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The 33rd British Mathematical Olympiad
1997年第33届英国奥林匹克数学竞赛
  1. M and N are 9-digit numbers. If any digit of M is replaced by the corresponding digit of N (eg the 10s digit of M replaced by the 10s digit of N), then the resulting integer is a multiple of 7. Show that if any digit of N is replaced by the corresponding digit of M, then the resulting integer must be a multiple of 7. Find d > 9, such that the result remains true when M and N are d-digit numbers.
  2. ABC is an acute-angled triangle. The median BM and the altitude CF have equal length, and ∠MBC = ∠FCA. Show that ABC must be equilateral.
  3. Find the number of polynomials of degree 5 with distinct coefficients from the set {1, 2, ... , 9} which are divisible by x2 - x + 1.
  4. Let S be the set {1/1, 1/2, 1/3, 1/4, ... }. The subset {1/20, 1/8, 1/5} is an arithmetic progression of length 3 and is maximal, because it cannot be extended (within S) to a longer arithmetic progression. Find a maximal arithmetic progression in S of length 1996. Is there a maximal arithmetic progression in S of length 1997 ?
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