2nd Balkan 1985


A1. ABC is a triangle. O is the circumcenter, D is the midpoint of AB, and E is the centroid of ACD. Prove that OE is perpendicular to CD iff AB = AC.
A2. The reals w, x, y, z all lie between -π/2 and π/2 and satisfy sin w + sin x + sin y + sin z = 1, cos 2w + cos 2x + cos 2y + cos 2z ≥ 10/3. Prove that they are all non-negative and at most π/6.
A3. Can we find an integer N such that if a and b are integers which are equally spaced either side of N/2 (so that N/2 - a = b - N/2), then exactly one of a, b can be written as 19m + 85n for some positive integers m, n?
A4. There are 1985 people in a room. Each speaks at most 5 languages. Given any three people, at least two of them have a language in common. Prove that there is a language spoken by at least 200 people in the room.

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