Each integer has one of three colors. Prove that there exists two distinct integers of the same color with the difference between them being a square.

score 343 · Answer 9 · 2021-03-22 13:32:42

343

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If $S\subseteq \{1,2,...,15\}$ has more than $5$ elements, then there are two distinct subsets of $S$ with the same element sum.

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score 249 · Answer 10 · 2026-03-27 12:17:36

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score 73 · Answer 11 · 2021-03-24 04:24:48

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score 139 · Answer 12 · 2026-03-31 03:30:52

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score 45 · Answer 13 · 2021-03-26 09:08:38

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score 63 · Answer 14 · 2021-03-26 17:19:44

63

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Let $S$ be the subset of $\{1, 2, \dots, 100\}$ where $|S|=26$. Then, $S$ contains an odd or there exists $x,y\in S$ such that $x|y$

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score 248 · Answer 15 · 2021-03-26 19:03:34

248

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$S \subset \mathbb{Z}$ where $|S|=n$ and $n \ge 3$. Show $\exists \ x ,y \in S$ distinct with $x-y \equiv 0 \mod n-1$

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Each integer has one of three colors. Prove that there exists two distinct integers of the same color with the difference between them being a square.

If $S\subseteq \{1,2,...,15\}$ has more than $5$ elements, then there are two distinct subsets of $S$ with the same element sum.

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Let $S$ be the subset of $\{1, 2, \dots, 100\}$ where $|S|=26$. Then, $S$ contains an odd or there exists $x,y\in S$ such that $x|y$

$S \subset \mathbb{Z}$ where $|S|=n$ and $n \ge 3$. Show $\exists \ x ,y \in S$ distinct with $x-y \equiv 0 \mod n-1$

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