The International Mathematical Olympiad (IMO) - 2011
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.
Start with a line ℓ going through a point P ∈ S
Rotate ℓ clockwise around the pivot P until the line contains another point Q of S.
The point Q now takes over as the new pivot.
Show that for a suitable P ∈ S and a suitable starting line ℓ containing P, the resulting windmill will visit each point of S as a pivot infinitely often.