Maths Olympiad Prep

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HMMT February

Geometry Difficulty 4.8 AIME Find the answer

Each square in the following hexomino has side length 1. Find the minimum area of any rectangle that contains the entire hexomino.

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

If a rectangle contains the entire hexomino, it must also contain its convex hull, which is an origin-symmetric hexagon. It is fairly clear that the smallest rectangle that contains such a hexagon must share one set of parallel sides with the hexagon. There are three such rectangles, and checking them all, we find that the one shown below is the smallest. It has area 212\frac{21}{2}.

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