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\def\Avec{{(A_{1}, \ldots, A_{n})}}
\def\Acup{{A_{1} \cup A_{2} \cup \cdots \cup A_{n}}}

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\begin{document}

\begin{center}
${\bf 10^{\mathrm{th}}}$ {\bf Asian Pacific Mathematical Olympiad}
\end{center}

\begin{center}
{\bf March, 1998}
\end{center}

\bigskip

\begin{enumerate}

\item
Let $F$ be the set of all $n$-tuples $\Avec$ such that each $A_{i}$ is a subset of $\{1, 2, \ldots, 1998\}$.  Let $|A|$ denote the number of elements of the set $A$.  Find
\[
\sum_{\Avec\in F} |\Acup|.
\]

\item
Show that for any positive integers $a$ and $b$, $(36a+b)(a+36b)$ cannot be a power of $2$.

\item
Let $a$, $b$, $c$ be positive real numbers.  Prove that
\[
\biggl(1+\frac{a}{b}\biggr) \biggl(1+\frac{b}{c}\biggr) \biggl(1+\frac{c}{a}\biggr) \ge 2
\biggl(1+\frac{a+b+c}{\sqrt[3]{abc}}\biggr).
\]

\item
Let $ABC$ be a triangle and $D$ the foot of the altitude from $A$. Let
$E$ and $F$ lie on a line passing through $D$ such that $AE$ is
perpendicular to $BE$, $AF$ is perpendicular to $CF$, and $E$ and $F$
are different from $D$. Let $M$ and $N$ be the midpoints of the segments
$BC$ and $EF$, respectively. Prove that $AN$ is perpendicular to $NM$.

\item
Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

\end{enumerate}


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