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The Seventh Canadian Mathematical Olympiads
1975年第七届加拿大数学奥林匹克
  1. Evaluate (1·2·4 + 2·4·8 + 3·6·12 + 4·8·16 + ... + n·2n·4n)1/3/(1·3·9 + 2·6·18 + 3·9·27 + 4·12·36 + ... + n·3n·9n)1/3.
  2. Define the real sequence a1, a2, a3, ... by a1 = 1/2, n2an = a1 + a2 + ... + an. Evaluate an.
  3. Sketch the points in the x, y plane for which [x]2 + [y]2 = 4.
  4. Find all positive real x such that x - [x], [x], x form a geometric progression.
  5. Four points on a circle divide it into four arcs. The four midpoints form a quadrilateral. Show that its diagonals are perpendicular.
  6. 15 guests with different names sit down at a circular table, not realizing that there is a name card at each place. Everyone is in the wrong place. Show that the table can be rotated so that at least two guests match their name cards. Give an example of an arrangement where just one guest is correct, but rotating the table does not improve the situation.
  7. Is sin(x2) periodic ?
  8. Find all real polynomials p(x) such that p(p(x) ) = p(x)n for some positive integer n.
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