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Furthest and nearest point have the same color as P

Source: Own. Malaysian IMO TST 2022 P2

May 7, 2022
combinatorial geometrycombinatorics

Problem Statement

Let S\mathcal{S} be a set of 20232023 points in a plane, and it is known that the distances of any two different points in SS are all distinct. Ivan colors the points with kk colors such that for every point P∈SP \in \mathcal{S}, the closest and the furthest point from PP in S\mathcal{S} also have the same color as PP.
What is the maximum possible value of kk?
Proposed by Ivan Chan Kai Chin