IGO 2022 I4

We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$ , there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.)

Proposed by Hesam Rajabzadeh