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The 31st British Mathematical Olympiad
1995年第31届英国奥林匹克数学竞赛
  1. Find all positive integers a ≥ b ≥ c such that (1 + 1/a)(1 + 1/b)(1 + 1/c) = 2.
  2. ABC is a triangle. D, E, F are the midpoints of BC, CA, AB. Show that ∠DAC = ∠ABE iff ∠AFC = ∠ADB.
  3. x, y, z are real numbers such that x < y < z, x + y + z = 6 and xy + yz + zx = 9. Show that 0 < x < 1 < y < 3 < z < 4.
  4. (1) How many ways can 2n people be grouped into n teams of 2 ?
    (2) Show that (mn)! (mn)! is divisible by m!n+1 n!m+1 for all positive integers m, n.
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