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GARA NAZIONALE di MATEMATICA

Combinatorics Difficulty 4.3 AIME Prove it Italy

Problem:

Prove that in every convex polyhedron there are at least two faces with the same number of sides.

Solution

Solution:

Let $n$ be the number of faces of the polyhedron. Since distinct sides of a face border distinct faces, each face can have a number of sides between $3$ and $(n-1)$ . It follows that there are at least two faces with the same number of sides.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.