data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "How many positive odd integers greater than 1 and less than 200 are square-free? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 8700,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( n = 7 \\), which of the following expressions is equal to an even integer: \\( 9n, n+8, n^2, n(n-2), 8n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "8n"
} | {
"split": "train",
"index": 8701,
"solution": "When \\( n=7 \\), we have \\( 9n=63, n+8=15, n^2=49, n(n-2)=35, 8n=56 \\). Therefore, \\( 8n \\) is even. For every integer \\( n \\), the expression \\( 8n \\) is equal to an even integer."
} |
deepscaler | [
{
"role": "user",
"content": "A board of size \\(2022 \\times 2022\\) is given. Lisa and Varya take turns painting \\(2 \\times 2\\) squares on the board with red and blue colors. They agreed that each cell can be painted no more than once in blue and no more than once in red. Cells that are painted blue an... | math | {
"style": "rule",
"ground_truth": "2021 \\cdot 2020"
} | {
"split": "train",
"index": 8702,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, we have $AC = BC = 10$, and $AB = 8$. Suppose that $D$ is a point on line $AB$ such that $B$ lies between $A$ and $D$ and $CD = 12$. What is $BD$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2\\sqrt{15}"
} | {
"split": "train",
"index": 8703,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the value of \\(1^{25}+2^{24}+3^{23}+\\ldots+24^{2}+25^{1}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "66071772829247409"
} | {
"split": "train",
"index": 8704,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a tetrahedron $P-ABC$, in the base $\\triangle ABC$, $\\angle BAC=60^{\\circ}$, $BC=\\sqrt{3}$, $PA\\perp$ plane $ABC$, $PA=2$, then the surface area of the circumscribed sphere of this tetrahedron is ______. Let's think step by step and put only the final answer within \... | math | {
"style": "rule",
"ground_truth": "8\\pi"
} | {
"split": "train",
"index": 8705,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the last three digits of $9^{105}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "049"
} | {
"split": "train",
"index": 8706,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Initially, the numbers 1 and 2 are written at opposite positions on a circle. Each operation consists of writing the sum of two adjacent numbers between them. For example, the first operation writes two 3's, and the second operation writes two 4's and two 5's. After each operat... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 8707,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The product of two positive consecutive integers is 506. What is their sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "45"
} | {
"split": "train",
"index": 8708,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many positive integers \\( n \\) is the sum\n\\[\n(-n)^{3} + (-n+1)^{3} + \\cdots + (n-2)^{3} + (n-1)^{3} + n^{3} + (n+1)^{3}\n\\]\nless than \\( 3129 \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 8709,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 8710,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\\sqrt{m} + \\sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ Let's think step by step and put only the final answer within \\bo... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 8711,
"solution": "1. **Identify Key Triangles and Midpoint**: Let $M$ be the midpoint of $CD$. Since each side of the pentagon and the internal segments are of length $2$, and given the symmetry and equal lengths, we can infer that triangles $AED$ and $ABC$ are isosceles with a verte... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle ABC$, $a$, $b$, and $c$ are the sides opposite to angles $A$, $B$, and $C$ respectively, and it is given that $\\sin\\left(A+ \\frac{\\pi}{3}\\right) = 4\\sin \\frac{A}{2}\\cos \\frac{A}{2}$. \n(Ⅰ) Find the magnitude of angle $A$; \n(Ⅱ) If $\\sin B= \\sqrt{3}\\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{4}"
} | {
"split": "train",
"index": 8712,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In writing the integers from 20 through 99 inclusive, how many times is the digit 7 written? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8713,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Lydia likes a five-digit number if none of its digits are divisible by 3. Find the total sum of the digits of all five-digit numbers that Lydia likes. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "174960"
} | {
"split": "train",
"index": 8714,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Charles has two six-sided die. One of the die is fair, and the other die is biased so that it comes up six with probability $\\frac{2}{3}$ and each of the other five sides has probability $\\frac{1}{15}$. Charles chooses one of the two dice at random and rolls it three times. G... | math | {
"style": "rule",
"ground_truth": "167"
} | {
"split": "train",
"index": 8715,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $A=3x^{2}-x+2y-4xy$ and $B=2x^{2}-3x-y+xy$.\n\n$(1)$ Simplify $2A-3B$.\n\n$(2)$ When $x+y=\\frac{6}{7}$ and $xy=-1$, find the value of $2A-3B$.\n\n$(3)$ If the value of $2A-3B$ is independent of the value of $y$, find the value of $2A-3B$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{49}{11}"
} | {
"split": "train",
"index": 8716,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common divisor of $5!$ and $\\frac{8!}{3!}?$ Express your answer as an integer. (Reminder: If $n$ is a positive integer, then $n!$ stands for the product $1\\cdot 2\\cdot 3\\cdot \\cdots \\cdot (n-1)\\cdot n$.) Let's think step by step and put only the fin... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 8717,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a Cartesian coordinate system, the points where both the x-coordinate and y-coordinate are integers are called lattice points. How many lattice points (x, y) satisfy the inequality \\((|x|-1)^{2}+(|y|-1)^{2}<2\\)? Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 8718,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $r=H_{1}$ be the answer to this problem. Given that $r$ is a nonzero real number, what is the value of $r^{4}+4 r^{3}+6 r^{2}+4 r ?$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 8719,
"solution": "Since $H_{1}$ is the answer, we know $r^{4}+4 r^{3}+6 r^{2}+4 r=r \\Rightarrow(r+1)^{4}=r+1$. Either $r+1=0$, or $(r+1)^{3}=1 \\Rightarrow r=0$. Since $r$ is nonzero, $r=-1$."
} |
deepscaler | [
{
"role": "user",
"content": "Two distinct numbers a and b are chosen randomly from the set $\\{3, 3^2, 3^3, ..., 3^{15}\\}$. What is the probability that $\\mathrm{log}_a b$ is an integer?\nA) $\\frac{1}{10}$\nB) $\\frac{2}{7}$\nC) $\\frac{1}{7}$\nD) $\\frac{1}{5}$ Let's think step by step and put only the... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{7}"
} | {
"split": "train",
"index": 8720,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest integer less than 100 for which the greatest common divisor of that integer and 12 is 4? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "92"
} | {
"split": "train",
"index": 8721,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a fixed circle $\\odot P$ with a radius of 1, the distance from the center $P$ to a fixed line $l$ is 2. Point $Q$ is a moving point on $l$, and circle $\\odot Q$ is externally tangent to circle $\\odot P$. Circle $\\odot Q$ intersects $l$ at points $M$ and $N$. For any d... | math | {
"style": "rule",
"ground_truth": "60"
} | {
"split": "train",
"index": 8722,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of $x^{70}$ in the expansion of\n\\[(x - 1)(x^2 - 2)(x^3 - 3) \\dotsm (x^{11} - 11)(x^{12} - 12).\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 8723,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the addition problem shown, $m, n, p$, and $q$ represent positive digits. What is the value of $m+n+p+q$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 8724,
"solution": "From the ones column, we see that $3 + 2 + q$ must have a ones digit of 2. Since $q$ is between 1 and 9, inclusive, then $3 + 2 + q$ is between 6 and 14. Since its ones digit is 2, then $3 + 2 + q = 12$ and so $q = 7$. This also means that there is a carry of 1 into... |
deepscaler | [
{
"role": "user",
"content": "If $a$, $b\\in \\{-1,1,2\\}$, then the probability that the function $f\\left(x\\right)=ax^{2}+2x+b$ has a zero point is ______. 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": 8725,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 3-digit positive integers have exactly one even digit? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "350"
} | {
"split": "train",
"index": 8726,
"solution": "We write a general three-digit positive integer in terms of its digits as $A B C$. There are 9 possible values for the digit $A$ (the digits 1 to 9) and 10 possible values for each of $B$ and $C$ (the digits 0 to 9). We want to count the number of such integers with... |
deepscaler | [
{
"role": "user",
"content": "Jessica has exactly one of each of the first 30 states' new U.S. quarters. The quarters were released in the same order that the states joined the union. The graph below shows the number of states that joined the union in each decade. What fraction of Jessica's 30 coins represe... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{6}"
} | {
"split": "train",
"index": 8727,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find out how many positive integers $n$ not larger than $2009$ exist such that the last digit of $n^{20}$ is $1$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "804"
} | {
"split": "train",
"index": 8728,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers between $500$ and $1000$ contain both the digits $3$ and $4$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 8729,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A [i]permutation[/i] of the set of positive integers $[n] = \\{1, 2, . . . , n\\}$ is a sequence $(a_1 , a_2 , \\ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P... | math | {
"style": "rule",
"ground_truth": "4489"
} | {
"split": "train",
"index": 8730,
"solution": "\nTo solve this problem, we will analyze the given condition involving permutations and perfect squares to determine the smallest \\( n \\) such that \\( P(n) \\), the number of permutations of \\([n] = \\{1, 2, \\ldots, n\\}\\) where \\( ka_k \\) is a perfect squar... |
deepscaler | [
{
"role": "user",
"content": "In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 8731,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In equiangular octagon $CAROLINE$, $CA = RO = LI = NE =$ $\\sqrt{2}$ and $AR = OL = IN = EC = 1$. The self-intersecting octagon $CORNELIA$ encloses six non-overlapping triangular regions. Let $K$ be the area enclosed by $CORNELIA$, that is, the total area of the six triangular ... | math | {
"style": "rule",
"ground_truth": "23"
} | {
"split": "train",
"index": 8732,
"solution": "We can draw $CORNELIA$ and introduce some points.\n The diagram is essentially a 3x3 grid where each of the 9 squares making up the grid have a side length of 1.\n In order to find the area of $CORNELIA$, we need to find 4 times the area of $\\bigtriangleup$$ACY$ an... |
deepscaler | [
{
"role": "user",
"content": "$A B C D$ is a cyclic quadrilateral in which $A B=4, B C=3, C D=2$, and $A D=5$. Diagonals $A C$ and $B D$ intersect at $X$. A circle $\\omega$ passes through $A$ and is tangent to $B D$ at $X . \\omega$ intersects $A B$ and $A D$ at $Y$ and $Z$ respectively. Compute $Y Z / B D... | math | {
"style": "rule",
"ground_truth": "\\frac{115}{143}"
} | {
"split": "train",
"index": 8733,
"solution": "Denote the lengths $A B, B C, C D$, and $D A$ by $a, b, c$, and $d$ respectively. Because $A B C D$ is cyclic, $\\triangle A B X \\sim \\triangle D C X$ and $\\triangle A D X \\sim \\triangle B C X$. It follows that $\\frac{A X}{D X}=\\frac{B X}{C X}=\\frac{a}{c}$ a... |
deepscaler | [
{
"role": "user",
"content": "The product of three even consecutive positive integers is twenty times their sum. What is the sum of the three integers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 8734,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The faces of a cubical die are marked with the numbers $1$, $2$, $3$, $3$, $4$, and $4$. Another die's faces are marked with $2$, $2$, $5$, $6$, $7$, and $8$. Find the probability that the sum of the top two numbers will be $7$, $9$, or $11$.\nA) $\\frac{1}{18}$\nB) $\\frac{1}{... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{9}"
} | {
"split": "train",
"index": 8735,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a, b$ and $c$ be positive real numbers such that $$\\begin{aligned} a^{2}+a b+b^{2} & =9 \\\\ b^{2}+b c+c^{2} & =52 \\\\ c^{2}+c a+a^{2} & =49 \\end{aligned}$$ Compute the value of $\\frac{49 b^{2}-33 b c+9 c^{2}}{a^{2}}$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "52"
} | {
"split": "train",
"index": 8736,
"solution": "Consider a triangle $A B C$ with Fermat point $P$ such that $A P=a, B P=b, C P=c$. Then $$A B^{2}=A P^{2}+B P^{2}-2 A P \\cdot B P \\cos \\left(120^{\\circ}\\right)$$ by the Law of Cosines, which becomes $$A B^{2}=a^{2}+a b+b^{2}$$ and hence $A B=3$. Similarly, $B C... |
deepscaler | [
{
"role": "user",
"content": "The mean age of Amanda's $4$ cousins is $8$, and their median age is $5$. What is the sum of the ages of Amanda's youngest and oldest cousins? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"split": "train",
"index": 8737,
"solution": "1. **Calculate the total sum of the ages**: Given that the mean age of Amanda's 4 cousins is 8, the total sum of their ages is calculated by multiplying the mean by the number of cousins:\n \\[\n \\text{Total sum} = 4 \\times 8 = 32\n \\]\n\n2. **Understand th... |
deepscaler | [
{
"role": "user",
"content": "Line segment $\\overline{AB}$ is extended past $B$ to $P$ such that $AP:PB = 10:3.$ Then\n\\[\\overrightarrow{P} = t \\overrightarrow{A} + u \\overrightarrow{B}\\]for some constants $t$ and $u.$ Enter the ordered pair $(t,u).$\n\n[asy]\nunitsize(1 cm);\n\npair A, B, P;\n\nA =... | math | {
"style": "rule",
"ground_truth": "\\left( -\\frac{3}{7}, \\frac{10}{7} \\right)"
} | {
"split": "train",
"index": 8738,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $60$ people are wearing sunglasses, $40$ people are wearing caps, and a randomly selected person wearing a cap has a $\\frac{2}{5}$ probability of also wearing sunglasses, find the probability that a person wearing sunglasses is also wearing a cap. Let's think step b... | math | {
"style": "rule",
"ground_truth": "\\frac{4}{15}"
} | {
"split": "train",
"index": 8739,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $\\frac{1^{4y-1}}{5^{-1}+3^{-1}}$ is equal to: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{15}{8}"
} | {
"split": "train",
"index": 8740,
"solution": "1. **Simplify the Numerator**: \n The expression $1^{4y-1}$ simplifies to $1$ for any real number $y$, because any power of $1$ is $1$. Thus, the expression becomes:\n \\[\n \\frac{1^{4y-1}}{5^{-1} + 3^{-1}} = \\frac{1}{5^{-1} + 3^{-1}}\n \\]\n\n2. **Simplif... |
deepscaler | [
{
"role": "user",
"content": "The mean of the set of numbers $\\{106, 102, 95, 103, 100, y, x\\}$ is 104. What is the median of this set of seven numbers? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "103"
} | {
"split": "train",
"index": 8741,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a$ and $b$ be integers such that $ab = 72.$ Find the minimum value of $a + b.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-73"
} | {
"split": "train",
"index": 8742,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $\\triangle ABC$, $a$, $b$, $c$ are the opposite sides of the internal angles $A$, $B$, $C$, respectively, and $\\sin ^{2}A+\\sin A\\sin C+\\sin ^{2}C+\\cos ^{2}B=1$. \n$(1)$ Find the measure of angle $B$; \n$(2)$ If $a=5$, $b=7$, find $\\sin C$. Let's think step ... | math | {
"style": "rule",
"ground_truth": "\\frac{3\\sqrt{3}}{14}"
} | {
"split": "train",
"index": 8743,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=3\\sin \\left(2x-\\frac{\\pi }{8}\\right)$, determine the horizontal shift required to transform the graph of the function $y=3\\sin 2x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{8}"
} | {
"split": "train",
"index": 8744,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Alicia, Brenda, and Colby were the candidates in a recent election for student president. The pie chart below shows how the votes were distributed among the three candidates. If Brenda received $36$ votes, then how many votes were cast all together? Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "120"
} | {
"split": "train",
"index": 8745,
"solution": "1. **Understanding the problem**: We are given that Brenda received $36$ votes, which represents $\\frac{3}{10}$ of the total votes in the election.\n\n2. **Calculating the total number of votes**:\n - Since $36$ votes is $\\frac{3}{10}$ of the total votes, we can... |
deepscaler | [
{
"role": "user",
"content": "Calculate 8 divided by $\\frac{1}{8}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "64"
} | {
"split": "train",
"index": 8746,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "From the numbers 1, 2, 3, 5, 7, 8, two numbers are randomly selected and added together. Among the different sums that can be obtained, let the number of sums that are multiples of 2 be $a$, and the number of sums that are multiples of 3 be $b$. Then, the median of the sample 6... | math | {
"style": "rule",
"ground_truth": "5.5"
} | {
"split": "train",
"index": 8747,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The parabolas defined by the equations $y=2x^2-4x+4$ and $y=-x^2-2x+4$ intersect at points $(a,b)$ and $(c,d)$, where $c\\ge a$. What is $c-a$? 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{2}{3}"
} | {
"split": "train",
"index": 8748,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Suppose a cube has a side length of $8$. Its vertices are alternately colored black and green. What is the volume of the tetrahedron whose vertices are the green vertices of the cube? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{512}{3}"
} | {
"split": "train",
"index": 8749,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $f(x)= \\frac{1}{2^{x}+ \\sqrt {2}}$, use the method for deriving the sum of the first $n$ terms of an arithmetic sequence to find the value of $f(-5)+f(-4)+…+f(0)+…+f(5)+f(6)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 \\sqrt {2}"
} | {
"split": "train",
"index": 8750,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the slope of a line parallel to $2x+4y=-17$? 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}{2}"
} | {
"split": "train",
"index": 8751,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many positive integers $N$ less than $1000$ are there such that the equation $x^{\\lfloor x\\rfloor} = N$ has a solution for $x$? (The notation $\\lfloor x\\rfloor$ denotes the greatest integer that is less than or equal to $x$.) Let's think step by step and put only the fi... | math | {
"style": "rule",
"ground_truth": "412"
} | {
"split": "train",
"index": 8752,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three dice with faces numbered 1 through 6 are placed vertically in a stack as described. The visible numbers on these dice are 2, 2, 3, 3, 4, 5, 6. Calculate the total number of dots that are NOT visible on these dice. Let's think step by step and put only the final answer wit... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"split": "train",
"index": 8753,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a given plane, points $A$ and $B$ are $10$ units apart. How many points $C$ are there in the plane such that the perimeter of $\\triangle ABC$ is $50$ units and the area of $\\triangle ABC$ is $100$ square units? Let's think step by step and put only the final answer within ... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 8754,
"solution": "1. **Identify the fixed elements and set up the problem**: Given that points $A$ and $B$ are $10$ units apart, we can place them at coordinates $(0,0)$ and $(10,0)$ respectively without loss of generality. This simplifies the problem to finding a point $C$ such that... |
deepscaler | [
{
"role": "user",
"content": "Let $\\pi$ be a randomly chosen permutation of the numbers from 1 through 2012. Find the probability that $\\pi(\\pi(2012))=2012$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{1006}"
} | {
"split": "train",
"index": 8755,
"solution": "There are two possibilities: either $\\pi(2012)=2012$ or $\\pi(2012)=i$ and $\\pi(i)=2012$ for $i \\neq 2012$. The first case occurs with probability $2011!/ 2012!=1 / 2012$, since any permutation on the remaining 2011 elements is possible. Similarly, for any fixed ... |
deepscaler | [
{
"role": "user",
"content": "Convert $12012_3$ to a base 10 integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "140"
} | {
"split": "train",
"index": 8756,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the sum of the coefficients of the polynomial $P(x)=x^{4}-29 x^{3}+a x^{2}+b x+c$, given that $P(5)=11, P(11)=17$, and $P(17)=23$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3193"
} | {
"split": "train",
"index": 8757,
"solution": "Define $Q(x)=P(x)-x-6=x^{4}-29 x^{3}+a x^{2}+(b-1)x+(c-6)$ and notice that $Q(5)=Q(11)=Q(17)=0$. $Q(x)$ has degree 4 and by Vieta's Formulas the sum of its roots is 29, so its last root is $29-17-11-5=-4$, giving us $Q(x)=(x-5)(x-11)(x-17)(x+4)$. This means that $P(... |
deepscaler | [
{
"role": "user",
"content": "In an acute-angled $\\triangle ABC$, find the minimum value of $3 \\tan B \\tan C + 2 \\tan A \\tan C + \\tan A \\tan B$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6 + 2\\sqrt{3} + 2\\sqrt{2} + 2\\sqrt{6}"
} | {
"split": "train",
"index": 8758,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $PQR,$ points $M$ and $N$ are on $\\overline{PQ}$ and $\\overline{PR},$ respectively, and angle bisector $\\overline{PS}$ intersects $\\overline{MN}$ at $T.$ If $PM = 2,$ $MQ = 6,$ $PN = 4,$ and $NR = 8,$ compute $\\frac{PT}{PS}.$\n\n[asy]\nunitsize(1 cm);\n\npair P... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 8759,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $ABCD,$ $AB=15$ and $AC=17.$ What is the area of rectangle $ABCD?$ Additionally, find the length of the diagonal $BD.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 8760,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Point \\( M \\) is the midpoint of side \\( BC \\) of the triangle \\( ABC \\), where \\( AB = 17 \\), \\( AC = 30 \\), and \\( BC = 19 \\). A circle is constructed with diameter \\( AB \\). A point \\( X \\) is chosen arbitrarily on this circle. What is the minimum possible le... | math | {
"style": "rule",
"ground_truth": "6.5"
} | {
"split": "train",
"index": 8761,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $a\\oplus b=3a+4b$ for all real numbers $a$ and $b$. Find $3\\oplus 1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 8762,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the range of the function\n\\[f(x) = \\frac{\\sin^3 x + 6 \\sin^2 x + \\sin x + 2 \\cos^2 x - 8}{\\sin x - 1},\\]as $x$ ranges over all real numbers such that $\\sin x \\neq 1.$ Enter your answer using interval notation. Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "[2,12)"
} | {
"split": "train",
"index": 8763,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Assign five volunteers randomly to three different positions A, B, and C for service, with each position having at least one volunteer. \n(1) Calculate the probability that exactly two volunteers are assigned to position A. \n(2) Let the random variable $\\xi$ represent the n... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{5}"
} | {
"split": "train",
"index": 8764,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $501^2 - 499^2$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 8765,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A ''super ball'' is dropped from a window 16 meters above the ground. On each bounce it rises $\\frac34$ the distance of the preceding high point. The ball is caught when it reached the high point after hitting the ground for the third time. To the nearest meter, how far has it... | math | {
"style": "rule",
"ground_truth": "65"
} | {
"split": "train",
"index": 8766,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$. Suppose $AD=BC=x$ and a circle with center on $\\overline{AB}$ is tangent to segments $\\overline{AD}$ and $\\overline{BC}$. If $m$ is the smallest possible value of $x$, then $m^2$=\n$\\text{(A) } 1369\\quad ... | math | {
"style": "rule",
"ground_truth": "1679"
} | {
"split": "train",
"index": 8767,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate $\\log_432$. 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": 8768,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\cos B$ and $\\sin A$ in the following right triangle where side $AB = 15$ units, and side $BC = 20$ units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"split": "train",
"index": 8769,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the maximum value of the following expression:\n$$\n|\\cdots|\\left|x_{1}-x_{2}\\right|-x_{3}\\left|-\\cdots-x_{1990}\\right|,\n$$\nwhere \\( x_{1}, x_{2}, \\cdots, x_{1990} \\) are distinct natural numbers from 1 to 1990. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "1989"
} | {
"split": "train",
"index": 8770,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A certain chemical reaction requires a catalyst to accelerate the reaction, but using too much of this catalyst affects the purity of the product. If the amount of this catalyst added is between 500g and 1500g, and the 0.618 method is used to arrange the experiment, then the am... | math | {
"style": "rule",
"ground_truth": "882"
} | {
"split": "train",
"index": 8771,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the answer to this problem. An urn contains white and black balls. There are $n$ white balls and at least two balls of each color in the urn. Two balls are randomly drawn from the urn without replacement. Find the probability, in percent, that the first ball drawn is... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 8772,
"solution": "Let the number of black balls in the urn be $k \\geq 2$. Then the probability of drawing a white ball first is $\\frac{n}{n+k}$, and the probability of drawing a black ball second is $\\frac{k}{n+k-1}$. This gives us the equation $$\\frac{nk}{(n+k)(n+k-1)}=\\frac{n}... |
deepscaler | [
{
"role": "user",
"content": "In the diagram below, lines $k$ and $\\ell$ are parallel. Find the measure of angle $x$ in degrees.\n\n[asy]\nsize(200);\nimport markers;\npair A = dir(-22)*(0,0);\npair B = dir(-22)*(4,0);\npair C = dir(-22)*(4,2);\npair D = dir(-22)*(0,2);\npair F = dir(-22)*(0,1.3);\npair G... | math | {
"style": "rule",
"ground_truth": "60^\\circ"
} | {
"split": "train",
"index": 8773,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If rectangle ABCD has area 72 square meters and E and G are the midpoints of sides AD and CD, respectively, then the area of rectangle DEFG in square meters is Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8774,
"solution": "1. **Identify the Midpoints**: Points E and G are the midpoints of sides AD and CD, respectively, in rectangle ABCD.\n\n2. **Properties of Midpoints in a Rectangle**: Since E and G are midpoints, segment EG is parallel to sides AB and CD, and its length is half the ... |
deepscaler | [
{
"role": "user",
"content": "If $S$, $H$, and $E$ are all distinct non-zero digits less than $5$ and the following is true, find the sum of the three values $S$, $H$, and $E$, expressing your answer in base $5$. $$\\begin{array}{c@{}c@{}c@{}c} &S&H&E_5\\\\ &+&H&E_5\\\\ \\cline{2-4} &S&E&S_5\\\\ \\end{array... | math | {
"style": "rule",
"ground_truth": "12_5"
} | {
"split": "train",
"index": 8775,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute $(\\cos 185^\\circ + i \\sin 185^\\circ)^{54}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-i"
} | {
"split": "train",
"index": 8776,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all functions $f: \\mathbb R \\to \\mathbb R$ such that \\[ f( xf(x) + f(y) ) = f^2(x) + y \\] for all $x,y\\in \\mathbb R$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "f(x) = x \\text{ or } f(x) = -x"
} | {
"split": "train",
"index": 8777,
"solution": "\nTo find all functions \\( f: \\mathbb{R} \\to \\mathbb{R} \\) satisfying the given functional equation:\n\n\\[\nf(xf(x) + f(y)) = f^2(x) + y \\quad \\text{for all } x, y \\in \\mathbb{R},\n\\]\n\nwe proceed with the following steps.\n\n### Step 1: Analyzing the Fu... |
deepscaler | [
{
"role": "user",
"content": "The numbers from $1$ to $8$ are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "18"
} | {
"split": "train",
"index": 8778,
"solution": "1. **Identify the total sum of numbers on the cube**: Each number from $1$ to $8$ is placed on a vertex of the cube. The sum of these numbers is $1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$. Using the formula for the sum of the first $n$ natural numbers, $\\frac{n(n+1)}{2}$, whe... |
deepscaler | [
{
"role": "user",
"content": "The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 8779,
"solution": "1. **Identify the equations**: We are given two equations:\n - Circle: \\(x^2 + y^2 = 9\\)\n - Horizontal lines: \\(y^2 = 9\\)\n\n2. **Solve the second equation**: To find the values of \\(y\\), we solve \\(y^2 = 9\\):\n \\[\n y = \\pm 3\n \\]\n\n3. **Subs... |
deepscaler | [
{
"role": "user",
"content": "Given that $a$, $b$, and $c$ are the sides opposite the angles $A$, $B$, and $C$ in $\\triangle ABC$ respectively, and the equation $\\sqrt{3}b\\sin A - a\\cos B - 2a = 0$ holds, then the measure of $\\angle B$ is ______. Let's think step by step and put only the final answer w... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 8780,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the terminal side of angle \\\\(\\alpha\\\\) passes through the point \\\\(P(m,2\\sqrt{2})\\\\), \\\\(\\sin \\alpha= \\frac{2\\sqrt{2}}{3}\\\\) and \\\\(\\alpha\\\\) is in the second quadrant. \n\\\\((1)\\\\) Find the value of \\\\(m\\\\); \n\\\\((2)\\\\) If \\\\(\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{2}}{11}"
} | {
"split": "train",
"index": 8781,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A conical flask contains some water. When the flask is oriented so that its base is horizontal and lies at the bottom (so that the vertex is at the top), the water is 1 inch deep. When the flask is turned upside-down, so that the vertex is at the bottom, the water is 2 inches d... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}+\\frac{\\sqrt{93}}{6}"
} | {
"split": "train",
"index": 8782,
"solution": "$\\frac{1}{2}+\\frac{\\sqrt{93}}{6}$. Let $h$ be the height, and let $V$ be such that $V h^{3}$ equals the volume of the flask. When the base is at the bottom, the portion of the flask not occupied by water forms a cone similar to the entire flask, with a height of ... |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a\\_n\\}$ with a common ratio $q > 1$, and it satisfies: $a\\_2 + a\\_3 + a\\_4 = 28$, and $a\\_3 + 2$ is the arithmetic mean of $a\\_2$ and $a\\_4$.\n(1) Find the general term formula of the sequence $\\{a\\_n\\}$;\n(2) If $b\\_n = a\\_n \\log\... | math | {
"style": "rule",
"ground_truth": "n = 6"
} | {
"split": "train",
"index": 8783,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $O$ be the origin. There exists a scalar $k$ so that for any points $A,$ $B,$ $C,$ and $D$ such that\n\\[3 \\overrightarrow{OA} - 2 \\overrightarrow{OB} + 5 \\overrightarrow{OC} + k \\overrightarrow{OD} = \\mathbf{0},\\]the four points $A,$ $B,$ $C,$ and $D$ are coplanar. ... | math | {
"style": "rule",
"ground_truth": "-6"
} | {
"split": "train",
"index": 8784,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangle has a perimeter of 80 inches and an area greater than 240 square inches. How many non-congruent rectangles meet these criteria? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"split": "train",
"index": 8785,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Triangles $\\triangle ABC$ and $\\triangle A'B'C'$ lie in the coordinate plane with vertices $A(0,0)$, $B(0,12)$, $C(16,0)$, $A'(24,18)$, $B'(36,18)$, $C'(24,2)$. A rotation of $m$ degrees clockwise around the point $(x,y)$ where $0<m<180$, will transform $\\triangle ABC$ to $\... | math | {
"style": "rule",
"ground_truth": "108"
} | {
"split": "train",
"index": 8786,
"solution": "After sketching, it is clear a $90^{\\circ}$ rotation is done about $(x,y)$. Looking between $A$ and $A'$, $x+y=18$ and $x-y=24$. Solving gives $(x,y)\\implies(21,-3)$. Thus $90+21-3=\\boxed{108}$. ~mn28407"
} |
deepscaler | [
{
"role": "user",
"content": "Estimate the product $(.331)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "0.037"
} | {
"split": "train",
"index": 8787,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider an octagonal lattice where each vertex is evenly spaced and one unit from its nearest neighbor. How many equilateral triangles have all three vertices in this lattice? Every side of the octagon is extended one unit outward with a single point placed at each extension, ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 8788,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute the product of $0.\\overline{123}$ and $9$, and write your result as a fraction in simplified form. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{41}{37}"
} | {
"split": "train",
"index": 8789,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "John needs to pay 2010 dollars for his dinner. He has an unlimited supply of 2, 5, and 10 dollar notes. In how many ways can he pay? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "20503"
} | {
"split": "train",
"index": 8790,
"solution": "Let the number of 2,5 , and 10 dollar notes John can use be $x, y$, and $z$ respectively. We wish to find the number of nonnegative integer solutions to $2 x+5 y+10 z=2010$. Consider this equation $\\bmod 2$. Because $2 x, 10 z$, and 2010 are even, $5 y$ must also b... |
deepscaler | [
{
"role": "user",
"content": "\nBaron Munchausen told a story. \"There were a whole crowd of us. We reached a crossroads. Then half of our group turned left, a third turned right, and a fifth went straight.\" \"But wait, the Duke remarked, the sum of half, a third, and a fifth isn't equal to one, so you are... | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 8791,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC,$ points $D$ and $E$ are on $\\overline{AB}$ and $\\overline{AC},$ respectively, and angle bisector $\\overline{AT}$ intersects $\\overline{DE}$ at $F.$ If $AD = 1,$ $DB = 3,$ $AE = 2,$ and $EC = 4,$ compute $\\frac{AF}{AT}.$\n\n[asy]\nunitsize(1 cm);\n\npair ... | math | {
"style": "rule",
"ground_truth": "\\frac{5}{18}"
} | {
"split": "train",
"index": 8792,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The fraction\n\\[\\frac{\\left(3^{2008}\\right)^2-\\left(3^{2006}\\right)^2}{\\left(3^{2007}\\right)^2-\\left(3^{2005}\\right)^2}\\]\nsimplifies to what? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9"
} | {
"split": "train",
"index": 8793,
"solution": "1. **Rewrite the expression using the difference of squares formula**:\n \\[\n \\frac{(3^{2008})^2 - (3^{2006})^2}{(3^{2007})^2 - (3^{2005})^2} = \\frac{(3^{2008} + 3^{2006})(3^{2008} - 3^{2006})}{(3^{2007} + 3^{2005})(3^{2007} - 3^{2005})}\n \\]\n\n2. **Simpl... |
deepscaler | [
{
"role": "user",
"content": "Divide every natural number with at least two digits by the sum of its digits! When will the quotient be the largest, and when will it be the smallest? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1.9"
} | {
"split": "train",
"index": 8794,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Determine the number of prime dates in a non-leap year where the day and the month are both prime numbers, and the year has one fewer prime month than usual. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"split": "train",
"index": 8795,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large... | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 8796,
"solution": "1. **Define Variables:**\n Let $N$ be the number of small semicircles, each with radius $r$. The diameter of the large semicircle is then the sum of the diameters of all small semicircles, which is $2Nr$.\n\n2. **Radius of Large Semicircle:**\n The radius of the... |
deepscaler | [
{
"role": "user",
"content": "Each cell of a $100 \\times 100$ board is painted in either blue or white. We call a cell balanced if it has an equal number of blue and white neighboring cells. What is the maximum number of balanced cells that can be found on the board? (Cells are considered neighbors if they... | math | {
"style": "rule",
"ground_truth": "9608"
} | {
"split": "train",
"index": 8797,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "At his usual rate a man rows 15 miles downstream in five hours less time than it takes him to return. If he doubles his usual rate, the time downstream is only one hour less than the time upstream. In miles per hour, the rate of the stream's current is: Let's think step by step... | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 8798,
"solution": "Let $r$ be the man's rowing speed in still water (in miles per hour) and $w$ be the speed of the stream's current (in miles per hour).\n\n1. **Formulate the equations based on the problem statement:**\n - The time taken to row 15 miles downstream (where the speed ... |
deepscaler | [
{
"role": "user",
"content": "In rectangle ABCD, AB=30 and BC=15. Let F be a point on AB such that ∠BCF=30°. Find CF. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 8799,
"solution": ""
} |
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