Competition problem — Combinatorics

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Combinatorics Difficulty 6.4 National Olympiad Prove it Hong Kong

The base of a pyramid is a convex polygon with $9$ sides. Each of the diagonals of the base and each of the edges on the lateral surface of the pyramid is coloured either black or white. Both colours are used. (Note that the sides of the base are not coloured.) Prove that there are three segments coloured the same colour which form a triangle.

Solution

Consider the $9$ edges on the lateral surface. By the pigeonhole principle, $5$ of these edges $PA_1, PA_2, PA_3, PA_4, PA_5$ are of the same colour, say black. WLOG assume $A_1, A_2, A_3, A_4, A_5$ lie in this order at the base. Since the base has $9$ sides, WLOG assume $A_1A_2$ is a diagonal. Then $A_1A_4$ and $A_2A_4$ are also diagonals. If one of $A_1A_2, A_1A_4, A_2A_4$ is black, then it forms a monochromatic triangle with $P$ . Otherwise, all of $A_1A_2, A_1A_4, A_2A_4$ are white, so we still get a monochromatic triangle.

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