data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "In the polar coordinate system, the equation of curve C is $\\rho^2= \\frac{3}{1+2\\sin^2\\theta}$. Point R is at $(2\\sqrt{2}, \\frac{\\pi}{4})$. \nP is a moving point on curve C, and side PQ of rectangle PQRS, with PR as its diagonal, is perpendicular to the polar axis. Find... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{6}"
} | {
"split": "train",
"index": 28149,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given 100 real numbers, with their sum equal to zero. What is the minimum number of pairs that can be selected from them such that the sum of the numbers in each pair is non-negative? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "99"
} | {
"split": "train",
"index": 28150,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a weekend volleyball tournament, Team E plays against Team F, and Team G plays against Team H on Saturday. On Sunday, the winners of Saturday's matches face off in a final, while the losers compete for the consolation prize. Furthermore, there is a mini tiebreaker challenge ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28151,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $f(x)$ is an odd function defined on $\\mathbb{R}$ with a minimum positive period of 3, and for $x \\in \\left(-\\frac{3}{2}, 0\\right)$, $f(x)=\\log_{2}(-3x+1)$. Find $f(2011)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 28152,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $G$ be the set of points $(x, y)$ such that $x$ and $y$ are positive integers less than or equal to 20. Say that a ray in the coordinate plane is *ocular* if it starts at $(0, 0)$ and passes through at least one point in $G$ . Let $A$ be the set of angle meas... | math | {
"style": "rule",
"ground_truth": "1/722"
} | {
"split": "train",
"index": 28153,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the figure with circle $Q$, angle $KAT$ measures 42 degrees. What is the measure of minor arc $AK$ in degrees? [asy]\nimport olympiad; size(150); defaultpen(linewidth(0.8)); dotfactor=4;\ndraw(unitcircle);\ndraw(dir(84)--(-1,0)--(1,0));\ndot(\"$A$\",(-1,0),W); dot(\"$K$\",di... | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 28154,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sector $OAB$ is a quarter of a circle with a radius of 6 cm. A circle is inscribed within this sector, tangent to both the radius lines $OA$ and $OB$, and the arc $AB$. Determine the radius of the inscribed circle in centimeters. Express your answer in simplest radical form. Le... | math | {
"style": "rule",
"ground_truth": "6\\sqrt{2} - 6"
} | {
"split": "train",
"index": 28155,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Laura conducted a survey in her neighborhood about pest awareness. She found that $75.4\\%$ of the people surveyed believed that mice caused electrical fires. Of these, $52.3\\%$ incorrectly thought that mice commonly carried the Hantavirus. Given that these 31 people were misi... | math | {
"style": "rule",
"ground_truth": "78"
} | {
"split": "train",
"index": 28156,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)= \\overrightarrow{m} \\cdot \\overrightarrow{n} - \\frac{1}{2}$, where $\\overrightarrow{m}=( \\sqrt{3}\\sin x,\\cos x)$ and $\\overrightarrow{n}=(\\cos x,-\\cos x)$.\n1. Find the range of the function $y=f(x)$ when $x\\in[0, \\frac{\\pi}{2}]$.\n2. In $... | math | {
"style": "rule",
"ground_truth": "\\sqrt{7}"
} | {
"split": "train",
"index": 28157,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $128^3 = 16^x$, what is the value of $2^{-x}$? 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^{5.25}}"
} | {
"split": "train",
"index": 28158,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Evaluate the expression $3 - (-3)^{-\\frac{2}{3}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 - \\frac{1}{\\sqrt[3]{9}}"
} | {
"split": "train",
"index": 28159,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Construct a new sequence $\\{a_n\\}$ by extracting every 2nd, 4th, 8th, ..., $2^n$th, ... term from the sequence $\\{3n+\\log_2 n\\}$ in the original order. Find the general term $a_n$ for the new sequence $\\{a_n\\}$ and calculate the sum of the first five terms, $S_5$. Let's ... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 28160,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Read the text below and answer the questions. Everyone knows that $\\sqrt{2}$ is an irrational number, and irrational numbers are infinite non-repeating decimals. Therefore, we cannot write out all the decimal parts of $\\sqrt{2}$, but since $1 \\lt \\sqrt{2} \\lt 2$, the integ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{3} - 14"
} | {
"split": "train",
"index": 28161,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Rationalize the denominator of $\\frac{5}{\\sqrt{125}}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{5}}{5}"
} | {
"split": "train",
"index": 28162,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the least positive integer with exactly $12$ positive factors? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "72"
} | {
"split": "train",
"index": 28163,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A nine-digit number is formed by repeating a three-digit number three times; for example, $256256256$. Determine the common factor that divides any number of this form exactly. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1001001"
} | {
"split": "train",
"index": 28164,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, $AB = 6$, $AC = 8$, and $BC = 10$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{94}{15}"
} | {
"split": "train",
"index": 28165,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Henry's Hamburger Heaven offers its hamburgers with the following condiments: ketchup, mustard, mayonnaise, tomato, lettuce, pickles, cheese, and onions. A customer can choose one, two, or three meat patties, and any collection of condiments. How many different kinds of hamburg... | math | {
"style": "rule",
"ground_truth": "768"
} | {
"split": "train",
"index": 28166,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Steve guesses randomly on a 20-question multiple-choice test where each question has two choices. What is the probability that he gets at least half of the questions correct? Express your answer as a common fraction. Let's think step by step and put only the final answer within... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 28167,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The first term of a geometric sequence is 729, and the 7th term is 64. What is the positive, real value for the 5th term? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "144"
} | {
"split": "train",
"index": 28168,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\(\\frac{1}{4} + 4\\left(\\frac{1}{2013} + \\frac{1}{x}\\right) = \\frac{7}{4}\\), find the value of \\(1872 + 48 \\times \\left(\\frac{2013 x}{x + 2013}\\right)\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"split": "train",
"index": 28169,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $P$ be a point chosen on the interior of side $\\overline{BC}$ of triangle $\\triangle ABC$ with side lengths $\\overline{AB} = 10, \\overline{BC} = 10, \\overline{AC} = 12$ . If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $... | math | {
"style": "rule",
"ground_truth": "1936"
} | {
"split": "train",
"index": 28170,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. Given the line $l$: $ρ=- \\frac{6}{3\\cos θ +4\\sin θ}$, and the curve $C$: $\\begin{cases} ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 28171,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Of all positive integers between 10 and 100, what is the sum of the non-palindrome integers that take exactly eight steps to become palindromes? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "187"
} | {
"split": "train",
"index": 28172,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\(\\sin^{2}(18^{\\circ}) + \\cos^{2}(63^{\\circ}) + \\sqrt{2} \\sin(18^{\\circ}) \\cdot \\cos(63^{\\circ}) =\\) 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": 28173,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( 0 \\leq m-n \\leq 1 \\) and \\( 2 \\leq m+n \\leq 4 \\), when \\( m - 2n \\) reaches its maximum value, what is the value of \\( 2019m + 2020n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2019"
} | {
"split": "train",
"index": 28174,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\pi$ be a uniformly random permutation of the set $\\{1,2, \\ldots, 100\\}$. The probability that $\\pi^{20}(20)=$ 20 and $\\pi^{21}(21)=21$ can be expressed as $\\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$. (Here, $\\pi^{k}$ ... | math | {
"style": "rule",
"ground_truth": "1025"
} | {
"split": "train",
"index": 28175,
"solution": "We look at the cycles formed by $\\pi$ Let $\\operatorname{ord}_{\\pi}(n)$ denote the smallest $m$ such that $\\pi^{m}(n)=n$. In particular, the condition implies that $\\operatorname{ord}_{\\pi}(20) \\mid 20$ and $\\operatorname{ord}_{\\pi}(21) \\mid 21$. Claim 1.... |
deepscaler | [
{
"role": "user",
"content": "Find the coefficient of \\(x^5\\) in the expansion of \\(\\left(1+2x+3x^2+4x^3\\right)^5\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1772"
} | {
"split": "train",
"index": 28176,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For how many integer values of $b$ does the equation $$x^2 + bx + 12b = 0$$ have integer solutions for $x$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28177,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A square with an area of 40 is inscribed in a semicircle. If another square is inscribed in a full circle with the same radius, what is the area of this square? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 28178,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Maryam has a fair tetrahedral die, with the four faces of the die labeled 1 through 4. At each step, she rolls the die and records which number is on the bottom face. She stops when the current number is greater than or equal to the previous number. (In particular, she takes ... | math | {
"style": "rule",
"ground_truth": "625/256"
} | {
"split": "train",
"index": 28179,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest integer greater than 10 such that the sum of the digits in its base 17 representation is equal to the sum of the digits in its base 10 representation? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "153"
} | {
"split": "train",
"index": 28180,
"solution": "We assume that the answer is at most three digits (in base 10). Then our desired number can be expressed in the form $\\overline{a b c}{ }_{10}=\\overline{d e f}_{17}$, where $a, b, c$ are digits in base 10 , and $d, e, f$ are digits in base 17. These variables the... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sin ωx (ω > 0)$, if the minimum value of $f(x)$ in the interval $\\left[ -\\frac{\\pi}{3}, \\frac{\\pi}{4} \\right]$ is $-2$, find the minimum value of $ω$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{2}"
} | {
"split": "train",
"index": 28181,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 28182,
"solution": "1. **Identify the condition for the lines to be identical**: Two lines are identical if and only if they have the same slope and the same y-intercept.\n\n2. **Equations of the lines**: We are given the equations:\n - $3x + by + c = 0$\n - $cx - 2y + 12 = 0$\n\n... |
deepscaler | [
{
"role": "user",
"content": "Given an arithmetic sequence $\\{a_n\\}$ with the sum of its first $n$ terms denoted as $S_n$, and it is known that $a_1 > 0$, $a_3 + a_{10} > 0$, $a_6a_7 < 0$, determine the maximum natural number $n$ that satisfies $S_n > 0$. Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28183,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $m > n$ be positive integers such that $3(3mn - 2)^2 - 2(3m -3n)^2 = 2019$ . Find $3m + n$ .\n Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 28184,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n_{}$ but do not divide $n_{}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "589"
} | {
"split": "train",
"index": 28185,
"solution": "Let $n=p_1^{k_1}p_2^{k_2}$ for some prime $p_1,p_2$. Then $n^2$ has $\\frac{(2k_1+1)(2k_2+1)-1}{2}$ factors less than $n$.\n This simplifies to $\\frac{4k_1k_2+2k_1+2k_2}{2}=2k_1k_2+k_1+k_2$.\n The number of factors of $n$ less than $n$ is equal to $(k_1+1)(k_2+1)-... |
deepscaler | [
{
"role": "user",
"content": "For each positive integer $n$, let $f(n)$ be the sum of the digits in the base-four representation of $n$ and let $g(n)$ be the sum of the digits in the base-eight representation of $f(n)$. For example, $f(2020) = f(133210_{\\text{4}}) = 10 = 12_{\\text{8}}$, and $g(2020) = \\t... | math | {
"style": "rule",
"ground_truth": "151"
} | {
"split": "train",
"index": 28186,
"solution": "Let's work backwards. The minimum base-sixteen representation of $g(n)$ that cannot be expressed using only the digits $0$ through $9$ is $A_{16}$, which is equal to $10$ in base 10. Thus, the sum of the digits of the base-eight representation of the sum of the dig... |
deepscaler | [
{
"role": "user",
"content": "In $\\triangle XYZ$, $\\angle X = 90^\\circ$ and $\\tan Z = \\sqrt{3}$. If $YZ = 150$, what is $XY$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75\\sqrt{3}"
} | {
"split": "train",
"index": 28187,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the minimum value of\n\\[\\sqrt{x^2 + (1 - x)^2} + \\sqrt{(1 - x)^2 + (1 + x)^2}\\]over all real numbers $x.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 28188,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The sum of two fractions is $\\frac{11}{12}$ and their product is $\\frac{1}{6}$. What is the lesser of the two fractions? 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": 28189,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "On an island, there are 100 knights and 100 liars. Every resident has at least one friend. Knights always tell the truth, while liars always lie. One morning, each resident said either the phrase \"All my friends are knights\" or the phrase \"All my friends are liars,\" with ex... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 28190,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "There are $52$ people in a room. what is the largest value of $n$ such that the statement \"At least $n$ people in this room have birthdays falling in the same month\" is always true? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"split": "train",
"index": 28191,
"solution": "1. **Understanding the Problem:**\n We need to determine the largest number \\( n \\) such that in any group of 52 people, at least \\( n \\) people will have their birthdays in the same month.\n\n2. **Applying the Pigeonhole Principle:**\n The Pigeonhole Princ... |
deepscaler | [
{
"role": "user",
"content": "Adam and Sarah start on bicycle trips from the same point at the same time. Adam travels north at 10 mph and Sarah travels west at 5 mph. After how many hours are they 85 miles apart? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "7.6"
} | {
"split": "train",
"index": 28192,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nWrite the expression\n\n$$\nK=\\frac{\\frac{1}{a+b}-\\frac{2}{b+c}+\\frac{1}{c+a}}{\\frac{1}{b-a}-\\frac{2}{b+c}+\\frac{1}{c-a}}+\\frac{\\frac{1}{b+c}-\\frac{2}{c+a}+\\frac{1}{a+b}}{\\frac{1}{c-b}-\\frac{2}{c+a}+\\frac{1}{a-b}}+\\frac{\\frac{1}{c+a}-\\frac{2}{a+b}+\\frac{1}{b... | math | {
"style": "rule",
"ground_truth": "0.0625"
} | {
"split": "train",
"index": 28193,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Once upon a time, a team of Knights and a team of Liars met in the park and decided to ride a circular carousel that can hold 40 people (the \"Chain\" carousel, where everyone sits one behind the other). When they took their seats, each person saw two others: one in front and o... | math | {
"style": "rule",
"ground_truth": "26"
} | {
"split": "train",
"index": 28194,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find $\\begin{pmatrix} 3 \\\\ -7 \\end{pmatrix} + \\begin{pmatrix} -6 \\\\ 11 \\end{pmatrix}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} -3 \\\\ 4 \\end{pmatrix}"
} | {
"split": "train",
"index": 28195,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A box of chocolates in the shape of a cuboid was full of chocolates arranged in rows and columns. Míša ate some of them, and the remaining chocolates were rearranged to fill three entire rows completely, except for one space. Míša ate the remaining chocolates from another incom... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 28196,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $y=x^6-10x^5+29x^4-4x^3+ax^2$ lies above the line $y=bx+c$ except at three values of $x$, where the graph and the line intersect. What is the largest of these values? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 28197,
"solution": "\n1. **Identify the Polynomial and Intersection Points**:\n The problem states that the graph of the polynomial $y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$ intersects the line $y = bx + c$ at three values of $x$. These intersections are the roots of the polynomial $x... |
deepscaler | [
{
"role": "user",
"content": "Given a triangular pyramid \\( S-ABC \\) with a base that is an isosceles right triangle with \\( AB \\) as the hypotenuse, \\( SA = SB = SC = 2 \\), and \\( AB = 2 \\), suppose points \\( S, A, B, C \\) are all on a sphere with center \\( Q \\). Find the distance from point \\... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{3}"
} | {
"split": "train",
"index": 28198,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube has six faces, and each face has two diagonals. From these diagonals, choose two to form a pair. Among these pairs, how many form an angle of $60^\\circ$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 28199,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left( x \\right)=\\sin \\left( 2x+\\phi \\right)\\left(\\left| \\phi \\right| < \\dfrac{\\pi }{2} \\right)$ whose graph is symmetric about the point $\\left( \\dfrac{\\pi }{3},0 \\right)$, and $f\\left( {{x}_{1}} \\right)+f\\left( {{x}_{2}} \\right)=0$ wh... | math | {
"style": "rule",
"ground_truth": "-\\dfrac{\\sqrt{3}}{2}"
} | {
"split": "train",
"index": 28200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the $y$-intercept of the line $y = x + 4$ after it is translated down 6 units? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 28201,
"solution": "The line with equation $y = x + 4$ has a $y$-intercept of 4. When the line is translated 6 units downwards, all points on the line are translated 6 units down. This moves the $y$-intercept from 4 to $4 - 6 = -2$."
} |
deepscaler | [
{
"role": "user",
"content": "The sum of the digits of the integer equal to \\( 777777777777777^2 - 222222222222223^2 \\) can be found by evaluating the expression. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "74"
} | {
"split": "train",
"index": 28202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x) = \\sqrt{ax^2 + bx + c}$ $(a < 0)$, and all points $\\left(\\begin{matrix}s, f(t) \\end{matrix}\\right)$ (where $s, t \\in D$) form a square region, determine the value of $a$. Let's think step by step and put only the final answer within \\boxed{}."
... | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"split": "train",
"index": 28203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the area of the smallest region bounded by the graphs of $y=|x|$ and $x^2+y^2=4$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi"
} | {
"split": "train",
"index": 28204,
"solution": "1. **Identify the graphs and their intersections**:\n - The equation $x^2 + y^2 = 4$ represents a circle centered at the origin $(0,0)$ with radius $2$.\n - The equation $y = |x|$ represents two lines: $y = x$ for $x \\geq 0$ and $y = -x$ for $x \\leq 0$. These... |
deepscaler | [
{
"role": "user",
"content": "Convert the binary number $110110100_2$ to base 4. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "31220_4"
} | {
"split": "train",
"index": 28205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many zeros are at the end of the product \\( s(1) \\cdot s(2) \\cdot \\ldots \\cdot s(100) \\), where \\( s(n) \\) denotes the sum of the digits of the natural number \\( n \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "19"
} | {
"split": "train",
"index": 28206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "One of the angles in a triangle is $120^{\\circ}$, and the lengths of the sides form an arithmetic progression. Find the ratio of the lengths of the sides of the triangle. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3 : 5 : 7"
} | {
"split": "train",
"index": 28207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A cube with an edge length of 4 units has the same volume as a square-based pyramid with base edge lengths of 8 units and a height of $h$ units. What is the value of $h$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 28208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Define the sequence $\\{x_{i}\\}_{i \\geq 0}$ by $x_{0}=2009$ and $x_{n}=-\\frac{2009}{n} \\sum_{k=0}^{n-1} x_{k}$ for all $n \\geq 1$. Compute the value of $\\sum_{n=0}^{2009} 2^{n} x_{n}$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2009"
} | {
"split": "train",
"index": 28209,
"solution": "We have $-\\frac{n x_{n}}{2009}=x_{n-1}+x_{n-2}+\\ldots+x_{0}=x_{n-1}+\\frac{(n-1) x_{n-1}}{2009}$, which yields the recursion $x_{n}=\\frac{n-2010}{n} x_{n-1}$. Unwinding this recursion, we find $x_{n}=(-1)^{n} \\cdot 2009$. $\\binom{2008}{n}$. Thus $\\sum_{k=0}^{... |
deepscaler | [
{
"role": "user",
"content": "Two real numbers $x$ and $y$ are such that $8 y^{4}+4 x^{2} y^{2}+4 x y^{2}+2 x^{3}+2 y^{2}+2 x=x^{2}+1$. Find all possible values of $x+2 y^{2}$. 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": 28210,
"solution": "Writing $a=x+2 y^{2}$, the given quickly becomes $4 y^{2} a+2 x^{2} a+a+x=x^{2}+1$. We can rewrite $4 y^{2} a$ for further reduction to $a(2 a-2 x)+2 x^{2} a+a+x=x^{2}+1$, or $$\\begin{equation*} 2 a^{2}+\\left(2 x^{2}-2 x+1\\right) a+\\left(-x^{2}+x-1\\right)=0 \\... |
deepscaler | [
{
"role": "user",
"content": "In square \\(ABCD\\), \\(P\\) is the midpoint of \\(DC\\) and \\(Q\\) is the midpoint of \\(AD\\). If the area of the quadrilateral \\(QBCP\\) is 15, what is the area of square \\(ABCD\\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 28211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the sum of the first three terms of a geometric sequence $\\{a_n\\}$ is $3$ and the sum of the first nine terms is $39$, calculate the value of the sum of the first six terms. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The movie \"Thirty Thousand Miles in Chang'an\" allows the audience to experience the unique charm of Tang poetry that has been passed down for thousands of years and the beauty of traditional Chinese culture. In the film, Li Bai was born in the year $701$ AD. If we represent t... | math | {
"style": "rule",
"ground_truth": "-551"
} | {
"split": "train",
"index": 28213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Line $m$ in the coordinate plane has the equation $2x - 3y + 30 = 0$. This line is rotated $30^{\\circ}$ counterclockwise about the point $(10, 10)$ to form line $n$. Find the $x$-coordinate of the $x$-intercept of line $n$. Let's think step by step and put only the final answe... | math | {
"style": "rule",
"ground_truth": "\\frac{20\\sqrt{3} + 20}{2\\sqrt{3} + 3}"
} | {
"split": "train",
"index": 28214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Last summer, Pat worked at a summer camp. For each day that he worked, he earned \\$100 and he was not charged for food. For each day that he did not work, he was not paid and he was charged \\$20 for food. After 70 days, the money that he earned minus his food costs equalled \... | math | {
"style": "rule",
"ground_truth": "57"
} | {
"split": "train",
"index": 28215,
"solution": "Let \\( x \\) be the number of days on which Pat worked. On each of these days, he earned \\$100 and had no food costs, so he earned a total of \\( 100x \\) dollars. Since Pat worked for \\( x \\) of the 70 days, then he did not work on \\( 70-x \\) days. On each o... |
deepscaler | [
{
"role": "user",
"content": "The archipelago consists of $N \\geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that from each island, there are no more than 5 bridges, and among any 7 islands, there are always two islands connected by a bridge. What is the largest ... | math | {
"style": "rule",
"ground_truth": "36"
} | {
"split": "train",
"index": 28216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\\frac{7}{51}$ is $0.\\overline{23}_k = 0.232323..._k$. What is $k$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 28217,
"solution": "1. **Understanding the Problem**: We are given that the repeating base-$k$ representation of the fraction $\\frac{7}{51}$ is $0.\\overline{23}_k$. This means that in base $k$, the fraction $\\frac{7}{51}$ is represented as a repeating sequence of the digits 2 and 3... |
deepscaler | [
{
"role": "user",
"content": "The function $f(x),$ defined for $0 \\le x \\le 1,$ has the following properties:\n\n(i) $f(0) = 0.$\n(ii) If $0 \\le x < y \\le 1,$ then $f(x) \\le f(y).$\n(iii) $f(1 - x) = 1 - f(x)$ for all $0 \\le x \\le 1.$\n(iv) $f \\left( \\frac{x}{3} \\right) = \\frac{f(x)}{2}$ for $0 \... | math | {
"style": "rule",
"ground_truth": "\\frac{3}{8}"
} | {
"split": "train",
"index": 28218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Arrange the numbers $2011, \\sqrt{2011}, 2011^{2}$ in increasing order. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2011}, 2011, 2011^{2}"
} | {
"split": "train",
"index": 28219,
"solution": "Since $2011^{2}=4044121$ and $\\sqrt{2011} \\approx 44.8$, then the list of numbers in increasing order is $\\sqrt{2011}, 2011, 2011^{2}$. (If $n$ is a positive integer with $n>1$, then $n^{2}>n$ and $\\sqrt{n}<n$, so the list $\\sqrt{n}, n, n^{2}$ is always in inc... |
deepscaler | [
{
"role": "user",
"content": "Jack walked up a hill at a speed of $(x^2-11x-22)$ miles per hour. Meanwhile, Jill walked a total distance of $(x^2-3x-54)$ miles in $(x+6)$ hours. If Jack and Jill walked at the same speed, what is that speed, in miles per hour? Let's think step by step and put only the final ... | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 28220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the function \\( f(x) = \\left| \\log_{2} x \\right| \\). Real numbers \\( a \\) and \\( b \\) (where \\( a < b \\)) satisfy the following conditions:\n\\[\n\\begin{array}{l}\nf(a+1) = f(b+2), \\\\\nf(10a + 6b + 22) = 4\n\\end{array}\n\\]\n\nFind \\( ab \\). Let's think ste... | math | {
"style": "rule",
"ground_truth": "2/15"
} | {
"split": "train",
"index": 28221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In $\\Delta XYZ$, $\\overline{MN} \\parallel \\overline{XY}$, $XM = 5$ cm, $MY = 8$ cm, and $NZ = 9$ cm. What is the length of $\\overline{YZ}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "23.4"
} | {
"split": "train",
"index": 28222,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files. Let's think step ... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28223,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $y=\\cos(2x+\\frac{\\pi}{4})$, determine the $x$-coordinate of one of the symmetric centers of the translated graph after translating it to the left by $\\frac{\\pi}{6}$ units. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11\\pi}{24}"
} | {
"split": "train",
"index": 28224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Mrs. Thompson gives extra points on tests to her students with test grades that exceed the class median. Given that 101 students take the same test, what is the largest number of students who can be awarded extra points? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 28225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the expression $(1+x)^{56}$, the parentheses are expanded and like terms are combined. Find the coefficients of $x^8$ and $x^{48}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\binom{56}{8}"
} | {
"split": "train",
"index": 28226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Cara is sitting at a circular table with six friends. Assume there are three males and three females among her friends. How many different possible pairs of people could Cara sit between if each pair must include at least one female friend? Let's think step by step and put only... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"split": "train",
"index": 28227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider a 5x5 grid of squares. How many different squares can be traced using the lines in this grid? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "55"
} | {
"split": "train",
"index": 28228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the polynomial $f(x) = x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64$, calculate the value of $v_4$ when $x = 2$ using Horner's method. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 28229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the probability that the arrow stops on a shaded region if a circular spinner is divided into six regions, four regions each have a central angle of $x^{\\circ}$, and the remaining regions have central angles of $20^{\\circ}$ and $140^{\\circ}$? Let's think step by step... | math | {
"style": "rule",
"ground_truth": "\\frac{2}{3}"
} | {
"split": "train",
"index": 28230,
"solution": "The six angles around the centre of the spinner add to $360^{\\circ}$. \n Thus, $140^{\\circ}+20^{\\circ}+4x^{\\circ}=360^{\\circ}$ or $4x=360-140-20=200$, and so $x=50$. \n Therefore, the sum of the central angles of the shaded regions is $140^{\\circ}+50^{\\circ}... |
deepscaler | [
{
"role": "user",
"content": "Determine all polynomials $P(x)$ with real coefficients such that\n$P(x)^2 + P\\left(\\frac{1}{x}\\right)^2= P(x^2)P\\left(\\frac{1}{x^2}\\right)$ for all $x$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "P(x) = 0"
} | {
"split": "train",
"index": 28231,
"solution": "\nTo solve the problem, we need to determine all polynomials \\( P(x) \\) with real coefficients satisfying the equation:\n\n\\[\nP(x)^2 + P\\left(\\frac{1}{x}\\right)^2 = P(x^2)P\\left(\\frac{1}{x^2}\\right)\n\\]\n\nfor all \\( x \\).\n\n### Step 1: Analyze the Eq... |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=2\\sqrt{2}\\cos x\\sin\\left(x+\\frac{\\pi}{4}\\right)-1$. \n(I) Find the value of $f\\left(\\frac{\\pi}{4}\\right)$; \n(II) Find the maximum and minimum values of $f(x)$ in the interval $\\left[0, \\frac{\\pi}{2}\\right]$. Let's think step by step an... | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 28232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $f(x)=x^{3}+3 x-1$ have roots $a, b, c$. Given that $$\\frac{1}{a^{3}+b^{3}}+\\frac{1}{b^{3}+c^{3}}+\\frac{1}{c^{3}+a^{3}}$$ can be written as $\\frac{m}{n}$, where $m, n$ are positive integers and $\\operatorname{gcd}(m, n)=1$, find $100 m+n$. Let's think step by step and ... | math | {
"style": "rule",
"ground_truth": "3989"
} | {
"split": "train",
"index": 28233,
"solution": "We know that $a^{3}=-3 a+1$ and similarly for $b, c$, so $$\\frac{1}{a^{3}+b^{3}}=\\frac{1}{2-3 a-3 b}=\\frac{1}{2+3 c}=\\frac{1}{3(2 / 3+c)}$$ Now, $$f(x-2 / 3)=x^{3}-2 x^{2}+\\frac{13}{3} x-\\frac{89}{27}$$ has roots $a+2 / 3, b+2 / 3$, and $c+2 / 3$. Thus the an... |
deepscaler | [
{
"role": "user",
"content": "Find the least positive integer such that when its leftmost digit is deleted, the resulting integer is 1/19 of the original integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "95"
} | {
"split": "train",
"index": 28234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all values of \\( a \\) for which the quadratic trinomials \\( x^{2} - 6x + 4a \\) and \\( x^{2} + ax + 6 \\) each have two roots, with the sum of the squares of the roots of the first trinomial equal to the sum of the squares of the roots of the second trinomial. Let's th... | math | {
"style": "rule",
"ground_truth": "-12"
} | {
"split": "train",
"index": 28235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Consider the function\n\\[f(x) = \\max \\{-11x - 37, x - 1, 9x + 3\\}\\]defined for all real $x.$ Let $p(x)$ be a quadratic polynomial tangent to the graph of $f$ at three distinct points with $x$-coordinates $x_1,$ $x_2,$ $x_3.$ Find $x_1 + x_2 + x_3.$ Let's think step by st... | math | {
"style": "rule",
"ground_truth": "-\\frac{11}{2}"
} | {
"split": "train",
"index": 28236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given cos(π/4 - α) = 3/5 and sin(5π/4 + β) = -12/13, where α ∈ (π/4, 3π/4) and β ∈ (0, π/4), find tan(α)/tan(β). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-17"
} | {
"split": "train",
"index": 28237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Sara used $\\frac{5}{8}$ of a roll of wrapping paper to wrap four presents. She used an additional $\\frac{1}{24}$ of a roll on one of the presents for decorative purposes. How much wrapping paper did she use on each of the other three presents? Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\frac{7}{36}"
} | {
"split": "train",
"index": 28238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $w$ and $z$ are complex numbers such that $|w+z|=2$ and $|w^2+z^2|=28,$ find the smallest possible value of $|w^3+z^3|.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"split": "train",
"index": 28239,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A rectangular solid has three adjacent faces with areas of $1$, $2$, and $2$, respectively. All the vertices of the rectangular solid are located on the same sphere. Find the volume of this sphere. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{6}\\pi"
} | {
"split": "train",
"index": 28240,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute\n$$\\sum_{k=1}^{1000} k(\\lceil \\log_{\\sqrt{2}}{k}\\rceil- \\lfloor\\log_{\\sqrt{2}}{k} \\rfloor).$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "499477"
} | {
"split": "train",
"index": 28241,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The new individual income tax law has been implemented since January 1, 2019. According to the \"Individual Income Tax Law of the People's Republic of China,\" it is known that the part of the actual wages and salaries (after deducting special, additional special, and other leg... | math | {
"style": "rule",
"ground_truth": "9720"
} | {
"split": "train",
"index": 28242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A function \\( f(x) \\) defined on the interval \\([1,2017]\\) satisfies \\( f(1)=f(2017) \\), and for any \\( x, y \\in [1,2017] \\), \\( |f(x) - f(y)| \\leqslant 2|x - y| \\). If the real number \\( m \\) satisfies \\( |f(x) - f(y)| \\leqslant m \\) for any \\( x, y \\in [1,2... | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"split": "train",
"index": 28243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Today, Ivan the Confessor prefers continuous functions \\(f:[0,1] \\rightarrow \\mathbb{R}\\) satisfying \\(f(x)+f(y) \\geq|x-y|\\) for all pairs \\(x, y \\in[0,1]\\). Find the minimum of \\(\\int_{0}^{1} f\\) over all preferred functions. Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{4}"
} | {
"split": "train",
"index": 28244,
"solution": "The minimum of \\(\\int_{0}^{1} f\\) is \\(\\frac{1}{4}\\). Applying the condition with \\(0 \\leq x \\leq \\frac{1}{2}, y=x+\\frac{1}{2}\\) we get $$f(x)+f\\left(x+\\frac{1}{2}\\right) \\geq \\frac{1}{2}.$$ By integrating, $$\\int_{0}^{1} f(x) \\mathrm{d} x=\\int_... |
deepscaler | [
{
"role": "user",
"content": "Suppose that $x$ and $y$ are nonzero real numbers such that $\\frac{4x-3y}{x+4y} = 3$. Find the value of $\\frac{x-4y}{4x+3y}$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{11}{63}"
} | {
"split": "train",
"index": 28245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the units digit of $3^{2004}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"split": "train",
"index": 28246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that $O$ is a regular octahedron, that $C$ is the cube whose vertices are the centers of the faces of $O,$ and that the ratio of the volume of $O$ to that of $C$ is $\\frac mn,$ where $m$ and $n$ are relatively prime integers, find $m+n.$ Let's think step by step and put ... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 28247,
"solution": "Let the octahedron have vertices $(\\pm 3, 0, 0), (0, \\pm 3, 0), (0, 0, \\pm 3)$. Then the vertices of the cube lie at the centroids of the faces, which have coordinates $(\\pm 1, \\pm 1, \\pm 1)$. The cube has volume 8. The region of the octahedron lying in each ... |
deepscaler | [
{
"role": "user",
"content": "Find the number of integers $n$ such that \\[1+\\left\\lfloor\\dfrac{100n}{101}\\right\\rfloor=\\left\\lceil\\dfrac{99n}{100}\\right\\rceil.\\] Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10100"
} | {
"split": "train",
"index": 28248,
"solution": ""
} |
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