IMO 2006

Let $ABC$ be triangle with incenter $I$ . A point $P$ in the interior of the triangle satisfies $$\angle PBA+\angle PCA = \angle PBC+\angle PCB.$$ Show that $AP \geq AI$ , and that equality holds if and only if $P=I$ .

Let $P$ be a regular $2006$ -gon. A diagonal 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.

Determine the least real number $M$ such that the inequality $$|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}$$ holds for all real numbers $a$ , $b$ and $c$ .

Determine all pairs $(x, y)$ of integers such that $$1+2^{x}+2^{2x+1}= y^{2}.$$

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

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 $P$ .