data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
deepscaler | [
{
"role": "user",
"content": "Compute $\\frac{x^{10} - 32x^5 + 1024}{x^5 - 32}$ when $x=8$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32768"
} | {
"split": "train",
"index": 9200,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $Q$ be a point outside of circle $C$. A segment is drawn from $Q$, tangent to circle $C$ at point $R$, and a different secant from $Q$ intersects $C$ at points $D$ and $E$ such that $QD < QE$. If $QD = 5$ and the length of the tangent from $Q$ to $R$ ($QR$) is equal to $DE ... | math | {
"style": "rule",
"ground_truth": "\\frac{15 + 5\\sqrt{5}}{2}"
} | {
"split": "train",
"index": 9201,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In triangle $ABC$, where $AB = 6$ and $AC = 10$. Let $M$ be a point on $BC$ such that $BM : MC = 2:3$. If $AM = 5$, what is the length of $BC$?\nA) $7\\sqrt{2.2}$\nB) $5\\sqrt{6.1}$\nC) $10\\sqrt{3.05}$\nD) $15 - 3\\sqrt{6.1}$ Let's think step by step and put only the final ans... | math | {
"style": "rule",
"ground_truth": "5\\sqrt{6.1}"
} | {
"split": "train",
"index": 9202,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": " $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius $5$ , with $AB=6$ , $BC=7$ , $CD=8$ . Find $AD$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{51}"
} | {
"split": "train",
"index": 9203,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many ways can 8 teaching positions be allocated to three schools, given that each school receives at least one position, and School A receives at least two positions? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"split": "train",
"index": 9204,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that point \\(Z\\) moves on \\(|z| = 3\\) in the complex plane, and \\(w = \\frac{1}{2}\\left(z + \\frac{1}{z}\\right)\\), where the trajectory of \\(w\\) is the curve \\(\\Gamma\\). A line \\(l\\) passes through point \\(P(1,0)\\) and intersects the curve \\(\\Gamma\\) a... | math | {
"style": "rule",
"ground_truth": "-\\frac{25}{8}"
} | {
"split": "train",
"index": 9205,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An entrepreneur invested $\\$20,\\!000$ in a nine-month term deposit that paid a simple annual interest rate of $8\\%$. After the term ended, she reinvested all the proceeds into another nine-month term deposit. At the end of the second term, her total investment had grown to $... | math | {
"style": "rule",
"ground_truth": "7.840"
} | {
"split": "train",
"index": 9206,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a circle centered at $O$, points $A$ and $C$ lie on the circle, each of $\\overline{BA}$ and $\\overline{BC}$ are tangent to the circle. Triangle $ABC$ is isosceles with $AB = BC$ and $\\angle ABC = 100^\\circ$. The circle intersects $\\overline{BO}$ at $D$. Determine $\\fra... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 9207,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the real number \\( x \\), \\([x] \\) denotes the integer part that does not exceed \\( x \\). Find the positive integer \\( n \\) that satisfies:\n\n\\[\n\\left[\\log _{2} 1\\right] + \\left[\\log _{2} 2\\right] + \\left[\\log _{2} 3\\right] + \\cdots + \\left[\\log _{2}... | math | {
"style": "rule",
"ground_truth": "312"
} | {
"split": "train",
"index": 9208,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the complex number $z=-3\\cos \\theta + i\\sin \\theta$ (where $i$ is the imaginary unit). \n(1) When $\\theta= \\frac {4}{3}\\pi$, find the value of $|z|$; \n(2) When $\\theta\\in\\left[ \\frac {\\pi}{2},\\pi\\right]$, the complex number $z_{1}=\\cos \\theta - i\\sin \\t... | math | {
"style": "rule",
"ground_truth": "\\frac {2\\pi}{3}"
} | {
"split": "train",
"index": 9209,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can the digits of $45,\\!520$ be arranged to form a 5-digit number? (Remember, numbers cannot begin with 0.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"split": "train",
"index": 9210,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the greatest common factor of 30, 90 and 75? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"split": "train",
"index": 9211,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A line $y = -2$ intersects the graph of $y = 5x^2 + 2x - 6$ at points $C$ and $D$. Find the distance between points $C$ and $D$, expressed in the form $\\frac{\\sqrt{p}}{q}$ where $p$ and $q$ are positive coprime integers. What is $p - q$? Let's think step by step and put only ... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 9212,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The diagonals AC and CE of the regular hexagon ABCDEF are divided by inner points M and N respectively, so that AM/AC = CN/CE = r. Determine r if B, M, and N are collinear. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{\\sqrt{3}}"
} | {
"split": "train",
"index": 9213,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The slope angle of the tangent line to the curve $f\\left(x\\right)=- \\frac{ \\sqrt{3}}{3}{x}^{3}+2$ at $x=1$ is $\\tan^{-1}\\left( \\frac{f'\\left(1\\right)}{\\mid f'\\left(1\\right) \\mid} \\right)$, where $f'\\left(x\\right)$ is the derivative of $f\\left(x\\right)$. Let's ... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\pi}{3}"
} | {
"split": "train",
"index": 9214,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Members of the Rockham Soccer League buy socks and T-shirts. Socks cost $4 per pair and each T-shirt costs $5 more than a pair of socks. Each member needs one pair of socks and a shirt for home games and another pair of socks and a shirt for away games. If the total cost is $23... | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 9215,
"solution": "1. **Calculate the cost of one T-shirt**: \n Given that each T-shirt costs $5 more than a pair of socks, and each pair of socks costs $4, the cost of one T-shirt is:\n \\[\n 4 + 5 = 9 \\text{ dollars}\n \\]\n\n2. **Calculate the total cost for one member**:... |
deepscaler | [
{
"role": "user",
"content": "Two vertices of an obtuse triangle are $(6,4)$ and $(0,0)$. The third vertex is located on the negative branch of the $x$-axis. What are the coordinates of the third vertex if the area of the triangle is 30 square units? Let's think step by step and put only the final answer wi... | math | {
"style": "rule",
"ground_truth": "(-15, 0)"
} | {
"split": "train",
"index": 9216,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the volume of the right prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ is equal to the volume of the cylinder with the circumscribed circle of square $ABCD$ as its base, calculate the ratio of the lateral area of the right prism to that of the cylinder. Let's think step by step and pu... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 9217,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Read the following text and answer the questions:<br/>$\\because \\sqrt{1}＜\\sqrt{2}＜\\sqrt{4}$, which means $1＜\\sqrt{2}＜2$,<br/>$\\therefore$ The integer part of $\\sqrt{2}$ is $1$, and the decimal part is $\\sqrt{2}-1$.<br/>Please answer:<br/>$(1)$ The integer part of $\\sqr... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5} - 8"
} | {
"split": "train",
"index": 9218,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify the expression $\\frac{{2x+4}}{{{x^2}-1}}÷\\frac{{x+2}}{{{x^2}-2x+1}}-\\frac{{2x}}{{x+1}}$, then substitute an appropriate number from $-2$, $-1$, $0$, $1$ to evaluate. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 9219,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many three-digit whole numbers have at least one 8 or at least one 9 as digits? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "452"
} | {
"split": "train",
"index": 9220,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the data in the table, where the number of No. 5 batteries is $x$ and the number of No. 7 batteries is $y$, the masses of one No. 5 battery and one No. 7 battery are $x$ grams and $y$ grams, respectively. By setting up and solving a system of equations, express the value ... | math | {
"style": "rule",
"ground_truth": "24"
} | {
"split": "train",
"index": 9221,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $n$ to be a positive integer. Given a set $\\{ a_1, a_2, \\ldots, a_n \\} $ of integers, where $a_i \\in \\{ 0, 1, 2, 3, \\ldots, 2^n -1 \\},$ $\\forall i$, we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements e... | math | {
"style": "rule",
"ground_truth": "2^{n(n-1)/2}"
} | {
"split": "train",
"index": 9222,
"solution": "\nLet \\( n \\) be a positive integer, and consider the set of integers \\( \\{ a_1, a_2, \\ldots, a_n \\} \\) where each \\( a_i \\) is an element of the set \\( \\{ 0, 1, 2, \\ldots, 2^n - 1 \\} \\). We associate each subset of this set with the sum of its element... |
deepscaler | [
{
"role": "user",
"content": "How many pairs of integers $(a, b)$, with $1 \\leq a \\leq b \\leq 60$, have the property that $b$ is divisible by $a$ and $b+1$ is divisible by $a+1$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "106"
} | {
"split": "train",
"index": 9223,
"solution": "The divisibility condition is equivalent to $b-a$ being divisible by both $a$ and $a+1$, or, equivalently (since these are relatively prime), by $a(a+1)$. Any $b$ satisfying the condition is automatically $\\geq a$, so it suffices to count the number of values $b-a ... |
deepscaler | [
{
"role": "user",
"content": "For any real number $x$, the symbol $\\lfloor x \\rfloor$ represents the integer part of $x$, that is, $\\lfloor x \\rfloor$ is the largest integer not exceeding $x$. Calculate the value of $\\lfloor \\log_{2}1 \\rfloor + \\lfloor \\log_{2}2 \\rfloor + \\lfloor \\log_{2}3 \\rfl... | math | {
"style": "rule",
"ground_truth": "8204"
} | {
"split": "train",
"index": 9224,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the value of $\\frac{2023^3 - 2 \\cdot 2023^2 \\cdot 2024 + 3 \\cdot 2023 \\cdot 2024^2 - 2024^3 + 1}{2023 \\cdot 2024}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2023"
} | {
"split": "train",
"index": 9225,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In the geometric sequence $\\{a_n\\}$, if $a_n > a_{n+1}$, and $a_7 \\cdot a_{14} = 6, a_4 + a_{17} = 5$, calculate $\\frac{a_5}{a_{18}}$. 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": 9226,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let the complex numbers \\( z_1 \\) and \\( z_2 \\) correspond to the points \\( A \\) and \\( B \\) on the complex plane respectively, and suppose \\( \\left|z_1\\right| = 4 \\) and \\( 4z_1^2 - 2z_1z_2 + z_2^2 = 0 \\). Let \\( O \\) be the origin. Calculate the area of triang... | math | {
"style": "rule",
"ground_truth": "8\\sqrt{3}"
} | {
"split": "train",
"index": 9227,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many integers can be expressed as a sum of three distinct numbers if chosen from the set $\\{4, 7, 10, 13, \\ldots, 46\\}$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "37"
} | {
"split": "train",
"index": 9228,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as ... | math | {
"style": "rule",
"ground_truth": "21"
} | {
"split": "train",
"index": 9229,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In rectangle $LMNO$, points $P$ and $Q$ quadruple $\\overline{LN}$, and points $R$ and $S$ quadruple $\\overline{MO}$. Point $P$ is at $\\frac{1}{4}$ the length of $\\overline{LN}$ from $L$, and point $Q$ is at $\\frac{1}{4}$ length from $P$. Similarly, $R$ is $\\frac{1}{4}$ th... | math | {
"style": "rule",
"ground_truth": "0.75"
} | {
"split": "train",
"index": 9230,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Olympus Corporation has released a new version of its popular Vowel Soup. In this version, each vowel (A, E, I, O, U) appears 7 times in each bowl. How many six-letter \"words\" can be formed from this Olympus Vowel Soup? Note: Words do not need to be actual words in the Englis... | math | {
"style": "rule",
"ground_truth": "15625"
} | {
"split": "train",
"index": 9231,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A target is a triangle divided by three sets of parallel lines into 100 equal equilateral triangles with unit sides. A sniper shoots at the target. He aims at a triangle and hits either it or one of the adjacent triangles sharing a side. He can see the results of his shots and ... | math | {
"style": "rule",
"ground_truth": "25"
} | {
"split": "train",
"index": 9232,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "1. Given that the line $l$ passing through the point $M(-3,0)$ is intercepted by the circle $x^{2}+(y+2)^{2}=25$ to form a chord of length $8$, what is the equation of line $l$?\n2. Circles $C_{1}$: $x^{2}+y^{2}+2x+8y-8=0$ and $C_{2}$: $x^{2}+y^{2}-4x-4y-2=0$ intersect. What is... | math | {
"style": "rule",
"ground_truth": "16"
} | {
"split": "train",
"index": 9233,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given that the center of the hyperbola is at the origin and one focus is F<sub>1</sub>(-$$\\sqrt{5}$$, 0), if point P is on the hyperbola and the midpoint of segment PF<sub>1</sub> has coordinates (0, 2), then the equation of this hyperbola is _________ and its eccentricity is ... | math | {
"style": "rule",
"ground_truth": "\\sqrt{5}"
} | {
"split": "train",
"index": 9234,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $(3-2i)-(5-2i)$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-2"
} | {
"split": "train",
"index": 9235,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f\\left(x\\right)=\\frac{1}{3}a{x}^{3}-\\frac{1}{2}b{x}^{2}+x$, where $a$ and $b$ are the outcomes of rolling two dice consecutively, find the probability that the function $f'\\left( x \\right)$ takes its extreme value at $x=1$. Let's think step by step and... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{12}"
} | {
"split": "train",
"index": 9236,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Tanya wrote a certain two-digit number on a piece of paper; to Sveta, who was sitting opposite her, the written number appeared different and was 75 less. What number did Tanya write? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "91"
} | {
"split": "train",
"index": 9237,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In trapezoid $ABCD$, leg $\\overline{BC}$ is perpendicular to bases $\\overline{AB}$ and $\\overline{CD}$, and diagonals $\\overline{AC}$ and $\\overline{BD}$ are perpendicular. Given that $AB=\\sqrt{11}$ and $AD=\\sqrt{1001}$, find $BC^2$.\n Let's think step by step and put on... | math | {
"style": "rule",
"ground_truth": "110"
} | {
"split": "train",
"index": 9238,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The expression $2 + \\sqrt{2} + \\frac{1}{2 + \\sqrt{2}} + \\frac{1}{\\sqrt{2} - 2}$ equals: Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"split": "train",
"index": 9239,
"solution": "1. **Simplify $\\frac{1}{2 + \\sqrt{2}}$:** \n To remove the radical from the denominator, multiply the numerator and denominator by the conjugate of the denominator:\n \\[\n \\frac{1}{2 + \\sqrt{2}} \\cdot \\frac{2 - \\sqrt{2}}{2 - \\sqrt{2}} = \\frac{2 - \\... |
deepscaler | [
{
"role": "user",
"content": "A rectangle has length 8 cm and width $\\pi$ cm. A semi-circle has the same area as the rectangle. What is its radius? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"split": "train",
"index": 9240,
"solution": "A rectangle with length 8 cm and width $\\pi$ cm has area $8\\pi \\mathrm{cm}^{2}$. Suppose that the radius of the semi-circle is $r \\mathrm{~cm}$. The area of a circle with radius $r \\mathrm{~cm}$ is $\\pi r^{2} \\mathrm{~cm}^{2}$ and so the area of the semi-circ... |
deepscaler | [
{
"role": "user",
"content": "Pascal has a triangle. In the $n$th row, there are $n+1$ numbers $a_{n, 0}, a_{n, 1}, a_{n, 2}, \\ldots, a_{n, n}$ where $a_{n, 0}=a_{n, n}=1$. For all $1 \\leq k \\leq n-1, a_{n, k}=a_{n-1, k}-a_{n-1, k-1}$. Let $N$ be the value of the sum $$\\sum_{k=0}^{2018} \\frac{\\left|a_... | math | {
"style": "rule",
"ground_truth": "780.9280674537"
} | {
"split": "train",
"index": 9241,
"solution": "An estimate of $E>0$ earns \\left\\lfloor 20 \\cdot 2^{-|N-E| / 70}\\right\\rfloor$ points. A good estimate for this question is to use the fact that $$\\sum_{k=0}^{2018}\\left|a_{2018, k}\\right|=\\frac{2^{2018}+2}{3}$$ the answer to Guts 17 . This suggests that ea... |
deepscaler | [
{
"role": "user",
"content": "Given positive numbers $m$ and $n$ that satisfy $m^2 + n^2 = 100$, find the maximum or minimum value of $m + n$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "10\\sqrt{2}"
} | {
"split": "train",
"index": 9242,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $ABC$ be a right-angled triangle with $\\angle ABC=90^\\circ$ , and let $D$ be on $AB$ such that $AD=2DB$ . What is the maximum possible value of $\\angle ACD$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 9243,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Set $S_n = \\sum_{p=1}^n (p^5+p^7)$. Determine the greatest common divisor of $S_n$ and $S_{3n}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sum_{p=1}^{n}=\\frac{n^4(n+1)^4}{8}"
} | {
"split": "train",
"index": 9244,
"solution": "\nGiven the sequence defined as \\( S_n = \\sum_{p=1}^n (p^5 + p^7) \\), we need to determine the greatest common divisor (GCD) of \\( S_n \\) and \\( S_{3n} \\).\n\n### Calculating \\( S_n \\)\nThe expression for \\( S_n \\) is:\n\\[\nS_n = \\sum_{p=1}^{n} (p^5 + p... |
deepscaler | [
{
"role": "user",
"content": "Eight hockey teams are competing against each other in a single round to advance to the semifinals. What is the minimum number of points that guarantees a team advances to the semifinals? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"split": "train",
"index": 9245,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let points \\( A_{1}, A_{2}, A_{3}, A_{4}, A_{5} \\) be located on the unit sphere. Find the maximum value of \\( \\min \\left\\{A_{i} A_{j} \\mid 1 \\leq i < j \\leq 5 \\right\\} \\) and determine all cases where this maximum value is achieved. Let's think step by step and put... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"split": "train",
"index": 9246,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.\n\na) Write down any five-digit palindromic number that is divisible by 5.\n\nb) How many five-digit palindromic numbers are there that are... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"split": "train",
"index": 9247,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $T_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $5^n$ inclusive. Find the smallest positive integer $n$ for which $T_n$ is an integer. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "63"
} | {
"split": "train",
"index": 9248,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Isaac repeatedly flips a fair coin. Whenever a particular face appears for the $2n+1$ th time, for any nonnegative integer $n$ , he earns a point. The expected number of flips it takes for Isaac to get $10$ points is $\\tfrac ab$ for coprime positive integers $a$ and $... | math | {
"style": "rule",
"ground_truth": "201"
} | {
"split": "train",
"index": 9249,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the matrix that corresponds to reflecting over the $x$-axis. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\begin{pmatrix} 1 & 0 \\\\ 0 & -1 \\end{pmatrix}"
} | {
"split": "train",
"index": 9250,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The complex numbers $z$ and $w$ satisfy the system \\[z + \\frac{20i}w = 5+i\\] \\[w+\\frac{12i}z = -4+10i\\] Find the smallest possible value of $\\vert zw\\vert^2$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 9251,
"solution": "Multiplying the two equations together gives us \\[zw + 32i - \\frac{240}{zw} = -30 + 46i\\] and multiplying by $zw$ then gives us a quadratic in $zw$: \\[(zw)^2 + (30-14i)zw - 240 =0.\\] Using the quadratic formula, we find the two possible values of $zw$ to be $7i... |
deepscaler | [
{
"role": "user",
"content": "Alicia starts a sequence with $m=3$. What is the fifth term of her sequence following the algorithm: Step 1: Alicia writes down the number $m$ as the first term. Step 2: If $m$ is even, Alicia sets $n=\frac{1}{2} m$. If $m$ is odd, Alicia sets $n=m+1$. Step 3: Alicia writes dow... | math | {
"style": "rule",
"ground_truth": "43"
} | {
"split": "train",
"index": 9252,
"solution": "We follow Alicia's algorithm carefully: Step 1: Alicia writes down $m=3$ as the first term. Step 2: Since $m=3$ is odd, Alicia sets $n=m+1=4$. Step 3: Alicia writes down $m+n+1=8$ as the second term. Step 4: Alicia sets $m=8$. Step 2: Since $m=8$ is even, Alicia set... |
deepscaler | [
{
"role": "user",
"content": "Let $S$ denote the set of all triples $(i, j, k)$ of positive integers where $i+j+k=17$. Compute $$\\sum_{(i, j, k) \\in S} i j k$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "11628"
} | {
"split": "train",
"index": 9253,
"solution": "We view choosing five objects from a row of 19 objects in an unusual way. First, remove two of the chosen objects, the second and fourth, which are not adjacent nor at either end, forming three nonempty groups of consecutive objects. We then have $i, j$, and $k$ cho... |
deepscaler | [
{
"role": "user",
"content": "Adia writes a list in increasing order of the integers between 1 and 100, inclusive, that cannot be written as the product of two consecutive positive integers. What is the 40th integer in her list? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "46"
} | {
"split": "train",
"index": 9254,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A writer composed a series of essays totaling 60,000 words over a period of 150 hours. However, during the first 50 hours, she was exceptionally productive and wrote half of the total words. Calculate the average words per hour for the entire 150 hours and separately for the fi... | math | {
"style": "rule",
"ground_truth": "600"
} | {
"split": "train",
"index": 9255,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If $a$ and $b$ are the roots of the equation $x^2-2cx-5d = 0$ , $c$ and $d$ are the roots of the equation $x^2-2ax-5b=0$ , where $a,b,c,d$ are distinct real numbers, what is $a+b+c+d$ ? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "30"
} | {
"split": "train",
"index": 9256,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A four-digit number has the following properties:\r\n\r\n(a) It is a perfect square;\r\n\r\n(b) Its first two digits are equal\r\n\r\n(c) Its last two digits are equal.\r\n\r\nFind all such four-digit numbers. Let's think step by step and put only the final answer within \\boxe... | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"split": "train",
"index": 9257,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the remainder when $x^4 + 1$ is divided by $x^2 - 3x + 5.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-3x - 19"
} | {
"split": "train",
"index": 9258,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The overall idea is a common method in mathematical problem-solving. Below is the train of thought for factoring the polynomial $(a^{2}+2a)(a^{2}+2a+2)+1$: Consider \"$a^{2}+2a$\" as a whole, let $a^{2}+2a=x$, then the expression $=x(x+2)+1=x^{2}+2x+1=(x+1)^{2}$, then restore \... | math | {
"style": "rule",
"ground_truth": "2024"
} | {
"split": "train",
"index": 9259,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An inscribed dodecagon. A convex dodecagon is inscribed in a circle. The lengths of some six sides of the dodecagon are $\\sqrt{2}$, and the lengths of the remaining six sides are $\\sqrt{24}$. What is the radius of the circle? Let's think step by step and put only the final an... | math | {
"style": "rule",
"ground_truth": "\\sqrt{38}"
} | {
"split": "train",
"index": 9260,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A deck of forty cards consists of four $1$'s, four $2$'s,..., and four $10$'s. A matching pair (two cards with the same number) is removed from the deck. Given that these cards are not returned to the deck, let $m/n$ be the probability that two randomly selected cards also form... | math | {
"style": "rule",
"ground_truth": "758"
} | {
"split": "train",
"index": 9261,
"solution": "There are ${38 \\choose 2} = 703$ ways we can draw two cards from the reduced deck. The two cards will form a pair if both are one of the nine numbers that were not removed, which can happen in $9{4 \\choose 2} = 54$ ways, or if the two cards are the remaining two c... |
deepscaler | [
{
"role": "user",
"content": "The isosceles trapezoid shown has side lengths as labeled. How long is segment AC? [asy]\nunitsize(1.5mm);\ndefaultpen(linewidth(.7pt)+fontsize(10pt));\ndotfactor=3;\n\npair A=(0,0), B=(21,0), C=(15,8), D=(6,8);\npair[] dots={A,B,C,D};\n\ndraw(A--B--C--D--cycle);\ndot(dots);\nl... | math | {
"style": "rule",
"ground_truth": "17"
} | {
"split": "train",
"index": 9262,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given the function $f(x)=\\cos (2x-\\frac{\\pi }{3})+2\\sin^2x$.\n\n(Ⅰ) Find the period of the function $f(x)$ and the intervals where it is monotonically increasing;\n\n(Ⅱ) When $x \\in [0,\\frac{\\pi}{2}]$, find the maximum and minimum values of the function $f(x)$. Let's thi... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"split": "train",
"index": 9263,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Peyton puts 30 L of oil and 15 L of vinegar into a large empty can. He then adds 15 L of oil to create a new mixture. What percentage of the new mixture is oil? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "75\\%"
} | {
"split": "train",
"index": 9264,
"solution": "After Peyton has added 15 L of oil, the new mixture contains $30+15=45 \\mathrm{~L}$ of oil and 15 L of vinegar. Thus, the total volume of the new mixture is $45+15=60 \\mathrm{~L}$. Of this, the percentage that is oil is $\\frac{45}{60} \\times 100 \\%=\\frac{3}{4}... |
deepscaler | [
{
"role": "user",
"content": "Let $a,b$ be positive reals such that $\\frac{1}{a}+\\frac{1}{b}\\leq2\\sqrt2$ and $(a-b)^2=4(ab)^3$ . Find $\\log_a b$ . Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"split": "train",
"index": 9265,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given \\( x_{1}, x_{2}, \\cdots, x_{1993} \\) satisfy:\n\\[\n\\left|x_{1}-x_{2}\\right|+\\left|x_{2}-x_{3}\\right|+\\cdots+\\left|x_{1992}-x_{1993}\\right|=1993,\n\\]\nand\n\\[\ny_{k}=\\frac{x_{1}+x_{2}+\\cdots+x_{k}}{k} \\quad (k=1,2,\\cdots,1993),\n\\]\nwhat is the maximum po... | math | {
"style": "rule",
"ground_truth": "1992"
} | {
"split": "train",
"index": 9266,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A national team needs to select 4 out of 6 sprinters to participate in the 4×100 m relay at the Asian Games. If sprinter A cannot run the first leg and sprinter B cannot run the fourth leg, there are a total of ______ ways to participate. Let's think step by step and put only t... | math | {
"style": "rule",
"ground_truth": "252"
} | {
"split": "train",
"index": 9267,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given $x, y \\in \\mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:\n$$\n\\frac{9}{17}<\\frac{x}{x+y}<\\frac{8}{15}.\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "112"
} | {
"split": "train",
"index": 9268,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find the mathematical expectation of the area of the projection of a cube with edge of length $1$ onto a plane with an isotropically distributed random direction of projection. 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": 9269,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In how many ways can 6 purple balls and 6 green balls be placed into a $4 \\times 4$ grid of boxes such that every row and column contains two balls of one color and one ball of the other color? Only one ball may be placed in each box, and rotations and reflections of a single ... | math | {
"style": "rule",
"ground_truth": "5184"
} | {
"split": "train",
"index": 9270,
"solution": "In each row or column, exactly one box is left empty. There are $4!=24$ ways to choose the empty spots. Once that has been done, there are 6 ways to choose which two rows have 2 purple balls each. Now, assume without loss of generality that boxes $(1,1)$, $(2,2),(3,... |
deepscaler | [
{
"role": "user",
"content": "Find the area bounded by the graph of $y = \\arccos(\\sin x)$ and the $x$-axis on the interval $\\frac{\\pi}{2} \\le x \\le \\frac{5 \\pi}{2}.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\pi^2"
} | {
"split": "train",
"index": 9271,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find all triples of positive integers $(x, y, z)$ such that $x^{2}+y-z=100$ and $x+y^{2}-z=124$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "(12,13,57)"
} | {
"split": "train",
"index": 9272,
"solution": "Cancel $z$ to get $24=(y-x)(y+x-1)$. Since $x, y$ are positive, we have $y+x-1 \\geq 1+1-1>0$, so $0<y-x<y+x-1$. But $y-x$ and $y+x-1$ have opposite parity, so $(y-x, y+x-1) \\in\\{(1,24),(3,8)\\}$ yields $(y, x) \\in\\{(13,12),(6,3)\\}$. Finally, $0<z=x^{2}+y-100$ ... |
deepscaler | [
{
"role": "user",
"content": "Give the set of all positive integers $n$ such that $\\varphi(n)=2002^{2}-1$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\varnothing"
} | {
"split": "train",
"index": 9273,
"solution": "The empty set, $\\varnothing$. If $m$ is relatively prime to $n$ and $m<n$, then $n-m$ must likewise be relatively prime to $n$, and these are distinct for $n>2$ since $n / 2, n$ are not relatively prime. Therefore, for all $n>2, \\varphi(n)$ must be even. $2002^{2}... |
deepscaler | [
{
"role": "user",
"content": "Given the parametric equation of curve \\\\(C_{1}\\\\) as \\\\(\\begin{cases}x=3\\cos \\alpha \\\\ y=\\sin \\alpha\\end{cases} (\\alpha\\\\) is the parameter\\\\()\\\\), and taking the origin \\\\(O\\\\) of the Cartesian coordinate system \\\\(xOy\\\\) as the pole and the posit... | math | {
"style": "rule",
"ground_truth": "\\dfrac{6 \\sqrt{3}}{5}"
} | {
"split": "train",
"index": 9274,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given four positive integers \\(a, b, c,\\) and \\(d\\) satisfying the equations \\(a^2 = c(d + 20)\\) and \\(b^2 = c(d - 18)\\). Find the value of \\(d\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "180"
} | {
"split": "train",
"index": 9275,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $\\mathcal{S}_{n}$ be the set of strings with only 0's or 1's with length $n$ such that any 3 adjacent place numbers sum to at least 1. For example, $00100$ works, but $10001$ does not. Find the number of elements in $\\mathcal{S}_{11}$.\n Let's think step by step and put o... | math | {
"style": "rule",
"ground_truth": "927"
} | {
"split": "train",
"index": 9276,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Simplify $\\left( \\frac{1}{2k} \\right)^{-2} \\cdot (-k)^3$. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "-4k^5"
} | {
"split": "train",
"index": 9277,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Given a convex quadrilateral \\(ABCD\\) with \\(\\angle C = 57^{\\circ}\\), \\(\\sin \\angle A + \\sin \\angle B = \\sqrt{2}\\), and \\(\\cos \\angle A + \\cos \\angle B = 2 - \\sqrt{2}\\), find the measure of angle \\(D\\) in degrees. Let's think step by step and put only the ... | math | {
"style": "rule",
"ground_truth": "168"
} | {
"split": "train",
"index": 9278,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What's the largest number of elements that a set of positive integers between $1$ and $100$ inclusive can have if it has the property that none of them is divisible by another? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "50"
} | {
"split": "train",
"index": 9279,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Add $5_7 + 16_7.$ Express your answer in base $7.$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "24_7"
} | {
"split": "train",
"index": 9280,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "\nIn \"The Three-Body Problem,\" the operation involves intercepting the \"Judgment Day\" ship with a \"nano-blade\" material at the Panama Canal locks. As long as the \"Judgment Day\" ship passes through the \"nano-blade\" material completely undetected, the operation is a suc... | math | {
"style": "rule",
"ground_truth": "40"
} | {
"split": "train",
"index": 9281,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "What is the smallest prime whose digits sum to 23? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "1993"
} | {
"split": "train",
"index": 9282,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "An ellipse whose axes are parallel to the coordinate axes is tangent to the $x$-axis at $(6, 0)$ and tangent to the $y$-axis at $(0, 2)$. Find the distance between the foci of the ellipse. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "4\\sqrt{2}"
} | {
"split": "train",
"index": 9283,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If the sum of the coefficients of each term in the expansion of $(x- \\frac {4}{x})^{n}$ is $81$, then the constant term in the expansion is ________. Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "96"
} | {
"split": "train",
"index": 9284,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The three-digit number $2a3$ is added to the number $326$ to give the three-digit number $5b9$. If $5b9$ is divisible by 9, then $a+b$ equals Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"split": "train",
"index": 9285,
"solution": "1. **Identify the condition for divisibility by 9**: A number is divisible by 9 if the sum of its digits is a multiple of 9. Given the number $5b9$, the sum of its digits is $5 + b + 9$.\n\n2. **Calculate the sum of the digits**: \n \\[\n 5 + b + 9 = 14 + b\n ... |
deepscaler | [
{
"role": "user",
"content": "An infinite geometric series has common ratio $1/8$ and sum 60. What is the first term of the series? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{105}{2}"
} | {
"split": "train",
"index": 9286,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "In a zoo, oranges, bananas, and coconuts were brought to feed three monkeys, with an equal number of each type of fruit. The first monkey was fed only oranges and bananas, with the number of bananas being 40% more than the number of oranges. The second monkey was fed only banan... | math | {
"style": "rule",
"ground_truth": "1/2"
} | {
"split": "train",
"index": 9287,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Compute: \\(103 \\times 97\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "9991"
} | {
"split": "train",
"index": 9288,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Let $k$ be a positive integer, and the coefficient of the fourth term in the expansion of $(1+ \\frac{x}{k})^{k}$ is $\\frac{1}{16}$. Consider the functions $y= \\sqrt{8x-x^{2}}$ and $y= \\frac{1}{4}kx$, and let $S$ be the shaded region enclosed by their graphs. Calculate the p... | math | {
"style": "rule",
"ground_truth": "\\frac{\\pi}{4} - \\frac{1}{2}"
} | {
"split": "train",
"index": 9289,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Three one-inch squares are placed with their bases on a line. The center square is lifted out and rotated 45 degrees, as shown. Then it is centered and lowered into its original location until it touches both of the adjoining squares. How many inches is the point $B$ from the l... | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}+\\frac{1}{2}"
} | {
"split": "train",
"index": 9290,
"solution": "1. **Understanding the Problem Setup**: We have three 1-inch squares aligned horizontally. The middle square is rotated by 45 degrees and then lowered until it touches the other two squares. We need to find the vertical distance from point $B$ (the top vertex of the... |
deepscaler | [
{
"role": "user",
"content": "In the Cartesian coordinate system, the coordinates of the two foci of an ellipse are $F_{1}(-2\\sqrt{2},0)$ and $F_{2}(2\\sqrt{2},0)$. The minimum distance from a point on ellipse $C$ to the right focus is $3-2\\sqrt{2}$.\n\n$(1)$ Find the equation of ellipse $C$; \n$(2)$ Sup... | math | {
"style": "rule",
"ground_truth": "\\dfrac{3}{2}"
} | {
"split": "train",
"index": 9291,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "Find a monic cubic polynomial $P(x)$ with integer coefficients such that\n\\[P(\\sqrt[3]{2} + 1) = 0.\\](A polynomial is monic if its leading coefficient is 1.) Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "x^3 - 3x^2 + 3x - 3"
} | {
"split": "train",
"index": 9292,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A right circular cylinder with radius 2 is inscribed in a hemisphere with radius 5 so that its bases are parallel to the base of the hemisphere. What is the height of this cylinder? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{21}"
} | {
"split": "train",
"index": 9293,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "If \\( x \\) is positive, find the minimum value of \\(\\frac{\\sqrt{x^{4}+x^{2}+2 x+1}+\\sqrt{x^{4}-2 x^{3}+5 x^{2}-4 x+1}}{x}\\). Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"split": "train",
"index": 9294,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "How many 4-digit positive integers, where each digit is odd, are divisible by 3? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "208"
} | {
"split": "train",
"index": 9295,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "The graph of $x^{4}=x^{2} y^{2}$ is a union of $n$ different lines. What is the value of $n$? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"split": "train",
"index": 9296,
"solution": "The equation $x^{4}-x^{2} y^{2}=0$ factors as $x^{2}(x+y)(x-y)=0$, so its graph is the union of the three lines $x=0, x+y=0$, and $x-y=0$."
} |
deepscaler | [
{
"role": "user",
"content": "Compute the sum of all possible distinct values of \\( m+n \\) if \\( m \\) and \\( n \\) are positive integers such that\n$$\n\\operatorname{lcm}(m, n) + \\operatorname{gcd}(m, n) = 2(m+n) + 11\n$$ Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "32"
} | {
"split": "train",
"index": 9297,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "A $37$-gon $Q_1$ is drawn in the Cartesian plane, and the sum of the $x$-coordinates of its $37$ vertices equals $185$. The midpoints of the sides of $Q_1$ form a second $37$-gon, $Q_2$. Then, the midpoints of the sides of $Q_2$ form a third $37$-gon, $Q_3$. Find the sum of the... | math | {
"style": "rule",
"ground_truth": "185"
} | {
"split": "train",
"index": 9298,
"solution": ""
} |
deepscaler | [
{
"role": "user",
"content": "For real numbers \\( x \\) and \\( y \\) within the interval \\([0, 12]\\):\n\n$$\nxy = (12 - x)^2 (12 - y)^2\n$$\n\nWhat is the maximum value of the product \\( xy \\)? Let's think step by step and put only the final answer within \\boxed{}."
}
] | math | {
"style": "rule",
"ground_truth": "81"
} | {
"split": "train",
"index": 9299,
"solution": ""
} |
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