Thailand Mathematical Olympiad

Consider a circle $\omega$ with center $O$. Without loss of generality, let $O$ be red. If there exist two red points on $\omega$ not belonging to the same diameter, then these points together with $O$ form a red isosceles triangle.

On the other hand, if $\omega$ contains at most 2 red points lying on a diameter, consider a regular pentagon inscribed in $\omega$ with either green or blue vertices. By the pigeonhole principle, at least 3 of its vertices must be the same color. These vertices form an isosceles triangle where all vertices are the same color.