MathDB

4

Part of 2015 European Mathematical Cup

Problems(2)

B', A', O are collinear

Source: European Mathematical Cup, 2015, Junior, P4

12/30/2016
Let ABCABC be an acute angled triangle. Let B′,A′B' , A' be points on the perpendicular bisectors of AC,BCAC, BC respectively such that B′A⊥ABB'A \perp AB and A′B⊥ABA'B \perp AB. Let PP be a point on the segment ABAB and OO the circumcenter of the triangle ABCABC. Let D,ED, E be points on BC,ACBC, AC respectively such that DP⊥BODP \perp BO and EP⊥AOEP \perp AO. Let O′O' be the circumcenter of the triangle CDECDE. Prove that B′,A′B', A' and O′O' are collinear.
Steve Dinh
geometry
Making everyone content even with a brick wall in-between

Source: European Mathematical Cup, 2015, Senior, P4

12/30/2016
A group of mathematicians is attending a conference. We say that a mathematician is k−k-content if he is in a room with at least kk people he admires or if he is admired by at least kk other people in the room. It is known that when all participants are in a same room then they are all at least 3k+13k + 1-content. Prove that you can assign everyone into one of 22 rooms in a way that everyone is at least kk-content in his room and neither room is empty. Admiration is not necessarily mutual and no one admires himself.
Matija Bucić
combinatoricsgraph theoryDirected graphs