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 P Q be two segments perpendicular to the diagonal BD and such that the distance between them is d > BD/2, with M ∈ AD, N ∈ DC, P ∈ AB, and Q ∈ BC. Show that the perimeter of hexagon AMNCQP does not depend on the position of MN and P Q so long as the distance between them remains constant.

Question 2

Let m and n be positive integers such that n ≤ m. Prove that

2^{n}n! ≤ \frac{(m + n)!}{(m − n)!} ≤ (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_{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.

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 ≤ 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} ≤ \sqrt{a} + \sqrt{b} + \sqrt{c} ,

and determine when equality occurs.