International Mathematical Olympiad  – IMO 2006 Problems

Problem 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 $P=I$.

Problem 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.

Problem 3

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 $c$.

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 $Q(x) = P(P(\ldots P(P(x)) \ldots ))$, 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 $P$.