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JBMO Shortlist 2022 C2

Source: JBMO Shortlist 2022

June 26, 2023
combinatoricsgeometryJuniorBalkanshortlist

Problem Statement

Let n≥2n \ge 2 be an integer. Alex writes the numbers 1,2,...,n1, 2, ..., n in some order on a circle such that any two neighbours are coprime. Then, for any two numbers that are not comprime, Alex draws a line segment between them. For each such segment ss we denote by dsd_s the difference of the numbers written in its extremities and by psp_s the number of all other drawn segments which intersect ss in its interior. Find the greatest nn for which Alex can write the numbers on the circle such that ps≤∣ds∣p_s \le |d_s|, for each drawn segment ss.
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