MathDB

3

Part of 2010 Korea National Olympiad

Problems(2)

Three concurrent lines related to incircle

Source: Korea National Olympiad 2010 Problem 3

9/9/2012
Let I I be the incenter of triangle ABC ABC . The incircle touches BC,CA,AB BC, CA, AB at points P,Q,R P, Q, R . A circle passing through B,C B , C is tangent to the circle II at point X X , a circle passing through C,A C , A is tangent to the circle II at point Y Y , and a circle passing through A,B A , B is tangent to the circle II at point Z Z , respectively. Prove that three lines PX,QY,RZ PX, QY, RZ are concurrent.
geometryincenterinradiustrigonometrycircumcircleangle bisectorpower of a point
A simple double counting on a simple graph

Source: Korea National Olympiad 2010 Problem 7

9/9/2012
There are 2000 2000 people, and some of them have called each other. Two people can call each other at most 11 time. For any two groups of three people A A and B B which A∩B=∅ A \cap B = \emptyset , there exist one person from each of AA and BB that haven't called each other. Prove that the number of two people called each other is less than 201000 201000 .
combinatorics proposedcombinatorics