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The 21st British Mathematical Olympiad
1985年第21届英国奥林匹克数学竞赛
  1. Prove that ∑1n1n | xi - xj | ≤ n2 for all real xi such that 0 ≤ xi ≤ 2. When does equality hold ?
  2. (1) The incircle of the triangle ABC touches BC at L. LM is a diameter of the incircle. The ray AM meets BC at N. Show that | NL | = | AB - AC | .
    (2) A variable circle touches the segment BC at the fixed point T. The other tangents from B and C to the circle (apart from BC) meet at P. Find the locus of P .
  3. Let { x } denote the nearest integer to x, so that x - 1/2 ≤ { x } < x + 1/2. Define the sequence u1, u2, u3, ... by u1 = 1. un+1 = un + { un√2 }. So, for example, u2 = 2, u3 = 5, u4 = 12. Find the units digit of u1985 .
  4. A, B, C, D are points on a sphere of radius 1 such that the product of the six distances between the points is 512/27. Prove that ABCD is a regular tetrahedron .
  5. Let bn be the number of ways of partitioning the set {1, 2, ... , n} into non-empty subsets. For example, b3 = 5: 123; 12, 3; 13, 2; 23, 1; 1, 2, 3. Let cn be the number of partitions where each part has at least two elements. For example, c4 = 4: 1234; 12, 34; 13, 24; 14, 23. Show that cn = bn-1 - bn-2 + ... + (-1)nb1 .
  6. Find all non-negative integer solutions to 5a7b + 4 = 3c .
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