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The 39th British Mathematical Olympiad
2003年第39届英国奥林匹克数学竞赛
  1. Find all integers 0 < a < b < c such that b - a = c - b and none of a, b, c have a prime factor greater than 3.
  2. D is a point on the side AB of the triangle ABC such that AB = 4·AD. P is a point on the circumcircle such that angle ADP = angle C. Show that PB = 2·PD.
  3. f is a bijection on the positive integers. Show that there are three positive integers a0 < a1 < a2 in arithmetic progression such that f(a0) < f(a1) < f(a2). Is there necessarily an arithmetic progression a1 < a2 < ... < a2003 such that f(a0) < f(a1) < ... < f(a2003) ?
  4. Let X be the set of non-negative integers and f : X → X a map such that ( f(2n+1) )2 - ( f(2n) )2 = 6 f(n) + 1 and f(2n) ≥ f(n) for all n in X. How many numbers in f(X) are less than 2003 ?
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