MathDB

8

Part of 2007 IMO Shortlist

Problems(2)

Convex polygon and equilateral triangles

Source: ISL 2007, C8, AIMO 2008, TST 7, P3

6/3/2008
Given is a convex polygon P P with n n vertices. Triangle whose vertices lie on vertices of P P is called good if all its sides are unit length. Prove that there are at most 2n3 \frac {2n}{3} good triangles.
Author: Vyacheslav Yasinskiy, Ukraine
combinatoricscombinatorial geometrypolygonExtremal combinatoricsIMO Shortlist
A nice collinearity problem

Source: IMO Shortlist 2007, G8, AIMO 2008, TST 7, P2

7/13/2008
Point P P lies on side AB AB of a convex quadrilateral ABCD ABCD. Let ω \omega be the incircle of triangle CPD CPD, and let I I be its incenter. Suppose that ω \omega is tangent to the incircles of triangles APD APD and BPC BPC at points K K and L L, respectively. Let lines AC AC and BD BD meet at E E, and let lines AK AK and BL BL meet at F F. Prove that points E E, I I, and F F are collinear. Author: Waldemar Pompe, Poland
geometryquadrilateralincircleTriangleIMO Shortlist