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The 15th Balkan Mathematical Olympiad
1998年第15届巴尔干地区数学奥林匹克
  1. How many different integers can be written as [n2/1998] for n = 1, 2, ... , 1997 ?
  2. 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.
  3. Let S be the set of all points inside or on a triangle. Let T be the set S with one interior point excluded. Show that one can find points Pi, Qi such that Pi and Qi are distinct and the closed segments PiQi are all disjoint and have union T.
  4. Prove that there are no integers m, n satisfying m2 = n5 - 4.
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