Nice property of "almost" lattice points
Source: German TST 2004, IMO ShortList 2003, combinatorics problem 5
May 18, 2004
geometrycoordinate geometrycirclescoordinatesTrianglecombinatoricsIMO Shortlist
Problem Statement
Every point with integer coordinates in the plane is the center of a disk with radius .(1) Prove that there exists an equilateral triangle whose vertices lie in different discs.(2) Prove that every equilateral triangle with vertices in different discs has side-length greater than .Radu Gologan, Romania
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The "> 96" in (b) can be strengthened to "> 124". By the way, part (a) of this problem is the place where I used [url=http://mathlinks.ro/viewtopic.php?t=5537]the well-known "Dedekind" theorem.