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.

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