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{\centerline{\bf THE 1996 ASIAN PACIFIC MATHEMATICAL OLYMPIAD}}
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{\it Time allowed: 4 hours}
{\it NO calculators are to be used.}
{\it Each question is worth seven points.}
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{\bf Question 1}
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Let $ABCD$ be a quadrilateral $AB = BC = CD = DA$. Let $MN$ and $PQ$ be two
segments perpendicular to the diagonal $BD$ and such that the distance between
them is $d > BD/2$, with $M \in AD$, $N \in DC$, $P \in AB$, and $Q \in BC$.
Show that the perimeter of hexagon $AMNCQP$ does not depend on the position of
$MN$ and $PQ$ so long as the distance between them remains constant.
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{\bf Question 2}
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Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that $$2^n n!
\leq \frac{(m+n)!}{(m-n)!} \leq (m^2 + m)^n \ .$$
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{\bf Question 3}
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Let $P_1$, $P_2$, $P_3$, $P_4$ be four points on a circle, and let $I_1$ be
the incentre of the triangle $P_2 P_3 P_4$; $I_2$ be the incentre of the
triangle $P_1 P_3 P_4$; $I_3$ be the incentre of the triangle $P_1 P_2 P_4$;
$I_4$ be the incentre of the triangle $P_1 P_2 P_3$. Prove that $I_1$, $I_2$,
$I_3$, $I_4$ are the vertices of a rectangle.
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{\bf Question 4}
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The National Marriage Council wishes to invite $n$ couples to form 17
discussion groups under the following conditions:
\begin{enumerate}
\item All members of a group must be of the same sex; i.e. they are either all
male or all female.
\item The difference in the size of any two groups is 0 or 1.
\item All groups have at least 1 member.
\item Each person must belong to one and only one group.
\end{enumerate}
Find all values of $n$, $n \leq 1996$, for which this is possible. Justify
your answer.
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{\bf Question 5}
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Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Prove that
$$\sqrt{a+b-c} + \sqrt{b+c-a} + \sqrt{c+a-b} \leq \sqrt{a} + \sqrt{b} +
\sqrt{c} \ ,$$ and determine when equality occurs.
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