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The 36th Canadian Mathematical Olympiads
2004年第36届加拿大数学奥林匹克
  1. Find all ordered triples (x, y, z) of real numbers which satisfy the following system of equations:
    xy = z - x - y
    xz = y - x - z
    yz = x - y - z
  2. How many ways can 8 mutually non-attacking rooks be placed on a 9 × 9 chessboard so that all 8 rooks are on squares of the same colour ?
    [Two squares are said to be attacking each other if they are placed in the same row or column of the board.]
  3. Let A,B,C,D be four points on a circle (occurring in clockwise order), with AB < AD and BC > CD.
    Let the bisector of angle BAD meet the circle at X and the bisector of angle BCD meet the circle at Y . Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that BD must be a diameter of the circle.
  4. Let p be an odd prime. Prove that
    k from 1 to (p - 1) ( k^(2p+1)) ≡ p(p+1)/2 (mod p^2)
    [Note that a ≡ b (mod m) means that a - b is divisible by m.]
    (请原谅,我对有些公式使用了纯文本录入,但愿这没有影响您的阅读)
  5. Let T be the set of all positive integer divisors of 2004^100. What is the largest possible number of elements that a subset S of T can have if no element of S is an integer multiple of any other element of S ?
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