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\begin{document}
\begin{center}
${\bf 39}^{\mbox{\bf th}}$ {\bf International
Mathematical Olympiad} \\[.1in]
{\bf Taipei, Taiwan} \\ [.05in]
{\bf Day I}\\[.05in]
{\bf July 15, 1998}
\end{center}

\vspace*{.3in}

\begin{enumerate}
\item %% IMO 1, Luxembourg
In the convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$
are perpendicular and the opposite sides $AB$ and $DC$ are not parallel.
Suppose that the point $P$, where the perpendicular bisectors of $AB$ and $DC$
meet, is inside $ABCD$.  Prove that $ABCD$ is a cyclic quadrilateral if and
only if the triangles $ABP$ and $CDP$ have equal areas.

\item % IMO 2, India
In a competition, there are $a$ contestants and $b$ judges, where $b
\geq 3$ is an odd integer. Each judge rates each contestant as either
``pass'' or ``fail''. Suppose $k$ is a number such that, for any two
judges, their ratings coincide for at most $k$ contestants. Prove that
$k/a \geq (b-1)/(2b)$.

\item %% IMO3, Belarus
For any positive integer $n$, let $d(n)$ denote the number of
positive divisors of $n$ (including 1 and $n$ itself).
Determine all positive integers $k$ such that $d(n^2)/d(n) = k$ for some $n$.

\end{enumerate}

\pagebreak %% DAY 2
\begin{center}
${\bf 39}^{\mbox{\bf th}}$ {\bf International
Mathematical Olympiad} \\[.1in]
{\bf Taipei, Taiwai} \\ [.05in]
{\bf Day II}\\[.05in]
{\bf July 16, 1998}
\end{center}

\vspace*{.3in}

\begin{enumerate}
\setcounter{enumi}{3}
\item %% IMO4, UK
Determine all pairs $(a, b)$ of positive integers such that $ab^2+b+7$
divides $a^2 b+a+b$.

\item %% IMO5
Let $I$ be the incenter of triangle $ABC$.  Let the incircle of $ABC$
touch the sides $BC$, $CA$, and $AB$ at $K$, $L$, and $M$,
respectively.  The line through $B$ parallel to $MK$ meets the lines
$LM$ and $LK$ at $R$ and $S$, respectively.  Prove that angle $RIS$ is acute.

\item %% IMO6, Bulgaria
Consider all functions $f$ from the set $N$ of all positive
integers into itself satisfying $f(t^2 f(s)) = s(f(t))^2$ for all $s$
and $t$ in $N$.  Determine the least possible value of $f(1998)$.


\end{enumerate}
\end{document}
