2006 IMO Problems
Let ABC be a triangle with incentre I. A point P in the interior of the triangle satisfies <PBA + <PCA = <PBC + <PCB. Show that AP ≥ AI, and that equality holds if and only if P = I.
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.