IMO Shortlist 2003 problem C5


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2. travnja 2012.
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Regard a plane with a Cartesian coordinate system; for each point with integer coordinates, draw a circular disk centered at this point and having the radius \frac{1}{1000}.

a) Prove the existence of an equilateral triangle whose vertices lie in the interior of different disks;

b) Show that every equilateral triangle whose vertices lie in the interior of different disks has a sidelength 96.

Radu Gologan, Romania
Remark
[The " 96" in (b) can be strengthened to " 124". By the way, part (a) of this problem is the place where I used the well-known "Dedekind" theorem.]
Izvor: Međunarodna matematička olimpijada, shortlist 2003