2000 IMO 第 3 题

C3 (COL) Let $n \geq 4$ be a fixed positive integer. Given a set $S=$ $\left\{P_{1}, P_{2}, \ldots, P_{n}\right\}$ of 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 $P_{i} P_{j} P_{k}$ 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 $m(S)=f(n)$.

C3 (COL) 令$n \geq 4$ 为固定正整数。给定平面上的一组点 $S=$ $\left\{P_{1}, P_{2}, \ldots, P_{n}\right\}$，其中没有三个点共线，也没有四个点同圈，设 $a_{t}, 1 \leq t \leq n$ 为内部包含 $P_{t}$ 的圆 $P_{i} P_{j} P_{k}$ 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$m(S)=f(n)$。