data_source stringclasses 1
value | prompt listlengths 1 1 | ability stringclasses 1
value | reward_model dict | extra_info dict |
|---|---|---|---|---|
DeepScaleR | [
{
"role": "user",
"content": "Define $x \\star y=\\frac{\\sqrt{x^{2}+3 x y+y^{2}-2 x-2 y+4}}{x y+4}$. Compute $$((\\cdots((2007 \\star 2006) \\star 2005) \\star \\cdots) \\star 1)$$"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{15}}{9}"
} | {
"index": 7400,
"solution": "Note that $x \\star 2=\\frac{\\sqrt{x^{2}+6 x+4-2 x-4+4}}{2 x+4}=\\frac{\\sqrt{(x+2)^{2}}}{2(x+2)}=\\frac{1}{2}$ for $x>-2$. Because $x \\star y>0$ if $x, y>0$, we need only compute $\\frac{1}{2} \\star 1=\\frac{\\sqrt{\\frac{1}{4}+\\frac{3}{2}+1-3+4}}{\\frac{1}{2}+4}=\\frac{\\sqrt{15}... |
DeepScaleR | [
{
"role": "user",
"content": "Define $x\\otimes y=x^3-y$. What is $h\\otimes (h\\otimes h)$?"
}
] | math | {
"style": "rule",
"ground_truth": "h"
} | {
"index": 7401,
"solution": "1. **Define the operation $\\otimes$:** Given $x \\otimes y = x^3 - y$.\n\n2. **Calculate $h \\otimes h$:** Using the definition, substitute $x = h$ and $y = h$:\n \\[\n h \\otimes h = h^3 - h.\n \\]\n\n3. **Calculate $h \\otimes (h \\otimes h)$:** Now, substitute $x = h$ and $y ... |
DeepScaleR | [
{
"role": "user",
"content": "Define $||x||$ $(x\\in R)$ as the integer closest to $x$ (when $x$ is the arithmetic mean of two adjacent integers, $||x||$ takes the larger integer). Let $G(x)=||x||$. If $G(\\frac{4}{3})=1$, $G(\\frac{5}{3})=2$, $G(2)=2$, and $G(2.5)=3$, then $\\frac{1}{G(1)}+\\frac{1}{G(2)}+... | math | {
"style": "rule",
"ground_truth": "\\frac{1334}{15}"
} | {
"index": 7402,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define \\( n! = 1 \\times 2 \\times \\ldots \\times n \\), for example \\( 5! = 1 \\times 2 \\times 3 \\times 4 \\times 5 \\). If \\(\\frac{n! \\times (n+1)!}{2}\\) (where \\( \\mathbf{n} \\) is a positive integer and \\( 1 \\leq n \\leq 100 \\)) is a perfect square, what is th... | math | {
"style": "rule",
"ground_truth": "273"
} | {
"index": 7403,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define \\(P(x) =(x-1^2)(x-2^2)\\cdots(x-100^2)\\). How many integers \\(n\\) are there such that \\(P(n)\\leq 0\\)?"
}
] | math | {
"style": "rule",
"ground_truth": "5100"
} | {
"index": 7404,
"solution": "We are given the polynomial \\( P(x) = (x-1^2)(x-2^2)\\cdots(x-100^2) \\) and need to determine how many integers \\( n \\) satisfy \\( P(n) \\leq 0 \\).\n\n1. **Evaluating \\( P(n) = 0 \\):**\n - \\( P(n) = 0 \\) when \\( n \\) is exactly one of the squares from \\( 1^2 \\) to \\( 1... |
DeepScaleR | [
{
"role": "user",
"content": "Define \\[P(x) =(x-1^2)(x-2^2)\\cdots(x-50^2).\\] How many integers $n$ are there such that $P(n)\\leq 0$?"
}
] | math | {
"style": "rule",
"ground_truth": "1300"
} | {
"index": 7405,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \\[a^m\\equiv a\\pmod n.\\] Find the number of tasty residues of $2016$ ."
}
] | math | {
"style": "rule",
"ground_truth": "831"
} | {
"index": 7406,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a \"digitized number\" as a ten-digit number $a_0a_1\\ldots a_9$ such that for $k=0,1,\\ldots, 9$ , $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers."
}
] | math | {
"style": "rule",
"ground_truth": "6210001000"
} | {
"index": 7407,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a \"hook\" to be a figure made up of six unit squares as shown below in the picture, or any of the figures obtained by applying rotations and reflections to this figure. \n\n[asy]\nunitsize(0.5 cm);\n\ndraw((0,0)--(1,0));\ndraw((0,1)--(1,1));\ndraw((2,1)--(3,1));\ndraw((... | math | {
"style": "rule",
"ground_truth": "\\[\n\\{1, 2, 5\\} \\not\\subseteq \\{m, n\\},\n\\]\n\\[\n3 \\mid m \\text{ or } 3 \\mid n,\n\\]\n\\[\n4 \\mid m \\text{ or } 4 \\mid n.\n\\]"
} | {
"index": 7408,
"solution": "\nTo solve this problem, we need to understand the structure and properties of the \"hook\" figure. The hook consists of six unit squares arranged in a specific pattern. We are tasked with determining which \\( m \\times n \\) rectangles can be completely covered using these hooks with... |
DeepScaleR | [
{
"role": "user",
"content": "Define a \"spacy\" set of integers such that it contains no more than one out of any four consecutive integers. How many subsets of $\\{1, 2, 3, \\dots, 15\\}$, including the empty set, are spacy?"
}
] | math | {
"style": "rule",
"ground_truth": "181"
} | {
"index": 7409,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\it{good\\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is ne... | math | {
"style": "rule",
"ground_truth": "8748"
} | {
"index": 7410,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\it{good\\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is nev... | math | {
"style": "rule",
"ground_truth": "94"
} | {
"index": 7411,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\it{good\\ word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ --- some of these letters may not appear in the sequence --- and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is nev... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 7412,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\it{good\\ word}$ as a sequence of letters that consists only of the letters A, B, C, and D — some of these letters may not appear in the sequence — where A is never immediately followed by B or D, B is never immediately followed by C, C is never immediately followed... | math | {
"style": "rule",
"ground_truth": "512"
} | {
"index": 7413,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\it{great\\ word}$ as a sequence of letters that consists only of the letters $D$, $E$, $F$, and $G$ --- some of these letters may not appear in the sequence --- and in which $D$ is never immediately followed by $E$, $E$ is never immediately followed by $F$, $F$ is n... | math | {
"style": "rule",
"ground_truth": "972"
} | {
"index": 7414,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\textit{better word}$ as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never imm... | math | {
"style": "rule",
"ground_truth": "2916"
} | {
"index": 7415,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a $\\text{good~word}$ as a sequence of letters that consists only of the letters $A$, $B$, and $C$ - some of these letters may not appear in the sequence - and in which $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, and $C$ is never i... | math | {
"style": "rule",
"ground_truth": "192"
} | {
"index": 7416,
"solution": "There are three choices for the first letter and two choices for each subsequent letter, so there are $3\\cdot2^{n-1}\\ n$-letter good words. Substitute $n=7$ to find there are $3\\cdot2^6=\\boxed{192}$ seven-letter good words. ~ aopsav (Credit to AoPS Alcumus)"
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a **valid sequence** as a sequence of letters that consists only of the letters $A$, $B$, $C$, and $D$ — some of these letters may not appear in the sequence — where $A$ is never immediately followed by $B$, $B$ is never immediately followed by $C$, $C$ is never immediat... | math | {
"style": "rule",
"ground_truth": "8748"
} | {
"index": 7417,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a T-grid to be a $3\\times3$ matrix which satisfies the following two properties:\nExactly five of the entries are $1$'s, and the remaining four entries are $0$'s.\nAmong the eight rows, columns, and long diagonals (the long diagonals are $\\{a_{13},a_{22},a_{31}\\}$ and... | math | {
"style": "rule",
"ground_truth": "68"
} | {
"index": 7418,
"solution": "The T-grid can be considered as a tic-tac-toe board: five $1$'s (or X's) and four $0$'s (or O's).\n There are only $\\dbinom{9}{5} = 126$ ways to fill the board with five $1$'s and four $0$'s. Now we just need to subtract the number of bad grids. Bad grids are ones with more than one p... |
DeepScaleR | [
{
"role": "user",
"content": "Define a domino to be an ordered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after the first equals the second coordinate of the immediately preceding pair, and in which $(i,j)$ and $(... | math | {
"style": "rule",
"ground_truth": "761"
} | {
"index": 7419,
"solution": "We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected with every other point. The connections represent the dominoes.\n You need to have all even number of segments coming from each point except 0 or 2 which have an odd ... |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $g :\\mathbb{N} \\rightarrow \\mathbb{R}$ Such that $g(x)=\\sqrt{4^x+\\sqrt {4^{x+1}+\\sqrt{4^{x+2}+...}}}$ .\nFind the last 2 digits in the decimal representation of $g(2021)$ ."
}
] | math | {
"style": "rule",
"ground_truth": "53"
} | {
"index": 7420,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $A(m, n)$ by \\[ A(m,n) = \\left\\{ \\begin{aligned} &n+1& \\text{ if } m = 0 \\\\ &A(m-1, 1) & \\text{ if } m > 0 \\text{ and } n = 0 \\\\ &A(m-1, A(m, n-1))&\\text{ if } m > 0 \\text{ and } n > 0. \\end{aligned} \\right.\\]Compute $A(2, 1).$"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7421,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $A(m, n)$ in line with the Ackermann function and compute $A(3, 2)$."
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 7422,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $B(m, n)$ by \\[ B(m,n) = \\left\\{ \\begin{aligned} &n+2& \\text{ if } m = 0 \\\\ &B(m-1, 2) & \\text{ if } m > 0 \\text{ and } n = 0 \\\\ &B(m-1, B(m, n-1))&\\text{ if } m > 0 \\text{ and } n > 0. \\end{aligned} \\right.\\]Compute $B(2, 2).$"
}
] | math | {
"style": "rule",
"ground_truth": "12"
} | {
"index": 7423,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $f$ by $f(1)=1$, $f(2)=2$, and for all integers $n \\geq 3$, \n\\[ f(n) = f(n-1) + f(n-2) + n. \\]\nDetermine $f(10)$."
}
] | math | {
"style": "rule",
"ground_truth": "420"
} | {
"index": 7424,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $f$ from nonnegative integers to real numbers, with $f(1) = 1$ and the functional equation:\n\\[ f(m+n) + f(m-n) = 3(f(m) + f(n)) \\]\nfor all nonnegative integers $m \\ge n$. Determine $f(10)$."
}
] | math | {
"style": "rule",
"ground_truth": "100"
} | {
"index": 7425,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. For $x \\in [-3,-1)$, $f(x)=-(x+2)^{2}$, and for $x \\in [-1,3)$, $f(x)=x$. Calculate the sum $f(1)+f(2)+f(3)+\\ldots+f(2015)$."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 7426,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3, -1]$, $f(x) = -(x+2)^2$, and when $x \\in [-1, 3)$, $f(x) = x$. Calculate the value of $f(1) + f(2) + f(3) + \\ldots + f(2015)$."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 7427,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3,-1)$, $f(x)=-(x+2)^{2}$, and when $x \\in [-1,3)$, $f(x)=x$. Find the value of $f(1)+f(2)+f(3)+\\ldots+f(2016)$."
}
] | math | {
"style": "rule",
"ground_truth": "336"
} | {
"index": 7428,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $f(x)$ on $\\mathbb{R}$ that satisfies $f(x+6)=f(x)$. When $x \\in [-3,3)$, $f(x)=\\begin{cases} -(x+2)^2, & -3 \\leqslant x < -1 \\\\ x, & -1 \\leqslant x < 3 \\end{cases}$. Find $f(4)$ and the value of $f(1)+f(2)+f(3)+\\ldots+f(2016)+f(2017)$."
}
] | math | {
"style": "rule",
"ground_truth": "337"
} | {
"index": 7429,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $g$ from the positive integers to the positive integers with the following properties:\n(i) $g$ is increasing.\n(ii) $g(mn) = g(m)g(n)$ for all positive integers $m$ and $n$.\n(iii) If $m \\neq n$ and $m^n = n^m$, then $g(m) = n$ or $g(n) = m$.\n\nCompute all ... | math | {
"style": "rule",
"ground_truth": "7744"
} | {
"index": 7430,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $g(x),$ for positive integer values of $x,$ by \n\\[\ng(x) = \\left\\{\n\\begin{aligned} \n\\log_3 x & \\quad \\text{if } \\log_3 x \\text{is an integer} \\\\\n1 + g(x + 1) & \\quad \\text{otherwise}. \n\\end{aligned} \n\\right.\n\\]\nCompute $g(50)$."
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 7431,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $g(x),$ for positive integer values of $x,$ by \\[g(x) = \\left\\{\\begin{aligned} \\log_3 x & \\quad \\text{ if } \\log_3 x \\text{ is an integer} \\\\ 1 + g(x + 1) & \\quad \\text{ otherwise}. \\end{aligned} \\right.\\]Compute $g(200).$"
}
] | math | {
"style": "rule",
"ground_truth": "48"
} | {
"index": 7432,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $g(x),$ for positive integer values of $x,$ by \\[g(x) = \\left\\{\\begin{aligned} \\log_3 x & \\quad \\text{ if } \\log_3 x \\text{ is an integer} \\\\ 1 + g(x + 2) & \\quad \\text{ otherwise}. \\end{aligned} \\right.\\] Compute $g(50).$"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7433,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $g(z) = (3 + i)z^2 + \\alpha z + \\gamma$ for all complex $z$, where $\\alpha$ and $\\gamma$ are complex numbers. Assume that $g(1)$ and $g(i)$ both yield real numbers. Determine the smallest possible value of $|\\alpha| + |\\gamma|$."
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 7434,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function $h(x),$ for positive integer values of $x,$ by \\[h(x) = \\left\\{\\begin{aligned} \\log_2 x & \\quad \\text{ if } \\log_2 x \\text{ is an integer} \\\\ 1 + h(x + 1) & \\quad \\text{ otherwise}. \\end{aligned} \\right.\\]Compute $h(100).$"
}
] | math | {
"style": "rule",
"ground_truth": "35"
} | {
"index": 7435,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function \\( f \\) on the set of positive integers \\( N \\) as follows:\n(i) \\( f(1) = 1 \\), \\( f(3) = 3 \\);\n(ii) For \\( n \\in N \\), the function satisfies\n\\[ \n\\begin{aligned}\n&f(2n) = f(n), \\\\\n&f(4n+1) = 2f(2n+1) - f(n), \\\\\n&f(4n+3) = 3f(2n+1) - 2f... | math | {
"style": "rule",
"ground_truth": "92"
} | {
"index": 7436,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function \\( f \\), whose domain is positive integers, such that:\n$$\nf(n)=\\begin{cases}\nn-3 & \\text{if } n \\geq 1000 \\\\\nf(f(n+7)) & \\text{if } n < 1000\n\\end{cases}\n$$\nFind \\( f(90) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "999"
} | {
"index": 7437,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a function \\( f(x) \\) on \\( \\mathbb{R} \\) that satisfies \\( f(-x) = -f(x) \\), \\( f(x-2) = f(x+2) \\), and when \\( x \\in (-1,0) \\), \\( f(x) = 2^x + \\frac{1}{5} \\). Then, evaluate \\( f(\\log_{2} 20) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 7438,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a modified Ackermann function \\( A(m, n) \\) with the same recursive relationships as the original problem: \n\\[ A(m,n) = \\left\\{\n\\begin{aligned} \n&n+1& \\text{ if } m = 0 \\\\\n&A(m-1, 1) & \\text{ if } m > 0 \\text{ and } n = 0 \\\\\n&A(m-1, A(m, n-1))&\\text{ i... | math | {
"style": "rule",
"ground_truth": "29"
} | {
"index": 7439,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a monic irreducible polynomial with integral coefficients to be a polynomial with leading coefficient 1 that cannot be factored, and the prime factorization of a polynomial with leading coefficient 1 as the factorization into monic irreducible polynomials. How many not n... | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7440,
"solution": "$x^{8}+x^{4}+1=\\left(x^{8}+2 x^{4}+1\\right)-x^{4}=\\left(x^{4}+1\\right)^{2}-\\left(x^{2}\\right)^{2}=\\left(x^{4}-x^{2}+1\\right)\\left(x^{4}+x^{2}+1\\right)=$ $\\left(x^{4}-x^{2}+1\\right)\\left(x^{2}+x+1\\right)\\left(x^{2}-x+1\\right)$, and $x^{8}+x+1=\\left(x^{2}+x+1\\right)\\le... |
DeepScaleR | [
{
"role": "user",
"content": "Define a natural number to be \"super-nice\" if it is either the product of exactly three distinct prime numbers or the fourth power of a prime number. What is the sum of the first eight super-nice numbers?"
}
] | math | {
"style": "rule",
"ground_truth": "520"
} | {
"index": 7441,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a new function $\\$N$ such that $\\$N = 0.75N + 2$. Calculate $\\$(\\$(\\$30))$."
}
] | math | {
"style": "rule",
"ground_truth": "17.28125"
} | {
"index": 7442,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a new operation $\\star$ such that for positive integers $a, b, c$, $a \\star b \\star c = \\frac{a \\times b + c}{a + b + c}$. Calculate the value of $4 \\star 8 \\star 2$.\n**A)** $\\frac{34}{14}$ **B)** $\\frac{16}{7}$ **C)** $\\frac{17}{7}$ **D)** $\\frac{32}{14}$ **... | math | {
"style": "rule",
"ground_truth": "\\frac{17}{7}"
} | {
"index": 7443,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a new operation: \\( x \\odot y = 18 + x - a \\times y \\), where \\( a \\) is a constant. For example:\n\\[ 1 \\odot 2 = 18 + 1 - a \\times 2. \\]\n\nIf \\( 2 \\odot 3 = 8 \\), then what is \\( 3 \\odot 5 \\) and \\( 5 \\odot 3 \\)?"
}
] | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 7444,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a number to be an anti-palindrome if, when written in base 3 as $a_{n} a_{n-1} \\ldots a_{0}$, then $a_{i}+a_{n-i}=2$ for any $0 \\leq i \\leq n$. Find the number of anti-palindromes less than $3^{12}$ such that no two consecutive digits in base 3 are equal."
}
] | math | {
"style": "rule",
"ground_truth": "126"
} | {
"index": 7445,
"solution": "Note once the middle digit/pair of digits is determined, it suffices to choose the digits in the left half of the number and ensure no pair of consecutive digits are equal. For a number with an even number of digits, the middle pair is 02 or 20 while for a number with an odd number of ... |
DeepScaleR | [
{
"role": "user",
"content": "Define a operation \"\\*\" between sets A and B, where A\\*B = {p | p = x + y, x ∈ A, y ∈ B}. If A = {1, 2, 3} and B = {1, 2}, then the sum of all elements in the set A\\*B is ____."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7446,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1500$ are not factorial tails?"
}
] | math | {
"style": "rule",
"ground_truth": "300"
} | {
"index": 7447,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?"
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"index": 7448,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails?"
}
] | math | {
"style": "rule",
"ground_truth": "500"
} | {
"index": 7449,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a positive integer $n^{}_{}$ to be a factorial tail if there is some positive integer $m^{}_{}$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $1992$ are not factorial tails?"
}
] | math | {
"style": "rule",
"ground_truth": "396"
} | {
"index": 7450,
"solution": "After testing various values of $m$ in $f(m)$ of solution 1 to determine $m$ for which $f(m) = 1992$, we find that $m \\in \\{7980, 7981, 7982, 7983, 7984\\}$. WLOG, we select $7980$. Furthermore, note that every time $k$ reaches a multiple of $25$, $k!$ will gain two or more additiona... |
DeepScaleR | [
{
"role": "user",
"content": "Define a power cycle to be a set $S$ consisting of the nonnegative integer powers of an integer $a$, i.e. $S=\\left\\{1, a, a^{2}, \\ldots\\right\\}$ for some integer $a$. What is the minimum number of power cycles required such that given any odd integer $n$, there exists some... | math | {
"style": "rule",
"ground_truth": "10"
} | {
"index": 7451,
"solution": "Partition the odd residues mod 1024 into 10 classes: Class 1: $1(\\bmod 4)$. Class $n(2 \\leq n \\leq 9): 2^{n}-1\\left(\\bmod 2^{n+1}\\right)$. Class 10: $-1(\\bmod 1024)$. Let $S_{a}$ be the power cycle generated by $a$. If $a$ is in class 1, all of $S_{a}$ is in class 1. If a is in ... |
DeepScaleR | [
{
"role": "user",
"content": "Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\\ldots, P_nP_1$ such that\nthe points $P_1, P_2,\\ldots, P_n$ are coplanar and no three of them are collinear,\neach of the $n$ line segments intersects at least one of the other line segmen... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 7452,
"solution": "We use the Principle of Inclusion-Exclusion (PIE).\n If we join the adjacent vertices of the regular $n$-star, we get a regular $n$-gon. We number the vertices of this $n$-gon in a counterclockwise direction: $0, 1, 2, 3, \\ldots, n-1.$\n A regular $n$-star will be formed if we choose ... |
DeepScaleR | [
{
"role": "user",
"content": "Define a regular $n$-pointed star to be the union of $n$ line segments $P_1P_2, P_2P_3,\\ldots, P_nP_1$ such that\nthe points $P_1, P_2,\\ldots, P_n$ are coplanar and no three of them are collinear,\neach of the $n$ line segments intersects at least one of the other line segmen... | math | {
"style": "rule",
"ground_truth": "199"
} | {
"index": 7453,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a regular \\(n\\)-pointed star as described in the original problem, but with a modification: the vertex connection rule skips by \\(m\\) steps where \\(m\\) is coprime with \\(n\\) and \\(m\\) is not a multiple of \\(3\\). How many non-similar regular 120-pointed stars ... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7454,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence $\\left\\{a_{n}\\right\\}$ by $a_{1}=1$ and $a_{n}=\\left(a_{n-1}\\right)!+1$ for every $n>1$. Find the least $n$ for which $a_{n}>10^{10}$."
}
] | math | {
"style": "rule",
"ground_truth": "6"
} | {
"index": 7455,
"solution": "We have $a_{2}=2, a_{3}=3, a_{4}=7, a_{5}=7!+1=5041$, and $a_{6}=5041!+1$. But $$5041!+1 \\gg 5041 \\cdot 5040 \\cdot 5039>10^{10}$$ Hence, the answer is 6."
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence $a_{i, j}$ of integers such that $a_{1, n}=n^{n}$ for $n \\geq 1$ and $a_{i, j}=a_{i-1, j}+a_{i-1, j+1}$ for all $i, j \\geq 1$. Find the last (decimal) digit of $a_{128,1}$."
}
] | math | {
"style": "rule",
"ground_truth": "4"
} | {
"index": 7456,
"solution": "By applying the recursion multiple times, we find that $a_{1,1}=1, a_{2, n}=n^{n}+(n+1)^{n+1}$, and $a_{3, n}=n^{n}+2(n+1)^{n+1}+(n+2)^{n+2}$. At this point, we can conjecture and prove by induction that $a_{m, n}=\\sum_{k=0}^{m-1}\\binom{m-1}{k}(n+k)^{n+k}=\\sum_{k \\geq 0}\\binom{m-1... |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence by \\( a_0 = \\frac{1}{3} \\) and \\( a_n = 1 + (a_{n-1} - 1)^3 \\). Compute the infinite product \\( a_0 a_1 a_2 \\dotsm \\)."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{3}{5}"
} | {
"index": 7457,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence of complex numbers by $z_1 = 0$ and\n\\[z_{n + 1} = z_n^2 + i\\]for all $n \\ge 1.$ In the complex plane, how far from the origin is $z_{111}$?"
}
] | math | {
"style": "rule",
"ground_truth": "\\sqrt{2}"
} | {
"index": 7458,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence of convex polygons \\( P_n \\) as follows. \\( P_0 \\) is an equilateral triangle with side length 1. \\( P_{n+1} \\) is obtained from \\( P_n \\) by cutting off the corners one-third of the way along each side (for example, \\( P_1 \\) is a regular hexagon wi... | math | {
"style": "rule",
"ground_truth": "\\frac{\\sqrt{3}}{7}"
} | {
"index": 7459,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence of integers by $T_1 = 2$ and for $n\\ge2$ , $T_n = 2^{T_{n-1}}$ . Find the remainder when $T_1 + T_2 + \\cdots + T_{256}$ is divided by 255.\n\n*Ray Li.*"
}
] | math | {
"style": "rule",
"ground_truth": "20"
} | {
"index": 7460,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence of polynomials as follows: let $a_{1}=3 x^{2}-x$, let $a_{2}=3 x^{2}-7 x+3$, and for $n \\geq 1$, let $a_{n+2}=\\frac{5}{2} a_{n+1}-a_{n}$. As $n$ tends to infinity, what is the limit of the sum of the roots of $a_{n}$ ?"
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{13}{3}"
} | {
"index": 7461,
"solution": "By using standard methods for solving linear recurrences, we see that this recurrence has a characteristic polynomial of $x^{2}-\\frac{5}{2} x+1=\\left(x-\\frac{1}{2}\\right)(x-2)$, hence $a_{n}(x)=c(x) \\cdot 2^{n}+d(x) \\cdot 2^{-n}$ for some polynomials $c$ and $d$. Plugging in $n=1... |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$. Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$."
}
] | math | {
"style": "rule",
"ground_truth": "101"
} | {
"index": 7462,
"solution": "First it will be shown by induction that the zeros of $f_n$ are the integers $a, {a+2,} {a+4,} \\dots, {a + n(n-1)}$, where $a = n - \\frac{n(n-1)}2.$\n This is certainly true for $n=1$. Suppose that it is true for $n = m-1 \\ge 1$, and note that the zeros of $f_m$ are the solutions of... |
DeepScaleR | [
{
"role": "user",
"content": "Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\\[t_n = \\frac{5t_{n-1}+1}{25t_{n-2}}\\]for all $n \\ge 3$. Then $t_{2020}$ can be expressed as $\\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$."
}
] | math | {
"style": "rule",
"ground_truth": "626"
} | {
"index": 7463,
"solution": "Let $t_n=\\frac{s_n}{5}$. Then, we have $s_n=\\frac{s_{n-1}+1}{s_{n-2}}$ where $s_1 = 100$ and $s_2 = 105$. By substitution, we find $s_3 = \\frac{53}{50}$, $s_4=\\frac{103}{105\\cdot50}$, $s_5=\\frac{101}{105}$, $s_6=100$, and $s_7=105$. So $s_n$ has a period of $5$. Thus $s_{2020}=s_... |
DeepScaleR | [
{
"role": "user",
"content": "Define a set \\( \\mathcal{T} \\) of distinct positive integers such that for every integer \\( y \\) in \\( \\mathcal{T}, \\) the geometric mean of the values obtained by omitting \\( y \\) from \\( \\mathcal{T} \\) remains a positive integer. In addition, assume that 1 is a m... | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7464,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define a set of integers \"spacy\" if it contains no more than one out of any three consecutive integers. How many subsets of $\\{1, 2, 3, \\dots, 10\\}$, including the empty set, are spacy?"
}
] | math | {
"style": "rule",
"ground_truth": "60"
} | {
"index": 7465,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define an odd function \\( f(x) \\) on \\( \\mathbb{R} \\) that satisfies \\( f(x+1) \\) is an even function, and when \\( x \\in [0,1] \\), \\( f(x) = x(3-2x) \\). Then, find the value of \\( f\\left(\\frac{31}{2}\\right) \\)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 7466,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define an odd function f(x) on ℝ that satisfies f(x+1) is an even function, and when x ∈ [0,1], f(x) = x(3-2x). Evaluate f(31/2)."
}
] | math | {
"style": "rule",
"ground_truth": "-1"
} | {
"index": 7467,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define an operation between sets A and B: $A*B = \\{x | x = x_1 + x_2, \\text{ where } x_1 \\in A, x_2 \\in B\\}$. If $A = \\{1, 2, 3\\}$ and $B = \\{1, 2\\}$, then the sum of all elements in $A*B$ is ____."
}
] | math | {
"style": "rule",
"ground_truth": "14"
} | {
"index": 7468,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define an ordered quadruple of integers $(a, b, c, d)$ as captivating if $1 \\le a < b < c < d \\le 15$, and $a+d > 2(b+c)$. How many captivating ordered quadruples are there?"
}
] | math | {
"style": "rule",
"ground_truth": "200"
} | {
"index": 7469,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define an ordered quadruple of integers $(a, b, c, d)$ as interesting if $1 \\le a<b<c<d \\le 10$, and $a+d>b+c$. How many interesting ordered quadruples are there?"
}
] | math | {
"style": "rule",
"ground_truth": "80"
} | {
"index": 7470,
"solution": "We first start out when the value of $a=1$.\n Doing casework, we discover that $d=5,6,7,8,9,10$. We quickly find a pattern.\n Now, doing this for the rest of the values of $a$ and $d$, we see that the answer is simply:\n$(1)+(2)+(1+3)+(2+4)+(1+3+5)+(2+4+6)+(1)+(2)+(1+3)+(2+4)$ $+(1+3+5... |
DeepScaleR | [
{
"role": "user",
"content": "Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \\cap B| = |B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. For example, $(\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally intersecting triple. Let $N$ be the number of minimal... | math | {
"style": "rule",
"ground_truth": "760"
} | {
"index": 7471,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define an ordered triple $(A, B, C)$ of sets to be minimally intersecting if $|A \\cap B| = |B \\cap C| = |C \\cap A| = 1$ and $A \\cap B \\cap C = \\emptyset$. For example, $(\\{1,2\\},\\{2,3\\},\\{1,3,4\\})$ is a minimally intersecting triple. Let $N$ be the number of minimal... | math | {
"style": "rule",
"ground_truth": "760"
} | {
"index": 7472,
"solution": "Let each pair of two sets have one element in common. Label the common elements as $x$, $y$, $z$. Set $A$ will have elements $x$ and $y$, set $B$ will have $y$ and $z$, and set $C$ will have $x$ and $z$. There are $7 \\cdot 6 \\cdot 5 = 210$ ways to choose values of $x$, $y$ and $z$. T... |
DeepScaleR | [
{
"role": "user",
"content": "Define an ordered triple $(D, E, F)$ of sets to be minimally intersecting if $|D \\cap E| = |E \\cap F| = |F \\cap D| = 1$ and $D \\cap E \\cap F = \\emptyset$. Let $M$ be the number of such ordered triples where each set is a subset of $\\{1,2,3,4,5,6,7,8\\}$. Find $M$ modulo ... | math | {
"style": "rule",
"ground_truth": "064"
} | {
"index": 7473,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define binary operations $\\diamondsuit$ and $\\heartsuit$ by $a \\diamondsuit b = a^{\\log_{7}(b)}$ and $a \\heartsuit b = a^{\\frac{1}{\\log_{7}(b)}}$ for all real numbers $a$ and $b$ for which these expressions are defined. The sequence $(a_n)$ is defined recursively by $a_3... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 7474,
"solution": "1. **Define the operations and initial condition**: We are given two binary operations $\\diamondsuit$ and $\\heartsuit$ defined by:\n \\[\n a \\diamondsuit b = a^{\\log_{7}(b)} \\quad \\text{and} \\quad a \\heartsuit b = a^{\\frac{1}{\\log_{7}(b)}}\n \\]\n The sequence $(a_n)$... |
DeepScaleR | [
{
"role": "user",
"content": "Define mutually externally tangent circles $\\omega_1$ , $\\omega_2$ , and $\\omega_3$ . Let $\\omega_1$ and $\\omega_2$ be tangent at $P$ . The common external tangents of $\\omega_1$ and $\\omega_2$ meet at $Q$ . Let $O$ be the center of $\\omega_3$ . If $QP ... | math | {
"style": "rule",
"ground_truth": "77"
} | {
"index": 7475,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the *bigness*of a rectangular prism to be the sum of its volume, its surface area, and the lengths of all of its edges. Find the least integer $N$ for which there exists a rectangular prism with integer side lengths and *bigness* $N$ and another one with integer side ... | math | {
"style": "rule",
"ground_truth": "55"
} | {
"index": 7476,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the *hotel elevator cubic*as the unique cubic polynomial $P$ for which $P(11) = 11$ , $P(12) = 12$ , $P(13) = 14$ , $P(14) = 15$ . What is $P(15)$ ?\n\n*Proposed by Evan Chen*"
}
] | math | {
"style": "rule",
"ground_truth": "13"
} | {
"index": 7477,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the Fibonacci numbers by $F_{0}=0, F_{1}=1, F_{n}=F_{n-1}+F_{n-2}$ for $n \\geq 2$. For how many $n, 0 \\leq n \\leq 100$, is $F_{n}$ a multiple of 13?"
}
] | math | {
"style": "rule",
"ground_truth": "15"
} | {
"index": 7478,
"solution": "The sequence of remainders modulo 13 begins $0,1,1,2,3,5,8,0$, and then we have $F_{n+7} \\equiv 8 F_{n}$ modulo 13 by a straightforward induction. In particular, $F_{n}$ is a multiple of 13 if and only if $7 \\mid n$, so there are 15 such $n$."
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the annoyingness of a permutation of the first \\(n\\) integers to be the minimum number of copies of the permutation that are needed to be placed next to each other so that the subsequence \\(1,2, \\ldots, n\\) appears. For instance, the annoyingness of \\(3,2,1\\) is 3... | math | {
"style": "rule",
"ground_truth": "\\frac{2023}{2}"
} | {
"index": 7479,
"solution": "For a given permutation \\(p_{1}, \\ldots, p_{n}\\), let \\(f_{k}(p)\\) be the smallest number of copies of \\(p\\) that need to be placed next to each other to have \\(1, \\ldots, k\\) appear as a subsequence. We are interested in finding the expectation of \\(f_{n}(p)\\). Notice that... |
DeepScaleR | [
{
"role": "user",
"content": "Define the derivative of the $(n-1)$th derivative as the $n$th derivative $(n \\in N^{*}, n \\geqslant 2)$, that is, $f^{(n)}(x)=[f^{(n-1)}(x)]'$. They are denoted as $f''(x)$, $f'''(x)$, $f^{(4)}(x)$, ..., $f^{(n)}(x)$. If $f(x) = xe^{x}$, then the $2023$rd derivative of the f... | math | {
"style": "rule",
"ground_truth": "-\\frac{2023}{2024}"
} | {
"index": 7480,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the determinant operation $\\begin{vmatrix} a_{1} & a_{2} \\\\ b_{1} & b_{2}\\end{vmatrix} =a_{1}b_{2}-a_{2}b_{1}$, and consider the function $f(x)= \\begin{vmatrix} \\sqrt {3} & \\sin x \\\\ 1 & \\cos x\\end{vmatrix}$. If the graph of this function is translated to the ... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5\\pi}{6}"
} | {
"index": 7481,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $f(x) = 2x - 5$. For what value of $x$ is $f(x)$ equal to $f^{-1}(x)$?"
}
] | math | {
"style": "rule",
"ground_truth": "5"
} | {
"index": 7482,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $f(x)$ on $\\mathbb{R}$ such that $f(0)=0$, $f(x)+f(1-x)=1$, $f\\left(\\frac{x}{5}\\right)=\\frac{1}{2}f(x)$, and for $0 \\leq x_1 < x_2 \\leq 1$, $f(x_1) \\leq f(x_2)$. Find the value of $f\\left(\\frac{1}{2007}\\right)$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{1}{32}"
} | {
"index": 7483,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $f(x)=\\frac{b}{2x-3}$. If $f(2)=f^{-1}(b+1)$, find the product of all possible values of $b$."
}
] | math | {
"style": "rule",
"ground_truth": "-\\frac{3}{2}"
} | {
"index": 7484,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $f: \\mathbb{R} \\rightarrow \\mathbb{R}$ by $$f(x)= \\begin{cases}\\frac{1}{x^{2}+\\sqrt{x^{4}+2 x}} & \\text { if } x \\notin(-\\sqrt[3]{2}, 0] \\\\ 0 & \\text { otherwise }\\end{cases}$$ The sum of all real numbers $x$ for which $f^{10}(x)=1$ can be writt... | math | {
"style": "rule",
"ground_truth": "932"
} | {
"index": 7485,
"solution": "If $x \\in(-\\sqrt[3]{2}, 0]$, it is evidently not a solution, so let us assume otherwise. Then, we find $$f(x)=\\frac{\\sqrt{x^{4}+2 x}-x^{2}}{2 x}$$ which implies that $x f(x)^{2}+x^{2} f(x)-1 / 2=0$, by reverse engineering the quadratic formula. Therefore, if $x>0, f(x)$ is the uniq... |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}\\cdots p_k^{e_k}$ is the prime factorization of $n>1$, then\n\\[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\\cdots (p_k+1)^{e_k-1}.\\]\nFor every $m\\ge 2$, let $f_m(n)=f_1(f_{m-1}(n... | math | {
"style": "rule",
"ground_truth": "18"
} | {
"index": 7486,
"solution": "1. **Define Interesting and Boring Numbers**: A number $n$ is termed as *interesting* if the sequence $\\{f_k(n)\\}$ is unbounded. Conversely, it is *boring* if the sequence is bounded.\n\n2. **Reduction to Prime Factorization**: For a prime $p$, $f_1(p) = 1$. If $n = p_1^{e_1} p_2^{e_... |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $g$ on the set of integers such that \\[g(n)= \\begin{cases} n-4 & \\mbox{if } n \\geq 2000 \\\\ g(g(n+6)) & \\mbox{if } n < 2000. \\end{cases}\\] Determine $g(172)$."
}
] | math | {
"style": "rule",
"ground_truth": "2000"
} | {
"index": 7487,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the function $g(x) = x^{-2} + \\frac{x^{-2}}{1+x^{-2}}$. Determine $g(g(3))$."
}
] | math | {
"style": "rule",
"ground_truth": "\\frac{72596100}{3034921}"
} | {
"index": 7488,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the function \\(f(n)\\) on the positive integers such that \\(f(f(n)) = 3n\\) and \\(f(3n + 1) = 3n + 2\\) for all positive integers \\(n\\). Find \\(f(729)\\)."
}
] | math | {
"style": "rule",
"ground_truth": "729"
} | {
"index": 7489,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the lengths of intervals $(m, n)$, $[m, n)$, $(m, n]$, and $[m, n]$ to be $n - m$ ($n, m \\in \\mathbf{R}$ and $n > m$). Find the sum of the lengths of the intervals for real numbers $x$ that satisfy the inequality\n\\[\n\\frac{1}{x-20}+\\frac{1}{x-17} \\geqslant \\frac{... | math | {
"style": "rule",
"ground_truth": "1024"
} | {
"index": 7490,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation \"□\" as: $a□b=a^2+2ab-b^2$. Let the function $f(x)=x□2$, and the equation related to $x$ is $f(x)=\\lg|x+2|$ ($x\\neq -2$) has exactly four distinct real roots $x_1$, $x_2$, $x_3$, $x_4$. Find the value of $x_1+x_2+x_3+x_4$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 7491,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation \"\" such that $ab = a^2 + 2ab - b^2$. Let the function $f(x) = x2$, and the equation $f(x) = \\lg|x + 2|$ (where $x \\neq -2$) has exactly four distinct real roots $x_1, x_2, x_3, x_4$. Find the value of $x_1 + x_2 + x_3 + x_4$."
}
] | math | {
"style": "rule",
"ground_truth": "-8"
} | {
"index": 7492,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation $\\S$ as follows: $a\\,\\S\\, b=3a+5b$. What is the value of $7\\,\\S\\,2$?"
}
] | math | {
"style": "rule",
"ground_truth": "31"
} | {
"index": 7493,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation $\\spadesuit$ as $a\\,\\spadesuit\\,b = |a- b|$ . What is the value of $2\\, \\spadesuit\\,(4\\,\\spadesuit\\,7)$?"
}
] | math | {
"style": "rule",
"ground_truth": "1"
} | {
"index": 7494,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation $a\\nabla b = 2 + b^a$. What is the value of $(1\\nabla 2) \\nabla 3$?"
}
] | math | {
"style": "rule",
"ground_truth": "83"
} | {
"index": 7495,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation: $\\begin{vmatrix} a_{1} & a_{2} \\\\ a_{3} & a_{4}\\end{vmatrix} =a_{1}a_{4}-a_{2}a_{3}$, and consider the function $f(x)= \\begin{vmatrix} \\sqrt {3} & \\sin \\omega x \\\\ 1 & \\cos \\omega x\\end{vmatrix} (\\omega > 0)$. If the graph of $f(x)$ is shifte... | math | {
"style": "rule",
"ground_truth": "\\dfrac{5}{4}"
} | {
"index": 7496,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operation: \\(a \\quad b = \\frac{a \\times b}{a + b}\\). Calculate the result of the expression \\(\\frac{20102010 \\cdot 2010 \\cdots 2010 \\cdot 201 \\text{ C}}{\\text{ 9 \" \"}}\\)."
}
] | math | {
"style": "rule",
"ground_truth": "201"
} | {
"index": 7497,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the operations: \n\\( a \\bigcirc b = a^{\\log _{7} b} \\),\n\\( a \\otimes b = a^{\\frac{1}{\\log ^{6} b}} \\), where \\( a, b \\in \\mathbb{R} \\). \n\nA sequence \\(\\{a_{n}\\} (n \\geqslant 4)\\) is given such that:\n\\[ a_{3} = 3 \\otimes 2, \\quad a_{n} = (n \\otim... | math | {
"style": "rule",
"ground_truth": "11"
} | {
"index": 7498,
"solution": ""
} |
DeepScaleR | [
{
"role": "user",
"content": "Define the polynomials $P_0, P_1, P_2 \\cdots$ by:\n\\[ P_0(x)=x^3+213x^2-67x-2000 \\]\n\\[ P_n(x)=P_{n-1}(x-n), n \\in N \\]\nFind the coefficient of $x$ in $P_{21}(x)$."
}
] | math | {
"style": "rule",
"ground_truth": "61610"
} | {
"index": 7499,
"solution": "\nTo find the coefficient of \\( x \\) in \\( P_{21}(x) \\), we need to evaluate the transformation of the polynomial \\( P_0(x) \\) through a series of substitutions as defined by the recurrence relation \\( P_n(x) = P_{n-1}(x-n) \\).\n\nInitially, we have:\n\\[ P_0(x) = x^3 + 213x^2 ... |
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