Let $ P$ be a regular polygon. A regular sub-polygon of $ P$ is a subset of vertices of $ P$ with at least two vertices such that divides the circumcircle to equal arcs. Prove that there is a subset of vertices of $ P$ such that its intersection with each regular sub-polygon has even number of vertices.
Problem
Source: Iranian National Olympiad (3rd Round) 2008
Tags: geometry, circumcircle, number theory proposed, number theory