APMO 1990

In $\triangle A B C$, let $D, E, F$ be the midpoints of $B C, A C, A B$ respectively and let $G$ be the centroid of the triangle. For each value of $\angle B A C$, how many non-similar triangles are there in which $A E G F$ is a cyclic quadrilateral?

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

$$S_{k} S_{n-k} \geq\binom{ n}{k}^{2} a_{1} a_{2} \ldots a_{n}, \text { for } k=1,2, \ldots, n-1$$

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

A set of 1990 persons is divided into non-intersecting subsets in such a way that (a) no one in a subset knows all the others in the subset; (b) among any three persons in a subset, there are always at least two who do not know each other; and (c) 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. (i) Prove that within each subset, every person has the same number of acquaintances. (ii) 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.

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