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The 5th Asian Pacific Mathematical Olympiad
1993年第5届亚太地区数学奥林匹克数学
  1. Let ABCD be a quadrilateral such that all sides have equal length and angle ABC is 60 deg.Let l be a line passing through D and not intersecting the quadrilateral (except at D). Let E and F be the points of intersection of l with AB and BC respectively. Let M be the pointof intersection of CE and AF.
    Prove that CA2 = CM × CE.
  2. Find the total number of different integer values the function
    f(x) = [x] + [2x] + [5x/3] + [3x] + [4x]
    takes for real numbers x with 0 ≤ x ≤ 100.
  3. Let f(x) = anxn + an-1xn-1 + ... + a0 and g(x) = cn+1xn+1 + cnxn + ... + c0
    be non-zero polynomials with real coefficients such that g(x) = (x + r)f(x) for some real number r. If a = max(|an|, ..., |a0|) and c = max(|cn+1|, ..., |c0|), prove that a/c ≤ n + 1.
  4. Determine all positive integers n for which the equation xn + (2 + x)n + (2 - x)n = 0 has an integer as a solution.
  5. Let P1, P2, ... , P1993 = P0 be distinct points in the xy-plane with the following properties:
    (i) both coordinates of Pi are integers, for i = 1, 2, ... , 1993;
    (ii) there is no point other than Pi and Pi+1 on the line segment joining Pi with Pi+1 whose coordinates are both integers, for i = 0, 1, ... , 1992.
    Prove that for some i, 0 ≤ i ≤ 1992, there exists a point Q with coordinates (qx, qy) on the
    line segment joining Pi with Pi+1 such that both 2qx and 2qy are odd integers.
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