Difference between revisions of "2011 IMO Problems/Problem 2"
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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. | 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 <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. | 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. | ||
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| + | ==Solution== | ||
| + | {{solution}} | ||
| + | |||
| + | ==See Also== | ||
| + | *[[IMO Problems and Solutions]] | ||
Revision as of 23:10, 10 October 2013
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.
Solution
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