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Partitioning vertices of 2n gons in n pairs with distinct distances

Source: Bundeswettbewerb Mathematik 2023, Round 2 - Problem 4

September 8, 2023
combinatoricscombinatorics proposedconstruction

Problem Statement

Exactly nn chords (i.e. diagonals and edges) of a regular 2n2n-gon are coloured red, satisfying the following two conditions:
(1) Each of the 2n2n vertices occurs exactly once as the endpoint of a red chord. (2) No two red chords have the same length.
For which positive integers n≥2n \ge 2 is this possible?
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