Let us choose arbitrarily $n$ vertices of a regular $2n$-gon and color them red. The remaining vertices are colored blue. We arrange all red-red distances into a nondecreasing sequence and do the same with the blue-blue distances. Prove that the two sequences thus obtained are identical.
Problem
Source: 2012 Romania JBMO TST 4.2, 10th Vojtech Jarnik Mathematical Competition, Ostrava, 2000
Tags: combinatorial geometry, combinatorics, Coloring