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\begin{document}

\begin{center}
${\bf 44\th}$ {\bf International Mathematical Olympiad} \\[.1in]
{\bf Tokyo, Japan} \\ [.05in]
{\bf Day I%\hspace{.25in} 9 a.m. - 1:30 p.m.
}\\[.05in]
{\bf July 13, 2003}
\end{center}

\vspace*{.3in}

\begin{enumerate}
\item %% IMO1 %%Brazil
Let $A$ be a 101-element subset of the set $S = \{1, 2, \dots,
1000000\}$. Prove that there exist numbers $t_1, t_2, \dots, t_{100}$
in $S$ such that the sets
\[
A_j = \{ x + t_j \mid x \in A\} \quad j=1, 2,\dots, 100
\]
are pairwise disjoint.

\item %% IMO2 %%Bulgaria
Determine all pairs of positive integers $(a,b)$ such that 
\[
\frac{a^2}{2ab^2-b^3+1}
\]
is a positive integer.

\item %% IMO3 %%Poland
A convex hexagon is given in which any two opposite sides have the following property: the distance between their midpoints is 
$\sqrt{3}/2$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

(A convex $ABCDEF$ has three pairs of opposite sides: $AB$ and $DE$, $BC$ and $EF$, $CD$ and $FA$.)

\end{enumerate}

\pagebreak %% DAY 2
\begin{center}
${\bf 44\th}$ {\bf International Mathematical Olympiad} \\[.1in]
{\bf Tokyo, Japan} \\ [.05in]
{\bf Day II%\hspace{.25in} 9 a.m. - 1:30 p.m.
}\\[.05in]
{\bf July 14, 2003}
\end{center}

\vspace*{.3in}

\begin{enumerate}
\setcounter{enumi}{3}
\item %% IMO4 %%Finland
Let $ABCD$ be a cyclic quadrilateral. Let $P, Q$ and $R$ be the feet of perpendiculars from $D$ to lines $BC, CA$ and $AB$, respectively. Show that $PQ = QR$ if and only if the bisectors of angles $ABC$ and $ADC$ meet on segment $AC$. 

%% Note: the condition that $ABCD$ be cyclic is not necessary.


\item %% IMO5 %%Ireland
Let $n$ be a positive integer and $x_1, x_2, \dots , x_n$ be real numbers with $x_1 \le x_2 \le \dots \le x_n$. 
\begin{enumerate} 
\item [(a)] 
Prove that 
\[ 
\left(\sum_{i=1}^{n}\sum_{j=1}^{n} |x_i - x_j|\right)^2 
\le 
\frac{2(n^2 - 1)}{3}\sum_{i=1}^{n}\sum_{j=1}^{n} (x_i - x_j)^2. 
\] 
\item [(b)] 
Show that the equality holds if and only if $x_1, x_2, \dots , x_n$ form an arithmetic sequence. 
\end{enumerate}

\item %% IMO6 %France
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p - p$ is not divisible by $q$. 


\end{enumerate}

\end{document}

 
                   

