Problem

Source: 2004 Romania JBMO TST 4.4

Tags: Coloring, combinatorics, Regular



A regular polygon with $1000$ sides has the vertices colored in red, yellow or blue. A move consists in choosing to adjiacent vertices colored differently and coloring them in the third color. Prove that there is a sequence of moves after which all the vertices of the polygon will have the same color. Marius Ghergu