4
Part of 2008 Tuymaada Olympiad
Problems(2)
The interceptions of the A-midline on the incircle
Source: Tuymaada 2008, Senior League, First Day, Problem 3.
7/17/2008
Point is the reflection of incentre of triangle across the side . The circumcircle of intersects the line again at point . It is known that lies outside the incircle of the triangle . Two tangents drawn from to the latter circle touch it at points and . Prove that the line contains a medial line of the triangle .
Author: L. Emelyanov
geometrygeometric transformationreflectionincentercircumcirclegeometry proposed
Members can be distributed to several rooms
Source: Tuymaada 2008, Senior League, First Day, Problem 4.
7/18/2008
A group of persons is called good if its members can be distributed to several rooms so that nobody is acquainted with any person in the same room
but it is possible to choose a person from each room so that all the chosen persons are acquainted with each other.
A group is called perfect if it is good and every set of its members is also good.
A perfect group planned a party. However one of its members, Alice, brought here acquaintance Bob, who was not originally expected, and introduced him to all her other acquaintances. Prove that the new group is also perfect.
Author: C. Berge
combinatorics