Question

Suppose that the vertices of a regular polygon of 20 sides are coloured with three colours - red, blue and green- such that there are exactly three red vertices . Prove that there are three vertices A,B,C of the polygon having the same colour such that triangle ABC is isosceles.

Answer 1

Since there are exactly three vertices, among the remaining 17 vertices there are nine of them of the same colour, say blue. We can divide the vertices of the regular 20-gon into four disjoint sets such that each set consists of vertices that form a regular pentagon. Since there are nine blue points, at least one of these sets have three blue points .Since any three points on a pentagon form an isosceles triangle, the statement follows.