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The 37th British Mathematical Olympiad
2001年第37届英国奥林匹克数学竞赛
  1. A has a marbles and B has b < a marbles. Starting with A each gives the other enough marbles to double the number he has. After 2n such transfers A has b marbles. Find a/b in terms of n .
  2. Find all integer solutions to m2n + 1 = m2 + 2mn + 2m + n .
  3. ABC is a triangle with AB greater than AC. AD is the angle bisector. E is the point on AB such that ED is perpendicular to BC. F is the point on AC such that DE bisects angle BEF. Show that ∠FDC = ∠BAD .
  4. n dwarfs with heights 1, 2, 3, ... , n stand in a circle. S is the sum of the (non-negative) differences between each adjacent pair of dwarfs. What are the maximum and minimum possible values of S ?
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