
IMO 2011 (Problem 2)
Let $\displaystyle \mathcal{S} $ be a finite set of at least two points in the plane. Assume that no three points of $\displaystyle \mathcal{S}$ are collinear. A windmill is a process that starts with a line $\displaystyle l$ going through a single point $\displaystyle P \in\mathcal{S}$. The line rotates clockwise about the pivot $\displaystyle P$ until the first time that the line meets some other point belonging to$\displaystyle \mathcal{S}$. This point, $\displaystyle Q$, takes over as the new pivot, and the line now rotates clockwise about $\displaystyle Q$, until it next meets a point of $\displaystyle \mathcal{S}$. This process continues indefinitely.
Show that we can choose a point$\displaystyle P$ in $\displaystyle \mathcal{S}$ and a line going through $\displaystyle P$ such that the resulting windmill uses each point of S as a pivot infinitely many times.