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# 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 APAI, 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(a2b2) + bc(b2c2) + ca(c2a2)|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

Previous news item: IMO team in Ljubljana (10 July 2006).

Next news item: IMO Paper 2 (13 July 2006).