Competition problem — Combinatorics

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Combinatorics Difficulty 7.8 National Olympiad, round 2 Prove it Hong Kong

Let $n$ be an integer, and let $A$ be a subset of $\{0, 1, 2, 3, \dots, 5^n\}$ consisting of $4n+2$ numbers. Prove that there exist $a, b, c \in A$ such that $a < b < c$ and $c + 2a > 3b$ .

Solution

(IMO Shortlist 2021 A1) See the official solution.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.