data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "For $\\{1, 2, 3, \\ldots, 10\\}$ and each of its non-empty subsets, a unique alternating sum is defined similarly as before. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum... | math | {
"style": "rule",
"ground_truth": "5120"
} | {
"index": 11300,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $\\{1, 2, 3, \\ldots, n\\}$ and each of its non-empty subsets a unique alternating sum is defined as follows. Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. For example, the altern... | math | {
"style": "rule",
"ground_truth": "448"
} | {
"index": 11301,
"solution": "Consider a given subset $T$ of $S$ that contains $7$; then there is a subset $T'$ which contains all the elements of $T$ except for $7$, and only those elements . Since each element of $T'$ has one fewer element preceding it than it does in $T$, their signs are opposite. Thus the sum ... |
DeepScaleR | [
{
"role": "user",
"content": "For $a$ a positive real number, let $x_{1}, x_{2}, x_{3}$ be the roots of the equation $x^{3}-a x^{2}+a x-a=0$. Determine the smallest possible value of $x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3 x_{1} x_{2} x_{3}$."
}
] | math | {
"style": "rule",
"ground_truth": "-4"
} | {
"index": 11302,
"solution": "Note that $x_{1}+x_{2}+x_{3}=x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}=a$. Then $$\\begin{aligned} & x_{1}^{3}+x_{2}^{3}+x_{3}^{3}-3 x_{1} x_{2} x_{3}=\\left(x_{1}+x_{2}+x_{3}\\right)\\left(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-\\left(x_{1} x_{2}+x_{2} x_{3}+x_{3} x_{1}\\right)\\right) \\\\ & \\qua... |
DeepScaleR | [
{
"role": "user",
"content": "For $i = 1, 2, \\cdots, n$, if $\\left|x_{i}\\right| < 1$ and $\\left|x_{1}\\right| + \\left|x_{2}\\right| + \\cdots + \\left|x_{n}\\right| = 2005 + \\left|x_{1} + x_{2} + \\cdots + x_{n} \\right|$, find the minimum value of the positive integer $n$."
}
] | math | {
"style": "rule",
"ground_truth": "2006"
} | {
"index": 11303,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $k > 0$, let $I_k = 10\\ldots 064$, where there are $k$ zeros between the $1$ and the $6$. Let $N(k)$ be the number of factors of $2$ in the prime factorization of $I_k$. What is the maximum value of $N(k)$?\n$\\textbf{(A)}\\ 6\\qquad \\textbf{(B)}\\ 7\\qquad \\textbf{(C)}\... | math | {
"style": "rule",
"ground_truth": "7"
} | {
"index": 11304,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $n \\ge 1$ call a finite sequence $(a_1, a_2 \\ldots a_n)$ of positive integers progressive if $a_i < a_{i+1}$ and $a_i$ divides $a_{i+1}$ for all $1 \\le i \\le n-1$. Find the number of progressive sequences such that the sum of the terms in the sequence is equal to $360$.... | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 11305,
"solution": "If the first term is $x$, then dividing through by $x$, we see that we can find the number of progressive sequences whose sum is $\\frac{360}{x} - 1$, and whose first term is not 1. If $a(k)$ denotes the number of progressive sequences whose sum is $k$ and whose first term is not 1, t... |
DeepScaleR | [
{
"role": "user",
"content": "For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. How many two-digit positive integers $n$ satisfy $R(n) = R(n+1)\\,?$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11306,
"solution": "We start by defining the function $R(n)$ as the sum of the remainders when $n$ is divided by each integer from $2$ to $10$. We are interested in finding two-digit integers $n$ such that $R(n) = R(n+1)$.\n\n#### Step 1: Define the Change in Remainder Function\nLet $\\Delta(n, k) = \\te... |
DeepScaleR | [
{
"role": "user",
"content": "For $n$ a positive integer, let $f(n)$ be the quotient obtained when the sum of all positive divisors of $n$ is divided by $n.$ For example, $f(14)=(1+2+7+14)\\div 14=\\frac{12}{7}$.\nWhat is $f(768)-f(384)?$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{192}"
} | {
"index": 11307,
"solution": "1. **Define the function $f(n)$**: \n Given $f(n) = \\frac{\\sigma(n)}{n}$, where $\\sigma(n)$ is the sum of all positive divisors of $n$.\n\n2. **Prime factorization and properties of $\\sigma(n)$**:\n If $n = \\prod_{i=1}^{k} p_i^{e_i}$ is the prime factorization of $n$, then $\... |
DeepScaleR | [
{
"role": "user",
"content": "For $p=1, 2, \\cdots, 10$ let $S_p$ be the sum of the first $40$ terms of the arithmetic progression whose first term is $p$ and whose common difference is $2p-1$; then $S_1+S_2+\\cdots+S_{10}$ is"
}
] | math | {
"style": "rule",
"ground_truth": "80200"
} | {
"index": 11308,
"solution": "1. **Identify the $40$th term of the sequence**: \n For an arithmetic progression (AP) with first term $a = p$ and common difference $d = 2p - 1$, the $n$th term of the AP is given by:\n \\[\n a_n = a + (n-1)d = p + (n-1)(2p-1).\n \\]\n Substituting $n = 40$, we get:\n \\[... |
DeepScaleR | [
{
"role": "user",
"content": "For $t = 1, 2, 3, 4$, define $S_t = \\sum_{i = 1}^{350}a_i^t$, where $a_i \\in \\{1,2,3,4\\}$. If $S_1 = 513$ and $S_4 = 4745$, find the minimum possible value for $S_2$."
}
] | math | {
"style": "rule",
"ground_truth": "905"
} | {
"index": 11309,
"solution": "Because the order of the $a$'s doesn't matter, we simply need to find the number of $1$s $2$s $3$s and $4$s that minimize $S_2$. So let $w, x, y,$ and $z$ represent the number of $1$s, $2$s, $3$s, and $4$s respectively. Then we can write three equations based on these variables. Since... |
DeepScaleR | [
{
"role": "user",
"content": "For $x > 0$, find the maximum value of $f(x) = 1 - 2x - \\frac{3}{x}$ and the value of $x$ at which it occurs."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{6}}{2}"
} | {
"index": 11310,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $x > 0$, the area of the triangle with vertices $(0, 0), (x, 2x)$, and $(x, 0)$ is 64 square units. What is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "8"
} | {
"index": 11311,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $x \\ge 0$ the smallest value of $\\frac {4x^2 + 8x + 13}{6(1 + x)}$ is:"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11312,
"solution": "1. **Define the function**: Let's denote the function by $f(x) = \\frac {4x^2 + 8x + 13}{6(1 + x)}$ for $x \\geq 0$.\n\n2. **Simplify the function**:\n \\[\n f(x) = \\frac {4x^2 + 8x + 13}{6(1 + x)} = \\frac{4(x^2+2x) + 13}{6(x+1)} = \\frac{4(x^2+2x+1-1)+13}{6(x+1)} = \\frac{4(x+1... |
DeepScaleR | [
{
"role": "user",
"content": "For $x$ a real number, let $f(x)=0$ if $x<1$ and $f(x)=2 x-2$ if $x \\geq 1$. How many solutions are there to the equation $f(f(f(f(x))))=x ?$"
}
] | math | {
"style": "rule",
"ground_truth": "2"
} | {
"index": 11313,
"solution": "Certainly 0,2 are fixed points of $f$ and therefore solutions. On the other hand, there can be no solutions for $x<0$, since $f$ is nonnegative-valued; for $0<x<2$, we have $0 \\leq f(x)<x<2$ (and $f(0)=0$ ), so iteration only produces values below $x$, and for $x>2, f(x)>x$, and iter... |
DeepScaleR | [
{
"role": "user",
"content": "For $x$ real, the inequality $1 \\le |x-2| \\le 7$ is equivalent to"
}
] | math | {
"style": "rule",
"ground_truth": "$-5 \\le x \\le 1$ or $3 \\le x \\le 9$"
} | {
"index": 11314,
"solution": "To solve the inequality $1 \\leq |x-2| \\leq 7$, we need to consider the absolute value inequality in two parts: the lower bound and the upper bound.\n\n1. **Solving the lower bound inequality $|x-2| \\geq 1$:**\n - The absolute value inequality $|x-2| \\geq 1$ implies that the dist... |
DeepScaleR | [
{
"role": "user",
"content": "For $y=\\frac{1-x}{2x+3}$ and $x\\neq-\\frac{3}{2}$, what is the value of $y$ that is not attainable? Express your answer as a common fraction."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac12"
} | {
"index": 11315,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For $y=\\frac{x+2}{5x-7}$, at what $x$-value is there a vertical asymptote?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{7}{5}"
} | {
"index": 11316,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For Beatrix's latest art installation, she has fixed a $2 \\times 2$ square sheet of steel to a wall. She has two $1 \\times 2$ magnetic tiles, both of which she attaches to the steel sheet, in any orientation, so that none of the sheet is visible and the line separating the tw... | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 11317,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For \\( n \\in \\mathbf{Z}_{+}, n \\geqslant 2 \\), let\n\\[ \nS_{n}=\\sum_{k=1}^{n} \\frac{k}{1+k^{2}+k^{4}}, \\quad T_{n}=\\prod_{k=2}^{n} \\frac{k^{3}-1}{k^{3}+1} \n\\]\n\nThen, \\( S_{n} T_{n} = \\) ."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{3}"
} | {
"index": 11318,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For \\( n=1,2,3, \\ldots \\), let \\( a_{n}=n^{2}+100 \\), and let \\( d_{n} \\) denote the greatest common divisor of \\( a_{n} \\) and \\( a_{n+1} \\). Find the maximum value of \\( d_{n} \\) as \\( n \\) ranges over all positive integers."
}
] | math | {
"style": "rule",
"ground_truth": "401"
} | {
"index": 11319,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For \\( x \\in \\mathbb{R} \\), the function \\( f(x) \\) satisfies \\( f(x+4) + f(x-4) = f(x) \\). Thus, it is a periodic function. The common minimum period of such functions is:"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11320,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For \\(0 \\leq x \\leq 1\\) and positive integer \\(n\\), let \\(f_0(x) = |1 - 2x|\\) and \\(f_n(x) = f_0(f_{n-1}(x))\\). How many solutions are there to the equation \\(f_{10}(x) = x\\) in the range \\(0 \\leq x \\leq 1\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "2048"
} | {
"index": 11321,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For \\(50 \\le n \\le 150\\), how many integers \\(n\\) are there such that \\(\\frac{n}{n+1}\\) is a repeating decimal and \\(n+1\\) is not divisible by 3?"
}
] | math | {
"style": "rule",
"ground_truth": "67"
} | {
"index": 11322,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a $5 \\times 5$ chessboard colored as shown below, place 5 different rooks on black squares such that no two rooks can attack each other (rooks attack if they are in the same row or column). How many different ways are there to do this?"
}
] | math | {
"style": "rule",
"ground_truth": "1440"
} | {
"index": 11323,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a $k$-element subset $T$ of the set $\\{1,2,\\cdots,242\\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$."
}
] | math | {
"style": "rule",
"ground_truth": "121"
} | {
"index": 11324,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a French class, I need to master a list of 600 vocabulary words for an upcoming test. The score on the test is based on the percentage of words I recall correctly. In this class, I have noticed that even when guessing the words I haven't studied, I have about a 10% chance o... | math | {
"style": "rule",
"ground_truth": "534"
} | {
"index": 11325,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a bijective function $g : R \\to R$ , we say that a function $f : R \\to R$ is its superinverse if it satisfies the following identity $(f \\circ g)(x) = g^{-1}(x)$ , where $g^{-1}$ is the inverse of $g$ . Given $g(x) = x^3 + 9x^2 + 27x + 81$ and $f$ is its super... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11326,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain complex number $c$, the polynomial\n\\[P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\\]has exactly 4 distinct roots. What is $|c|$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{10}"
} | {
"index": 11327,
"solution": "1. **Factorize the given polynomial**: \n The polynomial \\( P(x) \\) is given as:\n \\[\n P(x) = (x^2 - 2x + 2)(x^2 - cx + 4)(x^2 - 4x + 8)\n \\]\n We start by factorizing each quadratic term using the quadratic formula \\( x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a} \\).\n\... |
DeepScaleR | [
{
"role": "user",
"content": "For a certain hyperbola\n\\[\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1,\\]where $a > b,$ the angle between the asymptotes is $60^\\circ.$ Find $\\frac{a}{b}.$"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{3}"
} | {
"index": 11328,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain natural number $n$, $n^2$ gives a remainder of 4 when divided by 5, and $n^3$ gives a remainder of 2 when divided by 5. What remainder does $n$ give when divided by 5?"
}
] | math | {
"style": "rule",
"ground_truth": "3"
} | {
"index": 11329,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\\frac{1}{n}$ is $0.\\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\\frac{1}{n+6}$ is $0.\\overline{wxyz}$, a repeating decimal of period $4$. In which int... | math | {
"style": "rule",
"ground_truth": "[201,400]"
} | {
"index": 11330,
"solution": "1. **Understanding the repeating decimals**: Given that $\\frac{1}{n} = 0.\\overline{abcdef}$ has a repeating period of 6, it can be represented as $\\frac{abcdef}{999999}$. This implies that $n$ divides $999999$. Similarly, $\\frac{1}{n+6} = 0.\\overline{wxyz}$ has a repeating period... |
DeepScaleR | [
{
"role": "user",
"content": "For a certain positive integer $n,$ there exist real numbers $x_1,$ $x_2,$ $\\dots,$ $x_n$ such that\n\\begin{align*}\nx_1 + x_2 + x_3 + \\dots + x_n &= 1000, \\\\\nx_1^4 + x_2^4 + x_3^4 + \\dots + x_n^4 &= 512000.\n\\end{align*}Find the smallest positive integer $n$ for which ... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11331,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain square, two vertices lie on the line $y = 2x - 17,$ and the other two vertices lie on the parabola $y = x^2.$ Find the smallest possible area of the square."
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 11332,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain type of car, the purchase cost is $10$ ten thousand yuan, and the annual expenses for insurance, road maintenance, and car fees are about $0.9$ ten thousand yuan. The maintenance fee for the first year is $0.2$ ten thousand yuan, and it increases by $0.2$ ten thou... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11333,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain value of $k,$ the system\n\\begin{align*}\nx + ky + 3z &= 0, \\\\\n3x + ky - 2z &= 0, \\\\\n2x + 4y - 3z &= 0\n\\end{align*}has a solution where $x,$ $y,$ and $z$ are all nonzero. Find $\\frac{xz}{y^2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11334,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a certain weekend, the weatherman predicts that it will rain with a $40\\%$ probability on Saturday and a $50\\%$ probability on Sunday. Assuming these probabilities are independent, what is the probability that it rains over the weekend (that is, on at least one of the day... | math | {
"style": "rule",
"ground_truth": "70\\%"
} | {
"index": 11335,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a complex number $z \\neq 3$ , $4$ , let $F(z)$ denote the real part of $\\tfrac{1}{(3-z)(4-z)}$ . If \\[\n\t\\int_0^1 F \\left( \\frac{\\cos 2 \\pi t + i \\sin 2 \\pi t}{5} \\right) \\; dt = \\frac mn\n\\] for relatively prime positive integers $m$ and $n$ , find $... | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 11336,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a constant $c,$ in spherical coordinates $(\\rho,\\theta,\\phi),$ find the shape described by the equation\n\\[\\phi = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylinder\n(F) Cone\n\nEnter the letter of the correct option."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(F)}"
} | {
"index": 11337,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a designer suit, Daniel must specify his waist size in centimeters. If there are $12$ inches in a foot and $30.5$ centimeters in a foot, then what size should Daniel specify, in centimeters, if his waist size in inches is $34$ inches? (You may use a calculator on this prob... | math | {
"style": "rule",
"ground_truth": "86.4"
} | {
"index": 11338,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite graph $G$, let $f(G)$ be the number of triangles and $g(G)$ the number of tetrahedra formed by edges of $G$. Find the least constant $c$ such that \\[g(G)^3\\le c\\cdot f(G)^4\\] for every graph $G$.\n\n[i]"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{32}"
} | {
"index": 11339,
"solution": "\nLet \\( G \\) be a finite graph. We denote by \\( f(G) \\) the number of triangles and by \\( g(G) \\) the number of tetrahedra in \\( G \\). We seek to establish the smallest constant \\( c \\) such that\n\n\\[\ng(G)^3 \\le c \\cdot f(G)^4\n\\]\n\nfor every graph \\( G \\).\n\n### ... |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence $A=(a_1,a_2,\\dots,a_n)$ of numbers, the Cesaro sum of $A$ is defined to be\n\\[\\frac{S_1+\\cdots+S_n}{n},\\]where $S_k=a_1+\\cdots+a_k$ and $1\\leq k\\leq n$.\n\nIf the Cesaro sum of the 99-term sequence $(a_1,\\dots,a_{99})$ is 1000, what is the Cesaro... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"index": 11340,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_m)$ of numbers, the Cesaro sum of $B$ is defined to be\n\\[\\frac{T_1 + T_2 + \\dots + T_m}{m},\\]\nwhere $T_k = b_1 + b_2 + \\dots + b_k$ for $1 \\leq k \\leq m$.\n\nIf the Cesaro sum of the 50-term sequence $(b_1, b_2, \\dots, b... | math | {
"style": "rule",
"ground_truth": "492"
} | {
"index": 11341,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_n)$ of numbers, the Cesaro sum is defined as\n\\[\\frac{T_1 + T_2 + \\cdots + T_n}{n},\\] \nwhere $T_k = b_1 + b_2 + \\cdots + b_k$ for $1 \\leq k \\leq n$.\n\nIf the Cesaro sum of the 100-term sequence $(b_1, b_2, \\dots, b_{100}... | math | {
"style": "rule",
"ground_truth": "1191"
} | {
"index": 11342,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence $B = (b_1, b_2, \\dots, b_{50})$ of numbers, the Cesaro sum of $B$ is defined as\n\\[\\frac{T_1 + \\cdots + T_{50}}{50},\\]\nwhere $T_k = b_1 + \\cdots + b_k$ for $1 \\leq k \\leq 50$.\n\nIf the Cesaro sum of the 50-term sequence $(b_1, \\dots, b_{50})$ is... | math | {
"style": "rule",
"ground_truth": "198.078431372549"
} | {
"index": 11343,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence $B=(b_1,b_2,\\dots,b_{50})$ of numbers, the Cesaro sum of $B$ is defined to be\n\\[\\frac{T_1 + \\cdots + T_{50}}{50},\\] where $T_k = b_1 + \\cdots + b_k$ and $1 \\leq k \\leq 50$.\n\nIf the Cesaro sum of the 50-term sequence $(b_1,\\dots,b_{50})$ is 500,... | math | {
"style": "rule",
"ground_truth": "492"
} | {
"index": 11344,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence \\( B = (b_1, b_2, \\dots, b_{150}) \\) of numbers, the Cesaro sum of \\( B \\) is defined to be\n\\[\n\\frac{S_1 + \\cdots + S_{150}}{150},\n\\]\nwhere \\( S_k = b_1 + \\cdots + b_k \\) and \\( 1 \\leq k \\leq 150 \\).\n\nIf the Cesaro sum of the 150-term... | math | {
"style": "rule",
"ground_truth": "1194"
} | {
"index": 11345,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence \\( B = (b_1, b_2, \\dots, b_{50}) \\) of numbers, the Cesaro sum is defined as\n\\[\n\\frac{S_1 + \\cdots + S_{50}}{50},\n\\]\nwhere \\( S_k = b_1 + \\cdots + b_k \\) and \\( 1 \\leq k \\leq 50 \\).\n\nIf the Cesaro sum of the 50-term sequence \\( (b_1, \... | math | {
"style": "rule",
"ground_truth": "492"
} | {
"index": 11346,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence \\(P = \\left(p_1, p_2, \\cdots, p_n\\right)\\), the Cesaro sum (named after the mathematician Cesaro) is defined as \\(\\frac{1}{n}(S_1 + S_2 + \\cdots + S_n)\\), where \\(S_k = p_1 + p_2 + \\cdots + p_k\\) for \\(1 \\leq k \\leq n\\). If a sequence \\(\\... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"index": 11347,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a finite sequence \\(P = \\left(p_{1}, p_{2}, \\cdots, p_{n}\\right)\\), the Caesar sum (named after a mathematician Caesar) is defined as \\(\\frac{s_{1}+s_{2}+\\cdots+s_{n}}{n}\\), where \\(s_{k} = p_{1} + p_{2} + \\cdots + p_{k}\\) for \\(1 \\leq k \\leq n\\). If a seque... | math | {
"style": "rule",
"ground_truth": "991"
} | {
"index": 11348,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a four-digit natural number $A$, if the digit in the thousands place is $5$ more than the digit in the tens place, and the digit in the hundreds place is $3$ more than the digit in the units place, then $A$ is called a \"five-three number.\" For example, for the four-digit ... | math | {
"style": "rule",
"ground_truth": "5401"
} | {
"index": 11349,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a four-digit natural number $M$, if the digit in the thousands place is $6$ more than the digit in the units place, and the digit in the hundreds place is $2$ more than the digit in the tens place, then $M$ is called a \"naive number.\" For example, the four-digit number $7... | math | {
"style": "rule",
"ground_truth": "9313"
} | {
"index": 11350,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a four-digit natural number $M$, let the digit in the thousands place be $a$, in the hundreds place be $b$, in the tens place be $c$, and in the units place be $d$. The two-digit number formed by the thousands and units digits of $M$ is $A=10a+d$, and the two-digit number f... | math | {
"style": "rule",
"ground_truth": "8892"
} | {
"index": 11351,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a function $f(x)$ with domain $I$, if there exists an interval $\\left[m,n\\right]\\subseteq I$ such that $f(x)$ is a monotonic function on the interval $\\left[m,n\\right]$, and the range of the function $y=f(x)$ for $x\\in \\left[m,n\\right]$ is $\\left[m,n\\right]$, then... | math | {
"style": "rule",
"ground_truth": "\\frac{2\\sqrt{3}}{3}"
} | {
"index": 11352,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a geometric sequence $\\{b_n\\}$ with a common ratio of $4$, if $T_n$ is the product of the first $n$ terms of the sequence $\\{b_n\\}$, then $\\frac{T_{20}}{T_{10}}$, $\\frac{T_{30}}{T_{20}}$, $\\frac{T_{40}}{T_{30}}$ also form a geometric sequence with a common ratio of $... | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 11353,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a given arithmetic series the sum of the first $50$ terms is $200$, and the sum of the next $50$ terms is $2700$. The first term in the series is:"
}
] | math | {
"style": "rule",
"ground_truth": "-20.5"
} | {
"index": 11354,
"solution": "1. **Define the terms and expressions for the sums:**\n Let the first term of the arithmetic sequence be $a$ and the common difference be $d$. The $n$-th term of an arithmetic sequence can be expressed as $a + (n-1)d$.\n\n2. **Expression for the sum of the first 50 terms:**\n The ... |
DeepScaleR | [
{
"role": "user",
"content": "For a given integer $n\\ge 2$, let $a_0,a_1,\\ldots ,a_n$ be integers satisfying $0=a_0<a_1<\\ldots <a_n=2n-1$. Find the smallest possible number of elements in the set $\\{ a_i+a_j \\mid 0\\le i \\le j \\le n \\}$."
}
] | math | {
"style": "rule",
"ground_truth": "3n"
} | {
"index": 11355,
"solution": "\nFor a given integer \\( n \\ge 2 \\), let \\( a_0, a_1, \\ldots, a_n \\) be integers satisfying \\( 0 = a_0 < a_1 < \\ldots < a_n = 2n-1 \\). We aim to find the smallest possible number of elements in the set \\( \\{ a_i + a_j \\mid 0 \\le i \\le j \\le n \\} \\).\n\nFirst, we prove... |
DeepScaleR | [
{
"role": "user",
"content": "For a given list of three numbers, the operation \"changesum\" replaces each number in the list with the sum of the other two. For example, applying \"changesum\" to \\(3,11,7\\) gives \\(18,10,14\\). Arav starts with the list \\(20,2,3\\) and applies the operation \"changesum\... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11356,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a given list of three numbers, the operation \"changesum\" replaces each number in the list with the sum of the other two. For example, applying \"changesum\" to \\(3,11,7\\) gives \\(18,10,14\\). Arav starts with the list \\(20,2,3\\) and applies the operation \"changesum\... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 11357,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a given point $P$ on the curve $x^2 - y - \\ln x = 0$, what is the minimum distance from point $P$ to the line $y = x - 2$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 11358,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a given positive integer $ k$ denote the square of the sum of its digits by $ f_1(k)$ and let $ f_{n\\plus{}1}(k) \\equal{} f_1(f_n(k)).$ Determine the value of $ f_{1991}(2^{1990}).$"
}
] | math | {
"style": "rule",
"ground_truth": "256"
} | {
"index": 11359,
"solution": "Let \\( k \\) be a positive integer, and define the function \\( f_1(k) \\) as the square of the sum of the digits of \\( k \\). We are also given a recursive function \\( f_{n+1}(k) = f_1(f_n(k)) \\). We need to find the value of \\( f_{1991}(2^{1990}) \\).\n\n### Step-by-Step Soluti... |
DeepScaleR | [
{
"role": "user",
"content": "For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\\tfrac{N}{2}.$"
}
] | math | {
"style": "rule",
"ground_truth": "2k^3 + 3k^2 + 3k"
} | {
"index": 11360,
"solution": "\nGiven a positive integer \\( k \\), we need to find the minimum value of \\( N \\) such that there exists a set \\( S \\) of \\( 2k + 1 \\) distinct positive integers satisfying the following conditions:\n1. The sum of all elements in \\( S \\) is greater than \\( N \\).\n2. Every s... |
DeepScaleR | [
{
"role": "user",
"content": "For a given positive integer $n > 2^3$, what is the greatest common divisor of $n^3 + 3^2$ and $n + 2$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 11361,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a given positive integer $n$, we define $\\varphi(n)$ to be the number of positive integers less than or equal to $n$ which share no common prime factors with $n$. Find all positive integers $n$ for which $\\varphi(2019 n)=\\varphi\\left(n^{2}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "1346, 2016, 2019"
} | {
"index": 11362,
"solution": "Let $p_{1}, p_{2}, \\ldots, p_{k}$ be the prime divisors of $n$. Then it is known that $\\varphi(n)=n \\cdot \\frac{p_{1}-1}{p_{1}} \\ldots \\frac{p_{k}-1}{p_{k}}$. As $n^{2}$ and $n$ has the same set of prime divisors, it also holds that $\\varphi\\left(n^{2}\\right)=n^{2} \\cdot \\f... |
DeepScaleR | [
{
"role": "user",
"content": "For a given positive integer \\( k \\), let \\( f_{1}(k) \\) represent the square of the sum of the digits of \\( k \\), and define \\( f_{n+1}(k) = f_{1}\\left(f_{n}(k)\\right) \\) for \\( n \\geq 1 \\). Find the value of \\( f_{2005}\\left(2^{2006}\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "169"
} | {
"index": 11363,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a list of five positive integers, none greater than 100, the mean is 1.5 times the mode. If 31, 58, 98, $x$ and $x$ are the five integers, what is the value of $x$?"
}
] | math | {
"style": "rule",
"ground_truth": "34"
} | {
"index": 11364,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute th... | math | {
"style": "rule",
"ground_truth": "9100"
} | {
"index": 11365,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a modified toothpick pattern, the first stage is constructed using 5 toothpicks. If each subsequent stage is formed by adding three more toothpicks than the previous stage, what is the total number of toothpicks needed for the $15^{th}$ stage?"
}
] | math | {
"style": "rule",
"ground_truth": "47"
} | {
"index": 11366,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $b$ , let $N(b)$ denote the number of natural numbers $a$ for which the equation $x^2 + ax + b = 0$ has integer roots. What is the smallest value of $b$ for which $N(b) = 20$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "240"
} | {
"index": 11367,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $n \\ge 3$ , we draw $n - 3$ internal diagonals in a non self-intersecting, but not necessarily convex, n-gon, cutting the $n$ -gon into $n - 2$ triangles. It is known that the value (in degrees) of any angle in any of these triangles is a natural num... | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 11368,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $N$, if at least eight out of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called an \"Eight Immortals Number.\" What is the smallest \"Eight Immortals Number\" greater than $2000$?"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 11369,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $N$, if at least five out of the nine natural numbers $1-9$ can divide $N$, then $N$ is called a \"five-divisible number\". What is the smallest \"five-divisible number\" greater than 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"index": 11370,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a \"six-divisible number\". Among the natural numbers greater than $2000$, what is the smallest \"six-divisible number\"?"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 11371,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $n \\geq 1$, it satisfies: $2002 \\times n$ is a perfect cube, and $n \\div 2002$ is a perfect square. The smallest such $n$ is"
}
] | math | {
"style": "rule",
"ground_truth": "2002^5"
} | {
"index": 11372,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number $n$, if $n+6$ divides $n^3+1996$, then $n$ is called a lucky number of 1996. For example, since $4+6$ divides $4^3+1996$, 4 is a lucky number of 1996. Find the sum of all lucky numbers of 1996."
}
] | math | {
"style": "rule",
"ground_truth": "3720"
} | {
"index": 11373,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least eight out of the nine natural numbers from 1 to 9 can divide \\( N \\), then \\( N \\) is called a \"Ba Xian number\". What is the smallest \"Ba Xian number\" greater than 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 11374,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least five of the natural numbers from 1 to 9 can divide \\( N \\) evenly, then \\( N \\) is called a \"Five-Divisible Number.\" Find the smallest \"Five-Divisible Number\" that is greater than 2000."
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"index": 11375,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least five of the natural numbers from 1 to 9 can divide \\( N \\), then \\( N \\) is called a \"five-rule number.\" What is the smallest \"five-rule number\" greater than 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"index": 11376,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least five out of the nine natural numbers \\( 1 \\) through \\( 9 \\) can divide \\( N \\) evenly, then \\( N \\) is called a \"Five Sequential Number.\" What is the smallest \"Five Sequential Number\" greater than 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "2004"
} | {
"index": 11377,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least seven out of the nine natural numbers from 1 to 9 are factors of \\( N \\), \\( N \\) is called a \"seven-star number.\" What is the smallest \"seven-star number\" greater than 2000?"
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 11378,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( N \\), if at least six of the nine natural numbers from 1 to 9 are factors of \\( N \\), then \\( N \\) is called a “six-match number.” Find the smallest \"six-match number\" greater than 2000."
}
] | math | {
"style": "rule",
"ground_truth": "2016"
} | {
"index": 11379,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a natural number \\( x \\), five statements are made:\n\n$$\n3x > 91\n$$\n\n$$\n\\begin{aligned}\n& x < 120 \\\\\n& 4x > 37 \\\\\n& 2x \\geq 21 \\\\\n& x > 7\n\\end{aligned}\n$$\n\nIt is known that only three of these statements are true, and two are false. Determine \\( x ... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 11380,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$ and a strictly increasing sequence of real numbers $t_0,t_1,\\dots,t_n$, let $f(t)$ be the corresponding real-valued function defined for $t \\geq t_0$ by the following properties: \\begin{enumerate} \\item[(a)] $f(t)$ is continuous for $t \\geq t_... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 11381,
"solution": "The minimum value of $T$ is 29. Write $t_{n+1} = t_0+T$ and define $s_k = t_k-t_{k-1}$ for $1\\leq k\\leq n+1$. On $[t_{k-1},t_k]$, we have $f'(t) = k(t-t_{k-1})$ and so $f(t_k)-f(t_{k-1}) = \\frac{k}{2} s_k^2$. Thus if we define \\[ g(s_1,\\ldots,s_{n+1}) = \\sum_{k=1}^{n+1} ks_k^2, ... |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_7(3n)$ represent the remainder when $3n$ is divided by $7$. Determine the $22^{\\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 4~.$$"
}
] | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 11382,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_7(n)$ denote the remainder when $n$ is divided by $7.$ Determine the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 3.$$"
}
] | math | {
"style": "rule",
"ground_truth": "24"
} | {
"index": 11383,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_7(n)$ represent the remainder when $n$ is divided by $7$. Determine the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_7(3n)\\le 3.$$"
}
] | math | {
"style": "rule",
"ground_truth": "22"
} | {
"index": 11384,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_9(n)$ be the remainder when $n$ is divided by $9$. Consider all nonnegative integers $n$ for which $r_9(7n) \\leq 5$. Find the $15^{\\text{th}}$ entry in an ordered list of all such $n$."
}
] | math | {
"style": "rule",
"ground_truth": "21"
} | {
"index": 11385,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_9(n)$ stand for the remainder left when $n$ is divided by $9.$ For example, $r_9(25)=7.$\n\nWhat is the $22^{\\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_9(5n)\\le 4~?$$(Note that the first entry in... | math | {
"style": "rule",
"ground_truth": "38"
} | {
"index": 11386,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer $n$, let $r_{11}(n)$ represent the remainder when $n$ is divided by $11.$ What is the $15^{\\text{th}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_{11}(7n)\\le 3~?$$"
}
] | math | {
"style": "rule",
"ground_truth": "41"
} | {
"index": 11387,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a nonnegative integer \\(n\\), let \\(r_{11}(7n)\\) stand for the remainder left when \\(n\\) is divided by \\(11.\\) For example, \\(r_{11}(7 \\cdot 3) = 10.\\)\n\nWhat is the \\(15^{\\text{th}}\\) entry in an ordered list of all nonnegative integers \\(n\\) that satisfy $... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 11388,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a pair $ A \\equal{} (x_1, y_1)$ and $ B \\equal{} (x_2, y_2)$ of points on the coordinate plane, let $ d(A,B) \\equal{} |x_1 \\minus{} x_2| \\plus{} |y_1 \\minus{} y_2|$. We call a pair $ (A,B)$ of (unordered) points [i]harmonic[/i] if $ 1 < d(A,B) \\leq 2$. Determine the ... | math | {
"style": "rule",
"ground_truth": "3750"
} | {
"index": 11389,
"solution": "\nGiven a set of 100 points in the plane, we want to determine the maximum number of harmonic pairs, where a pair \\((A, B)\\) of points is considered harmonic if \\(1 < d(A, B) \\leq 2\\) and \\(d(A, B) = |x_1 - x_2| + |y_1 - y_2|\\).\n\nTo solve this problem, we can transform the di... |
DeepScaleR | [
{
"role": "user",
"content": "For a pair of integers \\((a, b)(0 < a < b < 1000)\\), a set \\(S \\subseteq \\{1, 2, \\cdots, 2003\\}\\) is called a \"jump set\" for the pair \\((a, b)\\) if for any pair of elements \\(\\left(s_{1}, s_{2}\\right)\\) in \\(S\\), \\(|s_{1}-s_{2}| \\notin\\{a, b\\}\\).\n\nLet \... | math | {
"style": "rule",
"ground_truth": "668"
} | {
"index": 11390,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a particular value of the angle $\\theta$ we can take the product of the two complex numbers $(8+i)\\sin\\theta+(7+4i)\\cos\\theta$ and $(1+8i)\\sin\\theta+(4+7i)\\cos\\theta$ to get a complex number in the form $a+bi$ where $a$ and $b$ are real numbers. Find t... | math | {
"style": "rule",
"ground_truth": "125"
} | {
"index": 11391,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a permutation $\\pi$ of the integers from 1 to 10, define\n\\[ S(\\pi) = \\sum_{i=1}^{9} (\\pi(i) - \\pi(i+1))\\cdot (4 + \\pi(i) + \\pi(i+1)), \\]\nwhere $\\pi (i)$ denotes the $i$ th element of the permutation. Suppose that $M$ is the maximum possible value of $S(... | math | {
"style": "rule",
"ground_truth": "40320"
} | {
"index": 11392,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a permutation $\\sigma$ of $1,2, \\ldots, 7$, a transposition is a swapping of two elements. Let $f(\\sigma)$ be the minimum number of transpositions necessary to turn $\\sigma$ into the permutation $1,2,3,4,5,6,7$. Find the sum of $f(\\sigma)$ over all permutations $\\sigm... | math | {
"style": "rule",
"ground_truth": "22212"
} | {
"index": 11393,
"solution": "To solve this problem, we use the idea of a cycle in a permutation. If $\\sigma$ is a permutation, we say that $\\left(a_{1} a_{2} \\cdots a_{k}\\right)$ is a cycle if $\\sigma\\left(a_{i}\\right)=\\sigma\\left(a_{i+1}\\right)$ for $1 \\leq i \\leq k-1$ and $\\sigma\\left(a_{k}\\right... |
DeepScaleR | [
{
"role": "user",
"content": "For a permutation $p = (a_1,a_2,\\ldots,a_9)$ of the digits $1,2,\\ldots,9$, let $s(p)$ denote the sum of the three $3$-digit numbers $a_1a_2a_3$, $a_4a_5a_6$, and $a_7a_8a_9$. Let $m$ be the minimum value of $s(p)$ subject to the condition that the units digit of $s(p)$ is $0$... | math | {
"style": "rule",
"ground_truth": "162"
} | {
"index": 11394,
"solution": "To minimize $s(p)$, the numbers $1$, $2$, and $3$ (which sum to $6$) must be in the hundreds places. For the units digit of $s(p)$ to be $0$, the numbers in the ones places must have a sum of either $10$ or $20$. However, since the tens digit contributes more to the final sum $s(p)$ t... |
DeepScaleR | [
{
"role": "user",
"content": "For a point $P = (a, a^2)$ in the coordinate plane, let $\\ell(P)$ denote the line passing through $P$ with slope $2a$ . Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2)$ , $P_2 = (a_2, a_2^2)$ , $P_3 = (a_3, a_3^2)$ , such that the intersections of ... | math | {
"style": "rule",
"ground_truth": "\\[\n\\boxed{y = -\\frac{1}{4}}\n\\]"
} | {
"index": 11395,
"solution": "Solution 1\nNote that the lines $l(P_1), l(P_2), l(P_3)$ are \\[y=2a_1x-a_1^2, y=2a_2x-a_2^2, y=2a_3x-a_3^2,\\] respectively. It is easy to deduce that the three points of intersection are \\[\\left(\\frac{a_1+a_2}{2},a_1a_2\\right),\\left(\\frac{a_2+a_3}{2},a_2a_3\\right), \\left(\\f... |
DeepScaleR | [
{
"role": "user",
"content": "For a point $P = (a,a^2)$ in the coordinate plane, let $l(P)$ denote the line passing through $P$ with slope $2a$. Consider the set of triangles with vertices of the form $P_1 = (a_1, a_1^2), P_2 = (a_2, a_2^2), P_3 = (a_3, a_3^2)$, such that the intersection of the lines $l(P_... | math | {
"style": "rule",
"ground_truth": "y = -\\frac{1}{4}"
} | {
"index": 11396,
"solution": "\nLet \\( P_1 = (a_1, a_1^2) \\), \\( P_2 = (a_2, a_2^2) \\), and \\( P_3 = (a_3, a_3^2) \\) be points in the coordinate plane. The lines \\( l(P_1) \\), \\( l(P_2) \\), and \\( l(P_3) \\) have equations with slopes equal to \\( 2a_1 \\), \\( 2a_2 \\), and \\( 2a_3 \\) respectively. T... |
DeepScaleR | [
{
"role": "user",
"content": "For a point $P=(x, y)$ in the Cartesian plane, let $f(P)=\\left(x^{2}-y^{2}, 2 x y-y^{2}\\right)$. If $S$ is the set of all $P$ so that the sequence $P, f(P), f(f(P)), f(f(f(P))), \\ldots$ approaches $(0,0)$, then the area of $S$ can be expressed as $\\pi \\sqrt{r}$ for some po... | math | {
"style": "rule",
"ground_truth": "133"
} | {
"index": 11397,
"solution": "For a point $P=(x, y)$, let $z(P)=x+y \\omega$, where $\\omega$ is a nontrivial third root of unity. Then $$\\begin{aligned} z(f(P))=\\left(x^{2}-y^{2}\\right)+\\left(2 x y-y^{2}\\right) \\omega=x^{2}+2 x y \\omega+y^{2} & (-1-\\omega) \\\\ & =x^{2}+2 x y \\omega+y^{2} \\omega^{2}=(x+... |
DeepScaleR | [
{
"role": "user",
"content": "For a polynomial $p(x),$ define its munificence as the maximum value of $|p(x)|$ on the interval $-1 \\le x \\le 1.$ For example, the munificence of the polynomial $p(x) = -x^2 + 3x - 17$ is 21, since the maximum value of $|-x^2 + 3x - 17|$ for $-1 \\le x \\le 1$ is 21, occurr... | math | {
"style": "rule",
"ground_truth": "\\frac{1}{2}"
} | {
"index": 11398,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "For a positive constant $c,$ in spherical coordinates $(\\rho,\\theta,\\phi),$ find the shape described by the equation\n\\[\\rho = c.\\](A) Line\n(B) Circle\n(C) Plane\n(D) Sphere\n(E) Cylinder\n(F) Cone\n\nEnter the letter of the correct option."
}
] | math | {
"style": "rule",
"ground_truth": "\\text{(D)}"
} | {
"index": 11399,
"solution": ""
} |
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