2000 IMO Shortlist S05

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 $m(S) = f(n).$

令 $n \geq 4$ 为固定正整数。给定平面上 $n$ 个点的集合 $S = \{P_1, P_2, \ldots, P_n\}$，其中没有三个点共线，也没有四个点同圈，令 $a_t,$ $1 \leq t \leq n,$ 为内部包含 $P_t$ 的圆 $P_iP_jP_k$ 的数量，并令$$m(S)=a_1+a_2+\cdots + a_n.$$证明存在一个正整数$f(n),$仅依赖于$n,$使得$S$的点是凸多边形的顶点当且仅当$m(S) = f(n).$