# 2006 IMO Problems

## Problem 1

Let be a triangle with incentre I. A point P in the interior of the triangle satisfies $\anglePBA$ (Error compiling LaTeX. ! Undefined control sequence.) + $\anglePCA$ (Error compiling LaTeX. ! Undefined control sequence.) = $\anglePBC$ (Error compiling LaTeX. ! Undefined control sequence.) + $\anglePCB$ (Error compiling LaTeX. ! Undefined control sequence.). Show that AP ≥ AI, and that equality holds if and only if P = I.

## Problem 2

Let be a regular 2006-gon. A diagonal of is called good if its endpoints divide the boundary of into two parts, each composed of an odd number of sides of . The sides of are also called good. Suppose has been dissected into triangles by 2003 diagonals, no two of which have a common point in the interior of . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.