IMO Problems 2006

IMO Problems 2006

Day 1

IMO 2006 Problem 1

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

IMO 2006 Problem 2

Let P be a regular 2006gon. A diagonal is called good if it’s 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.

IMO 2006 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.

Day 2

IMO 2006 Problem 4

Determine all pairs (x, y) of integers such that 1 + 2ˣ + 2²ˣ⁺¹ = y².

IMO 2006 Problem 5

Let P(x) be a polynomial of degree n > 1 with integer coefficients and let k be a positive integer. Consider the polynomial Q(x) = P(P(…P(x))…) ,where P occurs k times. Prove that there are at most n integers t such that Q(t) = t.

IMO 2006 Problem 6

Assign to each side b of a convex polygon P the maximum area of a triangle that has b as a side and is contained in P. Show that the sum of the areas assigned to the sides of P is at least twice the area of P.