \\), \\(\\angle ABC = 50^\\circ\\), \\(\\angle ACB = 30^\\circ\\), \\(M\\) is a point inside the triangle such that \\(\\angle MCB = 20^\\circ\\), \\(\\angle MAC = 40^\\circ\\). Find the measure of \\(\\angle MBC\\). \n\n(Problem 1208 from Mathema... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 3503,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $3 \\times 13 \\times 23 \\times 33 \\times \\ldots \\times 183 \\times 193$ is divided by $5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 3504,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expansion of \\((x+y+z)^{8}\\), what is the sum of the coefficients for all terms of the form \\(x^{2} y^{a} z^{b}\\) (where \\(a, b \\in \\mathbf{N})\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1792"
} | {
"split": "train",
"index": 3505,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m \\ge 3$ be an integer and let $S = \\{3,4,5,\\ldots,m\\}$. Find the smallest value of $m$ such that for every partition of $S$ into two subsets, at least one of the subsets contains integers $a$, $b$, and $c$ (not necessarily distinct) such that $ab = c$.\nNote: a partit... | math | {
"style": "rule",
"ground_truth": "243"
} | {
"split": "train",
"index": 3506,
"solution": "We claim that $243$ is the minimal value of $m$. Let the two partitioned sets be $A$ and $B$; we will try to partition $3, 9, 27, 81,$ and $243$ such that the $ab=c$ condition is not satisfied. Without loss of generality, we place $3$ in $A$. Then $9$ must be placed... |
deepscaler | [
{
"role": "user",
"content": "Find the largest prime factor of $15^3+10^4-5^5$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 3507,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\alpha, \\beta \\in \\left(0, \\frac{\\pi}{2}\\right)\\), \\(\\cos \\alpha = \\frac{4}{5}\\), \\(\\tan (\\alpha - \\beta) = -\\frac{1}{3}\\), find \\(\\cos \\beta\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{9 \\sqrt{10}}{50}"
} | {
"split": "train",
"index": 3508,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine how many different ways there are to assign the elements of the set \\( M = \\{1, 2, 3, 4, 5\\} \\) into three ordered sets \\( A, B,\\) and \\( C \\) such that the following conditions are satisfied: each element must belong to at least one of the sets, the intersect... | math | {
"style": "rule",
"ground_truth": "1230"
} | {
"split": "train",
"index": 3509,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the value of the expression \\[(5^{1003}+6^{1004})^2-(5^{1003}-6^{1004})^2\\] and express it in the form of $k\\cdot30^{1003}$ for some positive integer $k$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 3510,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Solve the equations:<br/>$(1)x^{2}-5x+1=0$;<br/>$(2) 2\\left(x-5\\right)^{2}+x\\left(x-5\\right)=0$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{10}{3}"
} | {
"split": "train",
"index": 3511,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two subsets of the set $T = \\{w, x, y, z, v\\}$ need to be chosen so that their union is $T$ and their intersection contains exactly three elements. How many ways can this be accomplished, assuming the subsets are chosen without considering the order? Let's think step by step ... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 3512,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $\frac{x-y}{z-y}=-10$, what is the value of $\frac{x-z}{y-z}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 3513,
"solution": "Since the problem asks us to find the value of $\frac{x-z}{y-z}$, then this value must be the same no matter what $x, y$ and $z$ we choose that satisfy $\frac{x-y}{z-y}=-10$. Thus, if we can find numbers $x, y$ and $z$ that give $\frac{x-y}{z-y}=-10$, then these num... |
deepscaler | [
{
"role": "user",
"content": "A rectangle PQRS has a perimeter of 24 meters and side PQ is fixed at 7 meters. Find the minimum diagonal PR of the rectangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{74}"
} | {
"split": "train",
"index": 3514,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "My co-worker Erich is very odd. He only likes numbers that are divisible by 5. How many different last digits are possible in numbers that Erich likes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 3515,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Points A and B lie on a circle centered at O, and ∠AOB = 60°. A second circle is internally tangent to the first and tangent to both OA and OB. What is the ratio of the area of the smaller circle to that of the larger circle? Let's think step by step and put only the final answ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{9}"
} | {
"split": "train",
"index": 3516,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When the municipal government investigated the relationship between changes in citizens' income and tourism demand, a sample of 5000 people was randomly selected using the independence test method. The calculation showed that $K^{2}=6.109$. Based on this data, the municipal gov... | math | {
"style": "rule",
"ground_truth": "97.5\\%"
} | {
"split": "train",
"index": 3517,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube? Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3518,
"solution": "1. **Calculate the volume of the cube**:\n The side length of the cube is given as $6$. Therefore, the volume of the cube ($V_{\\text{cube}}$) is calculated as:\n \\[\n V_{\\text{cube}} = 6^3 = 216.\n \\]\n\n2. **Calculate the volume of a ball of clay**:\n ... |
deepscaler | [
{
"role": "user",
"content": "A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 3519,
"solution": "1. **Determine the number of cans needed to feed 15 children**: \n Given that one can of soup can feed 5 children, we calculate the number of cans needed to feed 15 children as follows:\n \\[\n \\text{Number of cans} = \\frac{15 \\text{ children}}{5 \\text{ ch... |
deepscaler | [
{
"role": "user",
"content": "What is $\\dbinom{n}{n-1}$ for any positive integer $n$? (Answer in terms of $n$.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "n"
} | {
"split": "train",
"index": 3520,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a multiplication error involving two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the three-digit number $a$. The erroneous product obtained was $396$. Determine the correct value of the product $ab$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "693"
} | {
"split": "train",
"index": 3521,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.\n[asy] import graph; pointpen=black;pathpen=black+linewidth(0.7);pen f = fontsize(10); pair P=(-8,5),Q=(-15,-19),R=... | math | {
"style": "rule",
"ground_truth": "89"
} | {
"split": "train",
"index": 3522,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In right triangle $PQR$, we have $\\angle Q = \\angle R$ and $PR = 6\\sqrt{2}$. What is the area of $\\triangle PQR$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 3523,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the sum of all the integers from 100 to 2000? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1996050"
} | {
"split": "train",
"index": 3524,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a triangle with $\\angle BAC=40^\\circ $ , $O$ be the center of its circumscribed circle and $G$ is its centroid. Point $D$ of line $BC$ is such that $CD=AC$ and $C$ is between $B$ and $D$ . If $AD\\parallel OG$ , find $\\angle ACB$ . Let's thin... | math | {
"style": "rule",
"ground_truth": "70"
} | {
"split": "train",
"index": 3525,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that each of 5 different boxes contains exactly 2 fruits when 4 identical pears and 6 different apples are distributed into the boxes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.0074"
} | {
"split": "train",
"index": 3526,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first three stages of a pattern are shown below, in which each line segment represents a toothpick. If the pattern continues such that at each successive stage, three toothpicks are added to the previous arrangement, how many toothpicks are necessary to create the arrangeme... | math | {
"style": "rule",
"ground_truth": "751"
} | {
"split": "train",
"index": 3527,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "We have 10 springs, each originally $0.5 \\mathrm{~m}$ long with a spring constant of $200 \\mathrm{~N}/\\mathrm{m}$. A mass of $2 \\mathrm{~kg}$ is hung on each spring, and the springs, along with the masses, are hung in a series. What is the length of the resulting chain? (Ne... | math | {
"style": "rule",
"ground_truth": "10.39"
} | {
"split": "train",
"index": 3528,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of t... | math | {
"style": "rule",
"ground_truth": "34"
} | {
"split": "train",
"index": 3529,
"solution": "1. **Define the sequences**: Let the quarterly scores for the Raiders be $a, ar, ar^2, ar^3$ and for the Wildcats be $b, b+d, b+2d, b+3d$. Given that the sequences are increasing, we know $r > 1$ for the geometric sequence and $d > 0$ for the arithmetic sequence.\n\... |
deepscaler | [
{
"role": "user",
"content": "Given the function \\( f(x) = \\frac{2+x}{1+x} \\), let \\( f(1) + f(2) + \\cdots + f(1000) = m \\) and \\( f\\left(\\frac{1}{2}\\right) + f\\left(\\frac{1}{3}\\right) + \\cdots + f\\left(\\frac{1}{1000}\\right) = n \\). What is the value of \\( m + n \\)? Let's think step by s... | math | {
"style": "rule",
"ground_truth": "2998.5"
} | {
"split": "train",
"index": 3530,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tom's age is $T$ years, which is also the sum of the ages of his three children. His age $N$ years ago was twice the sum of their ages then. What is $T/N$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 3531,
"solution": "1. **Define the variables and set up the equation:**\n - Let $T$ be Tom's current age.\n - Let $N$ be the number of years ago when Tom's age was twice the sum of his children's ages.\n - The sum of the ages of Tom's three children is also $T$.\n\n2. **Calculat... |
deepscaler | [
{
"role": "user",
"content": "Given \\( x \\in \\mathbb{R} \\), find the maximum value of \\(\\frac{\\sin x(2-\\cos x)}{5-4 \\cos x}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 3532,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $S_{n}$ and $T_{n}$ represent the sum of the first $n$ terms of the arithmetic sequences ${a_{n}}$ and ${b_{n}}$, respectively. Given that $\\frac{S_{n}}{T_{n}} = \\frac{n+1}{2n-1}$, where $n \\in \\mathbb{N}^*$, find the value of $\\frac{a_{3} + a_{7}}{b_{1} + b_{9}}$. Let... | math | {
"style": "rule",
"ground_truth": "\\frac{10}{17}"
} | {
"split": "train",
"index": 3533,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 3534,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One day, Xiao Ming took 100 yuan to go shopping. In the first store, he bought several items of product A. In the second store, he bought several items of product B. In the third store, he bought several items of product C. In the fourth store, he bought several items of produc... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3535,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a middle-school mentoring program, a number of the sixth graders are paired with a ninth-grade student as a buddy. No ninth grader is assigned more than one sixth-grade buddy. If $\\frac{1}{3}$ of all the ninth graders are paired with $\\frac{2}{5}$ of all the sixth graders,... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{11}"
} | {
"split": "train",
"index": 3536,
"solution": "1. **Define Variables:**\n Let $s$ be the number of sixth graders, and $n$ be the number of ninth graders.\n\n2. **Set Up Equations:**\n According to the problem, $\\frac{1}{3}$ of all the ninth graders are paired with $\\frac{2}{5}$ of all the sixth graders. Th... |
deepscaler | [
{
"role": "user",
"content": "From noon till midnight, Clever Cat sleeps under the oak tree and from midnight till noon he is awake telling stories. A poster on the tree above him says \"Two hours ago, Clever Cat was doing the same thing as he will be doing in one hour's time\". For how many hours a day doe... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 3537,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the value of the following expression:\n\n$$\\left| \\, |{ -|{-1 + 1}| - 1 }| + 1\\right|.$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 3538,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many two-digit prime numbers can be formed by choosing two different digits from the set $\\{2, 7, 8, 9\\}$ to be used as the tens digit and units digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 3539,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If six geometric means are inserted between $16$ and $11664$, calculate the sixth term in the geometric series. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3888"
} | {
"split": "train",
"index": 3540,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $\\dbinom{20}{5}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11628"
} | {
"split": "train",
"index": 3541,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function $f(x) = \\cos(2x + \\frac{\\pi}{3}) + \\sqrt{3}\\sin(2x) + 2a$.\n\n(1) Find the intervals of monotonic increase for the function $f(x)$.\n\n(2) When $x \\in [0, \\frac{\\pi}{4}]$, the minimum value of $f(x)$ is 0. Find the maximum value of $f(x)$. Let's think s... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 3542,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two numbers is $30$. Their difference is $4$. What is the larger of the two numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 3543,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn the triangle \\(ABC\\), it is known that \\(AB=BC\\) and \\(\\angle BAC=45^\\circ\\). The line \\(MN\\) intersects side \\(AC\\) at point \\(M\\), and side \\(BC\\) at point \\(N\\). Given that \\(AM=2 \\cdot MC\\) and \\(\\angle NMC=60^\\circ\\), find the ratio of the are... | math | {
"style": "rule",
"ground_truth": "\\frac{7 - 3\\sqrt{3}}{11}"
} | {
"split": "train",
"index": 3544,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \\le a<b<c<d \\le 10$, and $a+d>b+c$. How many interesting ordered quadruples are there?\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 3545,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point $(x,y)$ is randomly picked from the rectangular region with vertices at $(0,0),(2010,0),(2010,2011),$ and $(0,2011)$. What is the probability that $x > 3y$? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "\\frac{670}{2011}"
} | {
"split": "train",
"index": 3546,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $XYZ$, $E$ lies on $\\overline{YZ}$ and $G$ lies on $\\overline{XY}$. Let $\\overline{XE}$ and $\\overline{YG}$ intersect at $Q.$\n\nIf $XQ:QE = 5:2$ and $GQ:QY = 3:4$, find $\\frac{XG}{GY}.$ Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 3547,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Calculate the perimeter of a triangle whose vertices are located at points $A(2,3)$, $B(2,10)$, and $C(8,6)$ on a Cartesian coordinate plane. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 + 2\\sqrt{13} + 3\\sqrt{5}"
} | {
"split": "train",
"index": 3548,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can four people sit in a row of five chairs? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 3549,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( 3x + 4 = x + 2 \\), what is the value of \\( x \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 3550,
"solution": "If \\( 3x + 4 = x + 2 \\), then \\( 3x - x = 2 - 4 \\) and so \\( 2x = -2 \\), which gives \\( x = -1 \\)."
} |
deepscaler | [
{
"role": "user",
"content": "Let $r(x)$ have a domain $\\{0,1,2,3\\}$ and a range $\\{1,3,5,7\\}$. Let $s(x)$ be defined on the domain $\\{1,2,3,4,5,6\\}$ with the function rule $s(x) = 2x + 1$. Determine the sum of all possible values of $s(r(x))$ where $r(x)$ outputs only odd numbers. Let's think step by... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 3551,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "When $\\sqrt[4]{2^7\\cdot3^3}$ is fully simplified, the result is $a\\sqrt[4]{b}$, where $a$ and $b$ are positive integers. What is $a+b$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "218"
} | {
"split": "train",
"index": 3552,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For a point $P=(x, y)$ in the Cartesian plane, let $f(P)=\\left(x^{2}-y^{2}, 2 x y-y^{2}\\right)$. If $S$ is the set of all $P$ so that the sequence $P, f(P), f(f(P)), f(f(f(P))), \\ldots$ approaches $(0,0)$, then the area of $S$ can be expressed as $\\pi \\sqrt{r}$ for some po... | math | {
"style": "rule",
"ground_truth": "133"
} | {
"split": "train",
"index": 3553,
"solution": "For a point $P=(x, y)$, let $z(P)=x+y \\omega$, where $\\omega$ is a nontrivial third root of unity. Then $$\\begin{aligned} z(f(P))=\\left(x^{2}-y^{2}\\right)+\\left(2 x y-y^{2}\\right) \\omega=x^{2}+2 x y \\omega+y^{2} & (-1-\\omega) \\\\ & =x^{2}+2 x y \\omega+y^... |
deepscaler | [
{
"role": "user",
"content": "John ate a total of 120 peanuts over four consecutive nights. Each night he ate 6 more peanuts than the night before. How many peanuts did he eat on the fourth night? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "39"
} | {
"split": "train",
"index": 3554,
"solution": "Suppose that John ate \\( x \\) peanuts on the fourth night. Since he ate 6 more peanuts each night than on the previous night, then he ate \\( x-6 \\) peanuts on the third night, \\((x-6)-6=x-12\\) peanuts on the second night, and \\((x-12)-6=x-18\\) peanuts on the... |
deepscaler | [
{
"role": "user",
"content": "Calculate the definite integral:\n$$\n\\int_{0}^{\\pi / 4} \\frac{5 \\operatorname{tg} x+2}{2 \\sin 2 x+5} d x\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2} \\ln \\left(\\frac{14}{5}\\right)"
} | {
"split": "train",
"index": 3555,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let \\( a, b, c \\) be real numbers such that \\( 9a^2 + 4b^2 + 25c^2 = 1 \\). Find the maximum value of \n\\[ 3a + 4b + 5c. \\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}"
} | {
"split": "train",
"index": 3556,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If a computer executes the following program:\n1. Initial values \\( x = 3 \\), \\( S = 0 \\)\n2. \\( x = x + 2 \\)\n3. \\( S = S + x \\)\n4. If \\( S \\geq 10000 \\), go to step 5; otherwise, repeat from step 2\n5. Print \\( x \\)\n6. Stop\n\nWhat value will be printed by step... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 3557,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "To ensure the safety of property during the Spring Festival holiday, an office needs to arrange for one person to be on duty each day for seven days. Given that there are 4 people in the office, and each person needs to work for either one or two days, the number of different d... | math | {
"style": "rule",
"ground_truth": "2520"
} | {
"split": "train",
"index": 3558,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The polynomial $Q(x) = 3x^3 + dx^2 + ex + f$ has the property that the mean of its zeros, the product of its zeros, and the sum of the coefficients are all equal. The $y$-intercept of the graph of $y = Q(x)$ is 9. What is $e$? Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "-42"
} | {
"split": "train",
"index": 3559,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are three video game systems: the Paystation, the WHAT, and the ZBoz2 \\pi, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBo... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{25}"
} | {
"split": "train",
"index": 3560,
"solution": "Since the games are not necessarily distinct, probabilities are independent. Multiplying the odds that each nephew receives a game he can play, we get $10 / 20 \\cdot 14 / 20 \\cdot 16 / 20=7 / 25$."
} |
deepscaler | [
{
"role": "user",
"content": "In the sequence $\\{a_n\\}$, $a_1 = 1$, $a_2 = 2$, and $a_{n+2} - a_n = 1 + (-1)^n$ $(n \\in \\mathbb{N}^*)$, then $S_{100} = \\_\\_\\_\\_\\_\\_\\_\\_$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2600"
} | {
"split": "train",
"index": 3561,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given triangle \\( \\triangle ABC \\) with \\( AB < AC \\), the altitude \\( AD \\), angle bisector \\( AE \\), and median \\( AF \\) are drawn from \\( A \\), with \\( D, E, F \\) all lying on \\(\\overline{BC}\\). If \\( \\angle BAD = 2 \\angle DAE = 2 \\angle EAF = \\angle F... | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 3562,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The formula for converting a Fahrenheit temperature $F$ to the corresponding Celsius temperature $C$ is $C = \\frac{5}{9}(F-32).$ An integer Fahrenheit temperature is converted to Celsius, rounded to the nearest integer, converted back to Fahrenheit, and again rounded to the ne... | math | {
"style": "rule",
"ground_truth": "539"
} | {
"split": "train",
"index": 3563,
"solution": "Examine $F - 32$ modulo 9.\nIf $F - 32 \\equiv 0 \\pmod{9}$, then we can define $9x = F - 32$. This shows that $F = \\left[\\frac{9}{5}\\left[\\frac{5}{9}(F-32)\\right] + 32\\right] \\Longrightarrow F = \\left[\\frac{9}{5}(5x) + 32\\right] \\Longrightarrow F = 9x + ... |
deepscaler | [
{
"role": "user",
"content": "A coin is flipped eight times, and the sequence of heads and tails occurring is recorded. How many distinct sequences are possible? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"split": "train",
"index": 3564,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Print 90,000 five-digit numbers\n$$\n10000, 10001, \\cdots, 99999\n$$\n\non cards, with each card displaying one five-digit number. Some numbers printed on the cards (e.g., 19806 when reversed reads 90861) can be read in two different ways and may cause confusion. How many card... | math | {
"style": "rule",
"ground_truth": "89100"
} | {
"split": "train",
"index": 3565,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A sequence $ (S_n), n \\geq 1$ of sets of natural numbers with $ S_1 = \\{1\\}, S_2 = \\{2\\}$ and\r\n\\[{ S_{n + 1} = \\{k \\in }\\mathbb{N}|k - 1 \\in S_n \\text{ XOR } k \\in S_{n - 1}\\}.\r\n\\]\r\nDetermine $ S_{1024}.$ Let's think step by step and put only the final... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"split": "train",
"index": 3566,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For an integer $n$, let $f_{9}(n)$ denote the number of positive integers $d \\leq 9$ dividing $n$. Suppose that $m$ is a positive integer and $b_{1}, b_{2}, \\ldots, b_{m}$ are real numbers such that $f_{9}(n)=\\sum_{j=1}^{m} b_{j} f_{9}(n-j)$ for all $n>m$. Find the smallest ... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3567,
"solution": "Let $M=9$. Consider the generating function $$F(x)=\\sum_{n \\geq 1} f_{M}(n) x^{n}=\\sum_{d=1}^{M} \\sum_{k \\geq 1} x^{d k}=\\sum_{d=1}^{M} \\frac{x^{d}}{1-x^{d}}$$ Observe that $f_{M}(n)=f_{M}\\left(n+M\\right.$ !) for all $n \\geq 1$ (in fact, all $n \\leq 0$ as... |
deepscaler | [
{
"role": "user",
"content": "Let $r,$ $s,$ and $t$ be the roots of the equation $x^3 - 20x^2 + 18x - 7 = 0.$ Find the value of $(1+r)(1+s)(1+t).$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 3568,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $p$ and $q$ be real numbers, and suppose that the roots of the equation \\[x^3 - 9x^2 + px - q = 0\\] are three distinct positive integers. Compute $p + q.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 3569,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the smallest integer $B$ such that there exists a sequence of consecutive integers, which must include at least one odd and one even number, adding up to 2023. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1011"
} | {
"split": "train",
"index": 3570,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the ellipse C: $mx^2+3my^2=1$ ($m>0$) with a major axis length of $2\\sqrt{6}$, and O as the origin. \n(1) Find the equation of ellipse C and its eccentricity. \n(2) Let point A be (3,0), point B be on the y-axis, and point P be on ellipse C, with point P on the right s... | math | {
"style": "rule",
"ground_truth": "3\\sqrt{3}"
} | {
"split": "train",
"index": 3571,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn trapezoid \\(A B C D\\), the base \\(A D\\) is four times larger than \\(B C\\). A line passing through the midpoint of diagonal \\(B D\\) and parallel to \\(A B\\) intersects side \\(C D\\) at point \\(K\\). Find the ratio \\(D K : K C\\). Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "2:1"
} | {
"split": "train",
"index": 3572,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( K \\) is the midpoint of edge \\( A A_{1} \\) of cube \\( A B C D A_{1} B_{1} C_{1} D_{1} \\), and point \\( L \\) lies on edge \\( B C \\). Segment \\( K L \\) touches the sphere inscribed in the cube. In what ratio does the point of tangency divide segment \\( K L \... | math | {
"style": "rule",
"ground_truth": "4/5"
} | {
"split": "train",
"index": 3573,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The real roots of the equations \\( x^{5} + x + 1 = 0 \\) and \\( x + \\sqrt[5]{x} + 1 = 0 \\) are \\(\\alpha\\) and \\(\\beta\\), respectively. What is the value of \\(\\alpha + \\beta\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 3574,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a five-digit number without repeated digits using 0, 1, 2, 3, and 4, with the condition that even and odd digits must be adjacent to each other. Find the total number of such five-digit numbers. Let's think step by step and put only the final answer within \\boxed{}."... | math | {
"style": "rule",
"ground_truth": "20"
} | {
"split": "train",
"index": 3575,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rational numbers $a$ and $b$ are chosen at random among all rational numbers in the interval $[0,2)$ that can be written as fractions $\\frac{n}{d}$ where $n$ and $d$ are integers with $1 \\le d \\le 5$. What is the probability that \\[(\\text{cos}(a\\pi)+i\\text{sin}(b\\pi))^4... | math | {
"style": "rule",
"ground_truth": "\\frac{6}{25}"
} | {
"split": "train",
"index": 3576,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $A(7,8)$, $B(10,4)$, $C(2,-4)$, then $S_{\\triangle ABC}$ is ______. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3577,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given:\n\\\\((1)y=x+ \\\\frac {4}{x}\\\\)\n\\\\((2)y=\\\\sin x+ \\\\frac {4}{\\\\sin x}(0 < x < π)\\\\)\n\\\\((3)y= \\\\frac {x^{2}+13}{ \\\\sqrt {x^{2}+9}}\\\\)\n\\\\((4)y=4⋅2^{x}+2^{-x}\\\\)\n\\\\((5)y=\\\\log \\_{3}x+4\\\\log \\_{x}3(0 < x < 1)\\\\)\n\nFind the function(s) w... | math | {
"style": "rule",
"ground_truth": "(4)"
} | {
"split": "train",
"index": 3578,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Express the quotient $2213_4 \\div 13_4$ in base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "53_4"
} | {
"split": "train",
"index": 3579,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all 4-digit numbers $\\overline{abcd}$ that are multiples of $11$ , such that the 2-digit number $\\overline{ac}$ is a multiple of $7$ and $a + b + c + d = d^2$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3454"
} | {
"split": "train",
"index": 3580,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\frac{\\cos 85^{\\circ} + \\sin 25^{\\circ} \\cos 30^{\\circ}}{\\cos 25^{\\circ}}$ simplify to a specific value. 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": 3581,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder. Express your answer as a common fraction. Let's thin... | math | {
"style": "rule",
"ground_truth": "\\frac{30}{11}"
} | {
"split": "train",
"index": 3582,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct numbers are selected simultaneously and at random from the set $\\{1, 2, 3, 4, 5\\}$. What is the probability that the smaller one divides the larger one? Express your answer as a common fraction. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 3583,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle \\(ABC\\), the sides \\(a\\), \\(b\\), and \\(c\\) are opposite to angles \\(A\\), \\(B\\), and \\(C\\) respectively, with \\(a-2b=0\\).\n1. If \\(B= \\dfrac{\\pi}{6}\\), find \\(C\\).\n2. If \\(C= \\dfrac{2}{3}\\pi\\) and \\(c=14\\), find the area of \\(\\triangle ... | math | {
"style": "rule",
"ground_truth": "14 \\sqrt{3}"
} | {
"split": "train",
"index": 3584,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "$\\frac{2}{25}=$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": ".08"
} | {
"split": "train",
"index": 3585,
"solution": "To convert the fraction $\\frac{2}{25}$ into a decimal, we can multiply the numerator and the denominator by a number that makes the denominator a power of 10, which simplifies the division process.\n\n1. **Choosing the multiplier**: We choose 4 because multiplying ... |
deepscaler | [
{
"role": "user",
"content": "The number \\( n \\) has exactly six divisors (including 1 and itself). These divisors are arranged in ascending order. It turns out that the third divisor is seven times larger than the second one, and the fourth divisor is 10 larger than the third one. What is \\( n \\)? Let'... | math | {
"style": "rule",
"ground_truth": "2891"
} | {
"split": "train",
"index": 3586,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify and evaluate:\n(1) Calculate the value of $\\frac {1}{\\log_{4}6}+6^{\\log_{6} \\sqrt {3}-1}-2\\log_{6} \\frac {1}{3}$;\n(2) Given $\\tan\\alpha=2$ and $\\sin\\alpha+\\cos\\alpha < 0$, find the value of $\\frac {\\tan(\\pi-\\alpha)\\cdot \\sin(-\\alpha+ \\frac {3\\pi}{... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 3587,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "ABCD is a rectangle, D is the center of the circle, and B is on the circle. If AD=4 and CD=3, then the area of the shaded region is between Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7 and 8"
} | {
"split": "train",
"index": 3588,
"solution": "1. **Identify the given information**: We are given a rectangle ABCD with D as the center of a circle and B on the circle. The lengths AD and CD are given as 4 and 3 respectively.\n\n2. **Calculate the diagonal AC using the Pythagorean Theorem**: Since ABCD is a rec... |
deepscaler | [
{
"role": "user",
"content": "Find the product of all constants \\(t\\) such that the quadratic \\(x^2 + tx + 12\\) can be factored in the form \\((x+a)(x+b)\\), where \\(a\\) and \\(b\\) are integers. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-530,784"
} | {
"split": "train",
"index": 3589,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There is a string of lights with a recurrent pattern of three blue lights followed by four yellow lights, spaced 7 inches apart. Determine the distance in feet between the 4th blue light and the 25th blue light, given that 1 foot equals 12 inches. Let's think step by step and p... | math | {
"style": "rule",
"ground_truth": "28"
} | {
"split": "train",
"index": 3590,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For any real number $a$, let $\\left[a\\right]$ denote the largest integer not exceeding $a$. For example, $\\left[4\\right]=4$, $[\\sqrt{3}]=1$. Now, for the number $72$, the following operations are performed: $72\\stackrel{1st}{→}[\\sqrt{72}]=8\\stackrel{2nd}{→}[\\sqrt{8}]=2... | math | {
"style": "rule",
"ground_truth": "6560"
} | {
"split": "train",
"index": 3591,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangle $ABC$ has its vertices $A$, $B$, and $C$ on the sides of a rectangle 4 units by 5 units as shown. What is the area of triangle $ABC$ in square units?\n\n[asy]\n\nfill((0,1)--(4,0)--(2,5)--cycle,lightgray);\n\nfor(int i=1; i < 5; ++i){\nfor(int k=1; k < 4; ++k){\ndraw((... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 3592,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Square $PQRS$ has a side length of $2$ units. Points $T$ and $U$ are on sides $PQ$ and $QR$, respectively, with $PT = QU$. When the square is folded along the lines $ST$ and $SU$, sides $PS$ and $RS$ coincide and lie along diagonal $RQ$. Exprress the length of segment $PT$ in t... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 3593,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Violet has one-half of the money she needs to buy her mother a necklace. After her sister gives her $\\$30$, she has three-quarters of the amount she needs. How much will Violet's father give her? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "$30"
} | {
"split": "train",
"index": 3594,
"solution": "Violet starts with one-half of the money that she needed to buy the necklace. After her sister gives her money, she has three-quarters of the amount that she needs. This means that her sister gave her $\\frac{3}{4}-\\frac{1}{2}=\\frac{1}{4}$ of the total amount that... |
deepscaler | [
{
"role": "user",
"content": "For positive numbers $x$ and $y$ the operation $\\spadesuit (x,y)$ is defined as \n\\[\\spadesuit (x,y) = x-\\dfrac{1}{y}\\]\nWhat is $\\spadesuit (2,\\spadesuit (2,2))$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{4}{3}"
} | {
"split": "train",
"index": 3595,
"solution": "1. **Understanding the operation $\\spadesuit$:** The operation $\\spadesuit(x, y)$ is defined as $x - \\frac{1}{y}$. \n\n2. **Calculate $\\spadesuit(2, 2)$:**\n \\[\n \\spadesuit(2, 2) = 2 - \\frac{1}{2} = 2 - 0.5 = 1.5\n \\]\n\n3. **Substitute $\\spadesuit(2... |
deepscaler | [
{
"role": "user",
"content": "How many positive integers less than $800$ are either a perfect cube or a perfect square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"split": "train",
"index": 3596,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.\nBrian: \"M... | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 3597,
"solution": "1. **Analyze the statements:**\n - Brian: \"Mike and I are different species.\"\n - Chris: \"LeRoy is a frog.\"\n - LeRoy: \"Chris is a frog.\"\n - Mike: \"Of the four of us, at least two are toads.\"\n\n2. **Consider the implications of Chris and LeRoy's st... |
deepscaler | [
{
"role": "user",
"content": "Given $a \\star b = a^2 + 2ab + b^2$, what is the value of $a \\star b$ when $a = 4$ and $b = 6?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 3598,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many values of \\(c\\) in the interval \\([0, 2000]\\) does the equation\n\\[8 \\lfloor x \\rfloor + 3 \\lceil x \\rceil = c\\]\nhave a solution for \\(x\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "363"
} | {
"split": "train",
"index": 3599,
"solution": ""
} |
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