RU-ALLRUSSIAN 1986 Problem 17

$\mathrm{A_1A_2\ldots A_n}$ is a regular n- gon and P is an arbitrary point in the plane. Show that if n is even we can choose signs so that the vector sum $\pm \mathrm{PA}_1\pm \mathrm{PA}_2\pm \ldots \pm \mathrm{PA}_\mathrm{n} = 0$ , but if n is odd, then this is only possible for finitely many points P.