19. (ISR 3) For which integers $n \geq 3$ does there exist a regular $n$-gon in the plane such that all its vertices have integer coordinates in a rectangular coordinate system?
19. Suppose that for some $n>6$ there is a regular $n$-gon with vertices having integer coordinates, and that $A_{1} A_{2} \ldots A_{n}$ is the smallest such $n$-gon, of side length $a$. If $O$ is the origin and $B_{i}$ the point such that $\overrightarrow{O B_{1}}=\overrightarrow{A_{1-1} A_{1}}$, $i=1,2, \ldots, n$ (where $A_{0}=A_{n}$ ), then $B_{1}$ has integer coordinates and $B_{1} B_{2} \ldots B_{n}$ is a regular polygon of side length $2 a \sin (\pi / n)<a$, which is impossible.
It remains to analyze the cases $n \leq 6$. If $\mathcal{P}$ 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 $\mathcal{P}$ around $C$ thrice through $90^{\circ}$, thus obtaining a regular 12 -gon or 20 -gon, which is impossible. Hence we must have $n=4$ which is indeed a solution.