MathDB

2

Part of 2004 239 Open Mathematical Olympiad

Problems(2)

colour plane and find triangle

Source: 239MO 2004, grade 8-9, problem 4, grade 10-11, problem 2

12/11/2004
Do there exist such a triangle TT, that for any coloring of a plane in two colors one may found a triangle T′T', equal to TT, such that all vertices of T′T' have the same color.
proposed by S. Berlov
probabilitygeometrygeometric transformationreflectioncombinatorics unsolvedcombinatorics
The incircle of a triangle ABC has centre I

Source: 239MO 2004, grade 8-9, problem 2

12/11/2004
The incircle of a triangle ABCABC has centre II and touches sides AB,BC,CAAB, BC, CA in points C1,A1,B1C_1, A_1, B_1 respectively. Denote by LL the foot of a bissector of angle BB, and by KK the point of intersecting of lines B1IB_1I and A1C1A_1C_1. Prove that KL∥BB1KL\parallel BB_1. proposed by L. Emelyanov, S. Berlov
geometryRussiabisector