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The 35th Canadian Mathematical Olympiads
2003年第35届加拿大数学奥林匹克
  1. The angle between the hour and minute hands of a standard 12-hour clock is exactly 1o. The time is an integral number n of minutes after noon (where 0 < n < 720). Find the possible values of n .
  2. What are the last three digits of 2003N, where N = 20022001 .
  3. Find all positive real solutions to x3 + y3 + z3 = x + y + z , x2 + y2 + z2 = xyz .
  4. Three fixed circles pass through the points A and B. X is a variable point on the first circle different from A and B. The line AX meets the other two circles at Y and Z (with Y between X and Z). Show that XY/YZ is independent of the position of X .
  5. S is any set of n distinct points in the plane. The shortest distance between two points of S is d. Show that there is a subset of at least n/7 points such that each pair is at least a distance d√3 apart .
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