AoPS - Problem

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Problem

Source: Mediterranean MO 1999

Tags: geometry, pigeonhole principle, analytic geometry, combinatorics unsolved, combinatorics

MinatoF

11.10.2012 13:10

A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that this figure can be placed onto a Cartesian plane so that it covers at least $n+1$ points with integer coordinates.

mavropnevma

11.10.2012 13:16

Place the figure arbitrarily. Translate each latticeal unit square containing any part of it to a unique (distinguished) latticeal unit square. Then there will exist a point there "covered" by at least $n+1$ translates - by a variation of the pigeonhole principle. Translate the lattice so that the origin moves to this point.