1990 APMO Problems

Problem 1

Given triangle $ABC$ , let $D$ , $E$ , $F$ be the midpoints of $BC$ , $AC$ , $AB$ respective and let $G$ be the centroid of the triangle. For each value of $\angle BAC$ , how many non-similar triangles are there in which $AEGF$ is a cyclic quadrilateral?

Solution

Problem 2

Let $a_1, a_2, \dots , a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1, a_2, \dots , a_n$ taken $k$ at a time. Show that

$S_kS_{n-k}\geq \binom{n}{k}^2 a_1a_2\cdots a_n$

for $k=1, 2, \dots , n-1$

Solution

Problem 3

Consider all the triangles $ABC$ which have a fixed base $AB$ and whose altitude from $C$ is a constant $h$ . For which triangles is the product of its altitudes a maximum?

Solution

Problem 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 now know each other, and

3. For any two persons in a subset who do now 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.