Mini Polymath

Once again it’s time for the International Math OlympiadAs we talked about last year, that’s an international competition for some of the best (high school aged) math problem solvers in the world.  It’s like the Putnam, but even more extreme!

As last year, Terence Tao is hosting a mini polymath group.  That’s a online group discussion to solve one of the IMO problems.  It started yesterday at 2pm Oklahoma time and already has 130+ comments!  If you’re interested in joining in, go here.  This year’s question is geometric:

Problem 2.  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 Q 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.
For the full list of IMO questions, go here.