JBMO 2007 Problema 3 - Triunghiuri scalene
Moderators: Laurian Filip, Filip Chindea, maky, Cosmin Pohoata
-
Laurian Filip
- Site Admin
- Posts: 344
- Joined: Sun Nov 25, 2007 2:34 am
- Location: Bucuresti/Arad
- Contact:
JBMO 2007 Problema 3 - Triunghiuri scalene
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.
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.
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.
I can't speak Romanian, so I'll write in English. Hope you understand me!