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## BMO 2010 strip cover problem

Suppose that a set if a $S$ in the plane containing $n$ points has the property that any three points can be covered by an infinite strip of width $1$. Prove that $S$ can be covered by a strip of width $2$.
Probably the easiest problem in the contest, maybe just junior level. Taking the triangle with the greatest area, $ABC$ (if the points are not all in a line), constructing the triangle who’s midpoints are the vertices of the given triangle. It’s easy to see that $S$ is contained in this triangle. Taking a homothety of center the gravity center of $ABC$ and ratio $-2$ the strip which contains $ABC$ is transformed in a strip which contains \$S\$, and has width $2$.
I think, that by a limit argument this can be generalized to infinitely countable $S$.