IMO 1985 SL 19

Origin: ISR

Problem

For which integers n \geq3 does there exist a regular n-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?

Solution

Suppose that for some n > 6 there is a regular n-gon with vertices having integer coordinates, and that A1A2 . . . An is the smallest such n-gon, of side length a. If O is the origin and Bi the point such that −−\to OBi = −−−−\to Ai−1Ai, i = 1, 2, . . . , n (where A0 = An), then Bi has integer coordinates and B1B2 . . . Bn is a regular polygon of side length 2a sin(\pi/n) < a, which is impossible. It remains to analyze the cases n \leq6. If P is a regular n-gon with n = 3, 5, 6, then its center C has rational coordinates. We may suppose that C also has integer coordinates and then rotate P around C thrice through 90◦, thus obtaining a regular 12-gon or 20-gon, which is impossible. Hence we must have n = 4 which is indeed a solution.