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 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.

(Ireland)

Time allowed: 4 hours 30 minutes
Each problem is worth 7 points

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