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The 34th British Mathematical Olympiad
1998年第34届英国奥林匹克数学竞赛
  1. A station issues 3800 tickets covering 200 destinations. Show that there are at least 6 destinations for which the number of tickets sold is the same. Show that this is not necessarily true for 7 .
  2. The triangle ABC has ∠A > ∠C. P lies inside the triangle so that ∠PAC = ∠C. Q is taken outside the triangle so that BQ parallel to AC and PQ is parallel to AB. R is taken on AC (on the same side of the line AP as C) so that ∠PRQ = ∠C. Show that the circles ABC and PQR touch .
  3. a, b, c are positive integers satisfying 1/a - 1/b = 1/c and d is their greatest common divisor. Prove that abcd and d(b - a) are squares .
  4. Show that:

      xy + yz + zx = 12
      xyz - x - y - z = 2

    have a unique solution in the positive reals. Show that there is a solution with x, y, z distinct reals.

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