Problem

Source: 2024 China TST P6

Tags: combinatorics, combinatorial geometry, 2024 CTST



Let $m,n>2$ be integers. A regular ${n}$-sided polygon region $\mathcal T$ on a plane contains a regular ${m}$-sided polygon region with a side length of ${}{}{}1$. Prove that any regular ${m}$-sided polygon region $\mathcal S$ on the plane with side length $\cos{\pi}/[m,n]$ can be translated inside $\mathcal T.$ In other words, there exists a vector $\vec\alpha,$ such that for each point in $\mathcal S,$ after translating the vector $\vec\alpha$ at that point, it fall into $\mathcal T.$ Note: The polygonal area includes both the interior and boundaries. Created by Bin Wang