# Difference between revisions of "2011 IMO Problems/Problem 2"

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− | 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 | + | Let <math>\mathcal{S}</math> be a finite set of at least two points in the plane. Assume that no three points of <math>\mathcal S</math> are collinear. A ''windmill'' is a process that starts with a line <math>\ell</math> going through a single point <math>P \in \mathcal S</math>. The line rotates clockwise about the ''pivot'' <math>P</math> until the first time that the line meets some other point belonging to <math>\mathcal S</math>. This point, <math>Q</math>, takes over as the new pivot, and the line now rotates clockwise about <math>Q</math>, until it next meets a point of <math>\mathcal S</math>. This process continues indefinitely. |

− | Show that we can choose a point P in S and a line | + | Show that we can choose a point <math>P</math> in <math>\mathcal S</math> and a line <math>\ell</math> going through <math>P</math> such that the resulting windmill uses each point of <math>\mathcal S</math> as a pivot infinitely many times. |

## Revision as of 13:37, 27 November 2011

Let be a finite set of at least two points in the plane. Assume that no three points of are collinear. A *windmill* is a process that starts with a line going through a single point . The line rotates clockwise about the *pivot* until the first time that the line meets some other point belonging to . This point, , takes over as the new pivot, and the line now rotates clockwise about , until it next meets a point of . This process continues indefinitely.
Show that we can choose a point in and a line going through such that the resulting windmill uses each point of as a pivot infinitely many times.