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 $AP\ge AI$, 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.

Problem 3. Determine the least number $M$ such that the inequality

$|ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)|\le M(a^2+b^2+c^2{)}^2$

holds for all real numbers $a$, $b$ and $c$.

Problem 4. Determine all pairs $(x,y)$ of integers such that

$1+2^x+2^{2x+1}=y^2.$

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

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 $P$.