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\begin{document}
\begin{center}
${\bf 40}^{\mbox{\bf th}}$ {\bf International
Mathematical Olympiad} \\[.1in]
{\bf Bucharest} \\ [.05in]
{\bf Day I}\\[.05in]
{\bf July 16, 1999}
\end{center}

\vspace*{.3in}

\begin{enumerate}
\item %% IMO 1, Luxembourg
Determine all finite sets $S$ of at least three points in the plane which satisfy the following condition: 
\begin{quote}
     for any two distinct points $A$ and $B$ in $S$, the perpendicular bisector of the line segment $AB$ 
     is an axis of symmetry for $S$.
\end{quote}

\item % IMO 2, India
Let $n$ be a fixed integer, with $n \geq 2$. 
\begin{quote}
     (a) Determine the least constant $C$ such that the inequality 
\begin{quote}
\[\sum_{1\leq i<j \leq n} x_i x_j (x_i^2 + x_j^2) \leq C \left(\sum_{1 \leq i \leq n}x_i\right)^4 \]                                                              \end{quote}        

     holds for all real numbers $x_1,\cdots,x_n \geq 0$. \\


     (b) For this constant $C$, determine when equality holds. 
\end{quote}
\item %% IMO3, Belarus
Consider an $n\; \times \;n$ square board, where $n$ is a fixed even positive integer. The board is divided into $n^2$ unit squares. We say that two
different squares on the board are adjacent if they have a common side. 

$N$ unit squares on the board are marked in such a way that every square (marked or unmarked) on the board is adjacent to at least
one marked square. 

Determine the smallest possible value of $N$. 

\end{enumerate}

\pagebreak %% DAY 2
\begin{center}
${\bf 40}^{\mbox{\bf th}}$ {\bf International
Mathematical Olympiad} \\[.1in]
{\bf Bucharest} \\ [.05in]
{\bf Day II}\\[.05in]
{\bf July 17, 1999}
\end{center}

\vspace*{.3in}

\begin{enumerate}
\setcounter{enumi}{3}
\item %% IMO4, UK
Determine all pairs $(n,p)$ of positive integers such that 
\begin{quote}
     $p$ is a prime, \\ 
     $n$ not exceeded $2p$, and \\
     $(p-1)^{n} + 1$ is divisible by $n^{p-1}$. 
\end{quote}

\item %% IMO5
Two circles $G_{1}$ and $G_{2}$ are contained inside the circle $G$, and are tangent to $G$ at the distinct points $M$ and $N$, respectively. $G_{1}$
passes through the center of $G_{2}$. The line passing through the two points of intersection of $G_{1}$ and $G_{2}$ meets $G$ at $A$ and $B$. The
lines $MA$ and $MB$ meet $G_{1}$ at $C$ and $D$, respectively. 

Prove that $CD$ is tangent to $G_{2}$. 

\item %% IMO6, Bulgaria
Determine all functions $f:{\textbf R} \longrightarrow {\textbf R}$ such that 
\begin{quote}
             $f(x-f(y))=f(f(y)) + x f(y)+f(x)-1$ 
\end{quote}
for all real numbers $x,y$. 

\end{enumerate}
\end{document}
