ISL 2000 C3

Let $n \geq 4$ be a fixed positive integer. Given a set $S = \{P_1, P_2, \ldots, P_n\}$ of $n$ points in the plane such that no three are collinear and no four concyclic, let $a_t,$ $1 \leq t \leq n,$ be the number of circles $P_iP_jP_k$ that contain $P_t$ in their interior, and let $$m(S)=a_1+a_2+\cdots + a_n.$$ Prove that there exists a positive integer $f(n),$ depending only on $n,$ such that the points of $S$ are the vertices of a convex polygon if and only if $m(S) = f(n).$