Question 3
C3 (COL). Let $n\ge4$ be a fixed positive integer. Given a set $S=\{P_1,P_2,\ldots,P_n\}$ of points in the plane such that no three are collinear and no four concyclic, let $a_t$, $1\le t\le n$, 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 $m(S)=f(n)$.
$$m(S)=a_1+a_2+\cdots+a_n.$$
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 $m(S)=f(n)$.