First Problem, Danube Mathematical Competition 2012

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Laurențiu Ploscaru
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First Problem, Danube Mathematical Competition 2012

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Given a positive integer $n$, determine the maximum number of lattice points in the plane a square of side length $n+\dfrac{1}{2n+1}$ may cover.
People are strange when you're a stranger,
Faces look ugly when you're alone.
Women seem wicked when you're unwanted,
Streets are uneven when you're down.
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