Question 44
$A_1A_2\ldots A_n$ 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 $\pm PA_1 \pm PA_2 \pm \ldots \pm PA_n = 0$, but if $n$ is odd, then this is only possible for finitely many points $P$.