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The 29th British Mathematical Olympiad
1993年第29届英国奥林匹克数学竞赛
  1. The angles in the diagram below are measured in some unknown unit, so that a, b, ... , k, l are all distinct positive integers. Find the smallest possible value of a + b + c and give the corresponding values of a, b, ... , k, l.
  2. p > 3 is prime. m = (4p - 1)/3. Show that 2m-1 = 1 mod m.
  3. P is a point inside the triangle ABC. x = ∠BPC - ∠A, y = ∠CPA - ∠B, z = ∠APB - ∠C. Show that PA sin A/sin x = PB sin B/sin y = PC sin C/sin z.
  4. For 0 < m < 10, let S(m, n) is the set of all positive integers with n 1s, n 2s, n 3s, ... , n ms. For a positive integer N let d(N) be the sum of the absolute differences between all pairs of adjacent digits. For example, d(122313) = 1 + 0 + 1 + 2 + 2 = 6. Find the mean value of d(N) for N in S(m, n).
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