Day 2

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

IMO 2006 Day2 Problem6

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.

IMO 2006 Day2 Problem5

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(P(x)) . . .)), where P
occurs k times. Prove that there are at most n integers t such that Q(t) = t.

IMO 2006 Day2 Problem4

Problem 4. Determine all pairs (x, y) of integers such that
1 + 2x + 22x+1 = y2.

Day 1

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

IMO 2006 Day1 Problem3

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.

IMO 2006 Day1 Problem2

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.

IMO 2006 Day1 Problem1

Problem 1. Let ABC be a triangle with incentre I. A point P in the interior of the
triangle satisfies
6 PBA + 6 PCA = 6 PBC + 6 PCB.
Show that AP  AI, and that equality holds if and only if P = I.