Question 166
Suppose for a convex 2010-gon, any 3 diagonals do not share a common point except for vertices. Let us consider a closed zig-zag line segments (abbr. by Z.Z.L.S.) which goes through each of the vertices of the 2010-gon once and only once. Determine the maximum possible number of the self-intersection points for such a closed Z.Z.L.S. Here, by a closed Z.Z.L.S. we mean contiguous line segments $P_1P_2\ldots P_nP_{n+1}$ for which $P_1 = P_{n+1}$ is satisfied.