## BMO 2010 strip cover problem

Suppose that a set if a in the plane containing points has the property that any three points can be covered by an infinite strip of width . Prove that can be covered by a strip of width .

*Balkan Mathematical Olympiad 2010*

Probably the easiest problem in the contest, maybe just junior level. Taking the triangle with the greatest area, (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 is contained in this triangle. Taking a homothety of center the gravity center of and ratio the strip which contains is transformed in a strip which contains $S$, and has width .

I think, that by a limit argument this can be generalized to infinitely countable .

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Categories: Combinatorics, Geometry, IMO, Olympiad, Problem Solving
IMO

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