2011 IMO Problems/Problem 2
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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