16th Balkan Mathematical Olympiad Problems 1999

16th Balkan Mathematical Olympiad Problems 1999A1.  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

15th Balkan Mathematical Olympiad Problems 1998

15th Balkan Mathematical Olympiad Problems 1998A1.  How many different integers can be written as [n2/1998] for n = 1, 2, ... , 1997? A2.  xi are distinct positive reals satisfying x1 < x2 < ... < x2n+1. Show that x1 - x2 + x3 - x4 + ... - x2n + x2n+1 < (x1n - x2n + ... - x2nn + x2n+1n)1/n. A2.  The reals w, x

1st Balkan Mathematical Olympiad Problems 1984

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