2011 IMO 第 2 题

Let $S$ be a finite set of at least two points in the plane. Assume that no three points of $S$ are collinear. A windmill is a process that starts with a line $\ell$ going through a single point $P\in S$ . The line rotates clockwise about the pivot $P$ until the first time that the line meets some other point belonging to $S$ . This point, $Q$ , takes over as the new pivot, and the line now rotates clockwise about $Q$ , until it next meets a point of $S$ . This process continues indefinitely.

Show that we can choose a point $P$ in $S$ and a line $\ell$ going through $P$ such that the resulting windmill uses each point of $S$ as a pivot infinitely many times.

设 $S$ 是平面上至少两个点的有限集。假设 $S$ 没有三点共线。风车是一个从线 $\ell$ 开始穿过单个点 $P\in S$ 的过程。该线围绕枢轴 $P$ 顺时针旋转，直到该线第一次遇到属于 $S$ 的其他点。该点 $Q$ 接管作为新的枢轴，并且该线现在绕 $Q$ 顺时针旋转，直到它下一个遇到 $S$ 点。这个过程无限期地持续下去。

证明我们可以在 $S$ 中选择一个点 $P$ 和一条穿过 $P$ 的线 $\ell$ ，这样生成的风车就会无限次地使用 $S$ 中的每个点作为枢轴。