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\begin{document}
\begin{center}
${\bf 47\th}$ {\bf International Mathematical Olympiad} \\[.1in]
{\bf Ljubljana, Slovenia} \\ [.05in]
{\bf Day I}\\[.05in]
{\bf July 12, 2006}
\end{center}
\vspace*{.3in}
\begin{enumerate}
\item
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 $P=I$.
\item
Let $P$ be a regular 2006-gon. A diagonal of $P$ is called \emph{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
\emph{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.
\item
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 $c$.
\end{enumerate}
\pagebreak %% DAY 2
\begin{center}
${\bf 47\th}$ {\bf International Mathematical Olympiad} \\[.1in]
{\bf Ljubljana, Slovenia} \\ [.05in]
{\bf Day II}\\[.05in]
{\bf July 13, 2006}
\end{center}
\vspace*{.3in}
\begin{enumerate}
\setcounter{enumi}{3}
\item
Determine all pairs $(x,y)$ of integers such that
\[
1 + 2^x + 2^{2x+1} = y^2.
\]
\item
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 $Q(t)=t$.
\item
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
$P$.
\end{enumerate}
\end{document}