16th Balkan 1999


A1. O is the circumcenter of the triangle ABC. XY is the diameter of the circumcircle perpendicular to BC. It meets BC at M. X is closer to M than Y. Z is the point on MY such that MZ = MX. W is the midpoint of AZ. Show that W lies on the circle through the midpoints of the sides of ABC. Show that MW is perpendicular to AY.
A2. p is an odd prime congruent to 2 mod 3. Prove that at most p-1 members of the set {m² - n³ - 1: 0 < m, n < p} are divisible by p.
A3. ABC is an acute-angled triangle area 1. Show that the triangle whose vertices are the feet of the perpendiculars from the centroid to AB, BC, CA has area between 4/27 and 1/4.
A4. 0 = a₁, a₂, a₃, ... is a non-decreasing, unbounded sequence of non-negative integers. Let the number of members of the sequence not exceeding n be b_n. Prove that (x₀ + x₁ + ... + x_m)(y₀ + y₁ + ... + y_n) ≥ (m + 1)(n + 1).

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