MathDB

7

Part of 2002 IMO Shortlist

Problems(2)

IMO ShortList 2002, geometry problem 7

Source: IMO ShortList 2002, geometry problem 7

9/28/2004
The incircle Ω \Omega of the acute-angled triangle ABC ABC is tangent to its side BC BC at a point K K. Let AD AD be an altitude of triangle ABC ABC, and let M M be the midpoint of the segment AD AD. If N N is the common point of the circle Ω \Omega and the line KM KM (distinct from K K), then prove that the incircle Ω \Omega and the circumcircle of triangle BCN BCN are tangent to each other at the point N N.
geometryincenterIMO ShortlistIran Lemmaradical axisPolarsgeometry solved
IMO ShortList 2002, combinatorics problem 7

Source: IMO ShortList 2002, combinatorics problem 7

9/28/2004
Among a group of 120 people, some pairs are friends. A weak quartet is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?
combinatoricsgraph theoryExtremal Graph TheoryExtremal combinatoricsIMO Shortlist