

News: IMO Paper 1 (12 July 2006)Problem 1. 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. (Korea) Problem 2. Let P be a regular 2006gon. 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. (Serbia and Montenegro) Problem 3. Determine the least real number M such that the inequality ab(a^{2}−b^{2}) + bc(b^{2}−c^{2}) + ca(c^{2}−a^{2}) ≤ M(a^{2}+b^{2}+c^{2})^{2} holds for all real numbers a, b and c. (Ireland) Time allowed: 4 hours 30 minutes Previous news item: IMO team in Ljubljana (10 July 2006). Next news item: IMO Paper 2 (13 July 2006). 