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 will have three blue points.
Since any three points on a pentagon form an isosceles triangle, the statement follows.