Combinatorics · Constructions · Perfect matchings · Complete graph on four points · General position · Non crossing conditions
A set \(\mathcal{A}\) of \(2000\) points in the plane contains no three collinear points. Prove that these points can be joined by \(1000\) blue, \(1000\) red and \(1000\) yellow segments in such a way that
(1) every point of \(\mathcal{A}\) is joined to exactly three other points of \(\mathcal{A}\);
(2) the segments leaving any given point of \(\mathcal{A}\) have three different colours;
(3) segments of different colours have no interior point in common.
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Serbian Republic Competition (Republicko takmicenje) 2001, high school grade I, category A, problem 5. Organized by the Mathematical Society of Serbia (DMS). Source