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Problem 972

AMC 12 late, AIME early

Combinatorics Difficulty 4.9 Prove it Saudi Arabia booklet · Saudi Arabia · 2024

A rhombus of side length 1 and internal angle 60° is called a diamond. Suppose that a regular hexagon of side length $n$ is dissected into diamonds. Prove that each main diagonal of the hexagon halves exactly $n$ diamonds.

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Divide the hexagon into $6n^2$ triangular cells (in each dissection every diamond will consist of two of them) and consider a half $H$ of the hexagon (on one side on a fixed main diagonal $d$ ).

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