JMO 1998 Problem 3

Let $P_{1}, \ldots P_{n}$ be the sequence of vertices of a closed polygons whose sides may properly intersect each other at points other than the vertices. The external angle at $P_{i}$ is defined as $180^\circ$ minus the angle of rotation about $P_{i}$ required to bring the ray $P_{i}P_{i-1}$ onto the ray $P_{i}P_{i+1}$, taken in the range ($0^\circ, 360^\circ$). (Here $P_{0}=P_{n}$ and $P_{1}=P_{n+1}$). Prove that if the sum of the external angles is a multiple of $720^\circ$, then the number of self-intersections is odd.