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JBMO 2007 Problema 3 - Triunghiuri scalene
Posted: Wed Apr 09, 2008 9:06 am
by Laurian Filip
Se considera 50 de puncte in plan, oricare trei necoliniare. Fiecare dintre acestea este colorat folosind una dintre patru culori date. Sa se arate ca exista o culoare si cel putin 130 de triunghiuri scalene cu varfurile in puncte de aceasta culoare.
Posted: Thu May 21, 2009 9:49 am
by Mole
By Pigeonhole, there are 13 points of one color (50:4=12.5).
From 13 points, there are \( \frac{13\cdot12}{2}=78 \) lines. From each line, we can make 2 isosceles triangles such that the given line is its base (if there are 3, those third points would lie on a line, contradiction). So we can make \( 78\cdot2=156 \) isosceles triangles, leaving us \( \frac{13\cdot12\cdot11}{6}-156=130 \) scalene triangles.