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 l going through a single point P S. The line rotates clockwise about the pivot P until the first time that the line meets some other point 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.
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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 l going through P such that the resulting windmill uses each point of S as a pivot infinitely many times.
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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 $\mathcal{S}$ 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.

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