KöMaL 2011 B4361

Let $a$ and $b$ denote the semi-axes of an ellipse. Let $O$ denote the centre, and let $P_1, P_2, \ldots,P_n$ be points on the curve such that each of the angles $P_i OP_{i+1}$ is $\frac\pi n$ . Prove that if $n$>1 then $\frac{2}{OP_1^2}+\frac{2}{OP_2^2}+ \dots +\frac{2}{OP_n^2}= \frac{n}{a^2}+\frac{n}{b^2}$.