Problem 3, 2020

Let \(n\) be a natural number divisible by \(6\). Prove that there exists a convex \(n\)-gon whose interior angles are all equal and which can be cut into finitely many pieces, each piece being of one of the following two types:

\((1^\circ)\) a regular \(k\)-gon, for some \(k < n\);

\((2^\circ)\) a triangular segment \(A_1A_2A_3\) of a regular \(\ell\)-gon \(A_1A_2 \dots A_\ell\), for some \(\ell < n\); that is, the triangle formed by three consecutive vertices of the regular \(\ell\)-gon, the piece cut off by the diagonal \(A_1A_3\).