Asian Pacific Mathematics Olympiad · 1990

Problems

Problem 1In triangle $ABC$ , let $D,E,F$ be the midpoints of $BC,CA,AB$ , respectively, and let $G$ be the centroid. For each value of $\angle BAC$ , how many non-similar triangles are there for which $AEGF$ is a cyclic quadrilateral?Solutions: 1
Problem 2Let $a_1,a_2,\ldots,a_n$ be positive real numbers, and let $S_r$ be the sum of all products of $r$ of them. Prove that $S_kS_{n-k}\ge\binom{n}{k}^{\!2}S_n$ for $k=1,2,\ldots,n-1$ .Solutions: 1
Problem 3Consider all triangles $ABC$ with a fixed base $AB$ and whose altitude from $C$ is a constant $h$ . For which of these triangles is the product of its three altitudes a maximum?Solutions: 1
Problem 4Let G be a graph with n vertices satisfying: (i) no vertex is joined to all the other vertices; (ii) there are no triangles; (iii) for every two nonjoined vertices A and B, there is exactly one vertex C joined to both. Prove that every vertex has the same degree, and find the smallest possible n.Solutions: 1
Problem 5Show that for every integer n>=6 there exists a convex hexagon that can be dissected into exactly n congruent triangles.Solutions: 1