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Combinatorial Geometry

Source: 2024 China TST P6

March 7, 2024
combinatoricscombinatorial geometry2024 CTST

Problem Statement

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