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.
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.