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"Find the greatest positive integer $m$ with the following property:\nFor every permutation $a_1, a_(...TRUNCATED) | "1. **Understanding the Problem:**\n We need to find the greatest positive integer \\( m \\) such (...TRUNCATED) | 3 | Number Theory | math-word-problem | aops_forum | ac83a1a6-e2f0-5a50-92df-fd4146d4b888 | [
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"Find the largest integer $k$ ($k \\ge 2$), for which there exists an integer $n$ ($n \\ge k$) such (...TRUNCATED) | "To solve this problem, we need to find the largest integer \\( k \\) such that for any collection o(...TRUNCATED) | 108 | Number Theory | math-word-problem | aops_forum | 899b636b-14e1-5f57-a6d3-6a0f61bbea74 | [
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"A capricious mathematician writes a book with pages numbered from $2$ to $400$. The pages are to be(...TRUNCATED) | "To determine the last page to be read, we need to follow the given reading order and identify the p(...TRUNCATED) | 397 | Number Theory | math-word-problem | aops_forum | 8f543768-14fb-5b81-8399-9e213344beec | [
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"## Task 2.\n\nLet $a \\geqslant 2018$ be a real number. In each of 2018 jars, there is a finite num(...TRUNCATED) | "## Solution.\n\nThe smallest number of balls of equal mass among the selected balls is 2018.\n\nWit(...TRUNCATED) | 2018 | Number Theory | math-word-problem | olympiads | e1ff9840-45dd-57b3-ac6d-88f353c465c7 | [
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"Let $a_{1}, a_{2}, \\ldots$ be a sequence of integers that has infinitely many positive and infinit(...TRUNCATED) | "Solution. In the sequence, no integer can appear more than once: if for some $i<j$ we have $a_{i}=a(...TRUNCATED) | 1 | Number Theory | math-word-problem | olympiads | 9c55ab12-6a18-5920-b75a-47522dd12c6f | [
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"5. In the cells of a $75 \\times 75$ table, pairwise distinct natural numbers are placed. Each of t(...TRUNCATED) | "Answer: 4218.\n\nSolution. We will say that a composite number $a$ serves a prime number $p$ if the(...TRUNCATED) | 4218 | Number Theory | math-word-problem | olympiads | 89a3d61c-62bd-5d5f-b356-5fb4c2e04308 | [
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"3. Find the smallest natural number $n$, such that among any $n$ irrational numbers, there always e(...TRUNCATED) | "Solution: Consider the four numbers $\\sqrt{2}, \\sqrt{2}, -\\sqrt{2}, -\\sqrt{2}$, we can see that(...TRUNCATED) | 5 | Number Theory | math-word-problem | olympiads | 4cb127e9-36f3-50d6-a1cc-bfad11a53dd6 | [
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] | null | null | 2 | [{"content":"3. Find the smallest natural number $n$, such that among any $n$ irrational numbers, th(...TRUNCATED) | "<think>\nOkay, so I need to find the smallest natural number n such that among any n irrational num(...TRUNCATED) | 61,442 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
"18th Chinese 2003 Problem A2 S is a subset of {1, 2, 3, ... , 100} such that given any two elements(...TRUNCATED) | "72 Solution Thanks to Polly T Wang Let S be the set of all integers n ≤ 100 divisible by just one(...TRUNCATED) | 72 | Number Theory | math-word-problem | olympiads | 77541473-9458-5064-8f47-4e9fe180e325 | [
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] | ["<think>\nAlright, let me try to figure out this problem. So, the question is about finding the lar(...TRUNCATED) | [
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"\n10.4. A positive integer is called simple if its ordinary decimal representation consists entirel(...TRUNCATED) | "\nSolution. We can always write $n=a_{l}+a_{2}+\\cdots+a_{9}$ where $a_{j}$ has 1 's in the places (...TRUNCATED) | 9 | Number Theory | math-word-problem | olympiads | b4ca4970-2037-514c-bc4d-5187854e705f | [
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"For example, $5 n$ positive integers $x_{1}, x_{2}, \\cdots, x_{n}$ have a sum of 2009. If these $n(...TRUNCATED) | "Let the 41 groups be $A_{1}, A_{2}, \\cdots, A_{41}$, where the sum of the numbers in each group is(...TRUNCATED) | 89 | Number Theory | math-word-problem | cn_contest | 681f4663-ad89-5bee-b2e1-f7f46cbbb8bc | [
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] | ["<think>\nAlright, so I need to find the minimum number of positive integers, n, such that their su(...TRUNCATED) | [
true,
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] | null | null | 2 | [{"content":"For example, $5 n$ positive integers $x_{1}, x_{2}, \\cdots, x_{n}$ have a sum of 2009.(...TRUNCATED) | "<think>\nOkay, so I need to solve this problem where there are n positive integers that add up to 2(...TRUNCATED) | 60,403 | "<|im_start|>system\nYou are Qwen, created by Alibaba Cloud. You are a helpful assistant.<|im_end|>\(...TRUNCATED) |
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