Home > Combinatorics, Geometry, IMO, Olympiad, Problem Solving > BMO 2010 strip cover problem

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.
Balkan Mathematical Olympiad 2010
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.

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