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Brazilian Mathematical Olympiad

Combinatorics Difficulty 4.6 AIME Prove it Brazil

Show that a convex polyhedron with an odd number of faces has at least one face with an even number of edges.

Solution

Count the number $N$ of pairs $(F, e)$ , where $F$ is a face and $e$ an edge belonging to $F$ . Each edge belongs to two faces, so $N$ is even. Hence the number of faces with an odd number of edges must be even. The total number of faces is odd, so the number of faces with an even number of edges must be odd. In particular, there is at least one.

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