MathDB

4

Part of 2015 Bundeswettbewerb Mathematik Germany

Problems(2)

Many friends in a social network

Source: BWM Germany Round 1, 2015 - Problem 4

4/8/2015
Many people use the social network "BWM". It is known that: By choosing any four people of that network there always is one that is a friend of the other three. Is it then true that by choosing any four people there always is one that is a friend of everyone in "BWM"?
Note: If member AA is a friend of member BB, then member BB also is a friend of member AA.
combinatoricsgraph theoryGermany
Sum of distances to sides => perpendicularity

Source: Bundeswettbewerb Mathematik 2015 - Round 2 - #4

9/4/2015
Let ABCABC be a triangle, such that its incenter II and circumcenter UU are distinct. For all points XX in the interior of the triangle let d(X)d(X) be the sum of distances from XX to the three (possibly extended) sides of the triangle. Prove: If two distinct points P,QP,Q in the interior of the triangle ABCABC satisfy d(P)=d(Q)d(P)=d(Q), then PQPQ is perpendicular to UIUI.
geometryincentercircumcircledistanceperpendicular