THE 1990 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

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

Question 1

Given triagnle 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 ∠BAC, how many non-similar triangles are there in which AEGF is a cyclic quadrilateral?

Question 2

Let a_{1}, a_{2}, …, a_{n} be positive real numbers, and let S_{k} be the sum of the products of a_{1}, a_{2}, …, a_{n} taken k at a time. Show that

S_{k}S_{n−k} ≥ \binom{n}{k}^{2}a_{1}a_{2} ⋯ a_{n}

for k = 1, 2, …, n − 1.

Question 3

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?

Question 4

A set of 1990 persons is divided into non-intersecting subsets in such a way that

1. No one in a subset knows all the others in the subset,

2. Among any three persons in a subset, there are always at least two who do not know each other, and

3. 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.

(a) Prove that within each subset, every person has the same number of acquaintances.

(b) Determine the maximum possible number of subsets. 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.

Question 5

Show that for every integer n ≥ 6, there exists a convex hexagon which can be dissected into exactly n congruent triangles.