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The 35th British Mathematical Olympiad
1999年第35届英国奥林匹克数学竞赛
  1. Let Xn = {1, 2, 3, ... , n}. For which n can we partition Xn into two parts with the same sum? For which n can we partition Xn into three parts with the same sum ?
  2. A circle is inscribed in a hexagon ABCDEF. It touches AB, CD and EF at their midpoints (L, M, N respectively) and touches BC, DE, FA at the points P, Q, R. Prove that LQ, MR, NP are concurrent .
  3. Show that xy + yz + zx ≤ 2/7 + 9xyz/7 for non-negative reals x, y, z with sum 1 .
  4. Find the smallest possible sum of digits for a number of the form 3n2 + n + 1 (where n is a positive integer). Does there exist a number of this form with sum of digits 1999 ?
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