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The 34th Canadian Mathematical Olympiads
2002年第34届加拿大数学奥林匹克
  1. What is the largest possible number of elements in a subset of {1, 2, 3, ... , 9} such that the sum of every pair (of distinct elements) in the subset is different ?
  2. We say that the positive integer m satisfies condition X if every positive integer less than m is a sum of distinct divisors of m. Show that if m and n satisfy condition X, then so does mn .
  3. Show that x3/(yz) + y3/(zx) + z3/(xy) ≥ x + y + z for any positive reals x, y, z. When do we have equality ?
  4. ABC is an equilateral triangle. C lies inside a circle center O through A and B. X and Y are points on the circle such that AB = BX and C lies on the chord XY. Show that CY = AO .
  5. Let X be the set of non-negative integers. Find all functions f: X → X such that x f(y) + y f(x) = (x + y) f(x2 + y2) for all x , y .
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