The International Mathematical Olympiad (IMO) - 2011

Youtube Video

Let S be a finite set of at least two points in the plane. Assume that no three points of S are collinear. By a windmill we mean a process as follows.

This process continues indefinitely, with the pivot always being a point from S.

Show that for a suitable PS and a suitable starting line containing P, the resulting windmill will visit each point of S as a pivot infinitely often.