# 2006 IMO Problems

## Problem 1

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