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Permutations inequality

Source: ISL 2023 A5

July 17, 2024
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Problem Statement

Let a1,a2,…,a2023a_1,a_2,\dots,a_{2023} be positive integers such that
[*] a1,a2,…,a2023a_1,a_2,\dots,a_{2023} is a permutation of 1,2,…,20231,2,\dots,2023, and [*] ∣a1−a2∣,∣a2−a3∣,…,∣a2022−a2023∣|a_1-a_2|,|a_2-a_3|,\dots,|a_{2022}-a_{2023}| is a permutation of 1,2,…,20221,2,\dots,2022.
Prove that max⁡(a1,a2023)≥507\max(a_1,a_{2023})\ge 507.
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