## THE 1996 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours
NO calculators are to be used.
Each question is worth seven points.

Question 1
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.

Question 2
Let $m$ and $n$ be positive integers such that $n \leq m$. Prove that
$2^nn! \leq \frac{(m + n)!}{(m- n)!} \leq (m^2 + m)^n$

Question 3
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 $P_2P_3P_4$; $I_2$ be the incentre of the triangle $P_1P_3P_4; I_3$ be the incentre of the triangle $P_1P_2P_4; I_4$ be the incentre of the triangle $P_1P_2P_3$. Prove that $I_1, I_2, I_3, I_4$ are the vertices of a rectangle.

Question 4
The National Marriage Council wishes to invite $n$ couples to form $17$ discussion groups under the following conditions:
1. All members of a group must be of the same sex; i.e. they are either all male or all female.
2. The difference in the size of any two groups is 0 or $1$.
3. All groups have at least $1$ member.
4. Each person must belong to one and only one group.
Find all values of $n, n \leq 1996$, for which this is possible. Justify your answer.

Question 5
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.