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The Nineteenth Canadian Mathematical Olympiads
1987年第十九届加拿大数学奥林匹克
  1. Find all positive integer solutions to n! = a2 + b2 for n < 14.
  2. Find all the ways in which the number 1987 can be written in another base as a three digit number with the digits having the same sum 25.
  3. ABCD is a parallelogram. X is a point on the side BC such that ACD, ACX and ABX are all isosceles. Find the angles of the parallelogram.
  4. n stationary people each fire a water pistol at the nearest person. They are arranged so that the nearest person is always unique. If n is odd, show that at least one person is not hit. Does one person always escape if n is even?
  5. Show that [√(4n + 1)] = [√(4n + 2)] = [√(4n + 3)] = [√n + √(n + 1)] for all positive integers n.
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