DE-BUNDESWETTBEWERB 2005 Problem 8

For any integer $n\geq 3$ , let $A\left(n\right)$ denote the maximal number of self-intersections a closed broken line $P_1P_2...P_nP_1$ can have; hereby, we assume that no three vertices of the broken line $P_1P_2...P_nP_1$ are collinear.

Prove that

(a) if n is odd, then $A\left(n\right)=\frac{n\left(n-3\right)}{2}$ ; (b) if n is even, then $A\left(n\right)=\frac{n\left(n-4\right)}{2}+1$ .

Note. A self-intersection of a broken line is a (non-ordered) pair of two distinct non-adjacent segments of the broken line which have a common point.