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Problem 1089

AIME late

Geometry Difficulty 5.0 Prove it Berkeley Math Circle: Monthly Contest 1 · United States

Determine the number of convex polygons all of whose sides are the square roots of positive integers which can be inscribed in a unit circle. Polygons that are rotations or reflections of each other are considered the same.

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

Solution:

Any chord in a unit circle has length at most $2$ (if it is a diameter), so all side lengths must be $1$ , $\sqrt{2}$ , $\sqrt{3}$ , or $2$ . We can do casework based on the number of sides:

- Triangles: The possible combinations of side lengths are $(\sqrt{2}, \sqrt{2}, 2)$ , $(1,1, \sqrt{3})$ , $(1, \sqrt{3}, 2)$ , and $(\sqrt{3}, \sqrt{3}, \sqrt{3})$ . This is $4$ possibilities.

- Quadrilaterals: The possible ordered combinations of side lengths are $(1,1,1,2)$ , $(1, \sqrt{3}, \sqrt{2}, \sqrt{2})$ , $(1, \sqrt{3}, 1, \sqrt{3})$ , $(1, \sqrt{3}, \sqrt{3}, 1)$ , $(\sqrt{2}, \sqrt{2}, \sqrt{2}, \sqrt{2})$ , and $(1, \sqrt{2}, \sqrt{3}, \sqrt{2})$ . This is $6$ possibilities.

- Pentagons: The possible combinations of side lengths are $(1,1,1,1,1, \sqrt{3})$ , $(1,1,1, \sqrt{2}, \sqrt{2})$ , and $(1,1, \sqrt{2}, 1, \sqrt{2})$ . This is $3$ possibilities.

- Hexagons: The only option is that all sides are $1$ , so there is $1$ possibility.

Putting these together, there are $4+6+3+1=14$ polygons total.

Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.