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.
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.
(Serbia and Montenegro)
Problem 3. Determine the least real number M such that the inequality
|ab(a2−b2) + bc(b2−c2) + ca(c2−a2)| ≤ M(a2+b2+c2)2
holds for all real numbers a, b and c.
Time allowed: 4 hours 30 minutes
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