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\begin{center}

${\bf 2^{nd}}$ {\bf Asian Pacific Mathematical Olympiad}

\end{center}



\begin{center}

{\bf March, 1990}

\end{center}



\bigskip



\begin{enumerate}



\item In $\triangle ABC$, let $D$, $E$, $F$ be the midpoints of $BC$,

$AC$, $AB$ respectively and let $G$ be the centroid of the triangle.



For each value of $\angle BAC$, how many non-similar triangles are

there in which $AEGF$ is a cyclic quadrilateral?



\item Let $a_{1}$, $a_{2}$, \dots, $a_{n}$ be positive real numbers,

and let $S_{k}$ be the sum of products of $a_{1}$, $a_{2}$, \dots,

$a_{n}$ taken $k$ at a time.



Show that

\[S_{k}S_{n-k} \ge {n \choose k}^{2}a_{1}a_{2}\ldots

a_{n},\quad\hbox{for $k = 1$, 2, \dots, $n-1$}\]



\item Consider all the triangles $ABC$ which have a fixed base $AB$

and whose altitude from $C$ is a constant $h$.  For which of these

triangles is the product of its altitudes a maximum?



\item A set of 1990 persons is divided into non-intersecting subsets in

such a way that

\begin{enumerate}

\item no one in a subset knows all the others in the subset;

\item among any three persons in a subset, there are always at least

two who do not know each other; and

\item for any two persons in a subset who do not know each other,

there is exactly one person in the same subset knowing both of them.

\end{enumerate}



\begin{enumerate}

\item[(i)] Prove that within each subset, every person has the same

number of acquaintances.

\item[(ii)] Determine the maximum possible number of subsets.

\end{enumerate}



Note: It is understood that if a person $A$ knows person $B$, then

person $B$ will know person $A$; an acquaintance is someone who is

known.  Every person is assumed to know one's self.



\item

Show that for every integer $n \ge 6$, there exists a convex hexagon

which can be dissected into exactly $n$ congruent triangles.



\end{enumerate}



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