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The Thirtieth Canadian Mathematical Olympiads
1998年第三十届加拿大数学奥林匹克
  1. How many real x satisfy x = [x/2] + [x/3] + [x/5] ?
  2. Find all real x equal to √(x - 1/x) + √(1 - 1/x) .
  3. Show that if n > 1 is an integer then (1 + 1/3 + 1/5 + ... + 1/(2n-1) )/(n+1) > (1/2 + 1/4 + ... + 1/2n)/n .
  4. The triangle ABC has ∠A = 40o and ∠B = 60o. X is a point inside the triangle such that ∠XBA = 20o and ∠XCA = 10o. Show that AX is perpendicular to BC .
  5. Show that non-negative integers a <= b satisfy (a2 + b2) = n2(ab + 1), where n is a positive integer, iff they are consecutive terms in the sequence ak defined by a0 = 0, a1 = n, ak+1 = n2ak - ak-1 .
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