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The 16th Balkan Mathematical Olympiad
1999年第16届巴尔干地区数学奥林匹克
  1. 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.
  2. 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.
  3. 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.
  4. 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 bn. Prove that (x0 + x1 + ... + xm)(y0 + y1 + ... + yn) ≥ (m + 1)(n + 1).
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