8th Balkan 1991


A1. The circumcircle of the acute-angled triangle ABC has center O. M lies on the minor arc AB. The line through M perpendicular to OA cuts AB at K and AC at L. The line through M perpendicular to OB cuts AB at N and BC at P. MN = KL. Find angle MLP in terms of angles A, B and C.
A2. Find an infinite set of incongruent triangles each of which has integral area and sides which are relatively prime integers, but none of whose altitudes are integral.
A3. A regular hexagon area H has its vertices on the perimeter of a convex polygon of area A. Prove that 2A ≤ 3H. When do we have equality?
A4. A is the set of positive integers and B is A ∪ {0}. Prove that no bijection f: A → B can satisfy f(mn) = f(m) + f(n) + 3 f(m) f(n) for all m, n.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.