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IMO 2006

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Day 1 - 12 July 2006

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

Day 2 - 13 July 2006

4	Determine all pairs $(x, y)$ of integers such that $1+2^{x}+2^{2x+1}= y^{2}.$	S
5	Let $P(x)$ be a polynomial of degree $n > 1$ with integer coefficients and let $k$ be a positive integer. Consider the polynomial $Q(x) = P(P(\ldots P(P(x)) \ldots ))$ , where $P$ occurs $k$ times. Prove that there are at most $n$ integers $t$ such that $Q(t) = t$ .	S
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 $P$ .	S

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