IMO Shortlist 2011 problem C3


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June 23, 2013
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Let \mathcal{S} be a finite set of at least two points in the plane. Assume that no three points of \mathcal S are collinear. A windmill is a process that starts with a line \ell going through a single point P \in \mathcal S. The line rotates clockwise about the pivot P until the first time that the line meets some other point belonging to \mathcal 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 \mathcal S. This process continues indefinitely.
Show that we can choose a point P in \mathcal S and a line \ell going through P such that the resulting windmill uses each point of \mathcal S as a pivot infinitely many times.

Proposed by Geoffrey Smith, United Kingdom
Source: Međunarodna matematička olimpijada, shortlist 2011