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The 10th Asian Pacific Mathematical Olympiad
1998年第10届亚太地区数学奥林匹克
  1. F is the set of all possible n-tuples (A1, A2, ... , An) where each Ai is a subset of {1, 2, ... , 1998}. For each member k of F let f(k) be the number of elements in the union of its n elements. Find the sum of f(k) over all k in F.
  2. Show that for any positive integers a and b,(36a + b)(a + 36b) cannot be a power of 2.
  3. Let a, b, c be positive real numbers. And d is the cube root of abc .Prove that
    (1 + a/b)(1 + b/c)(1 + c/a) ≥ 2 + 2(a + b + c)/d.
  4. Let ABC be a triangle and D the foot of the altitude from A. Let E and F be on a line through D
    such that AE is perpendicular to BE, AF is perpendicular to CF, and E and F are different from
    D. Let M and N be the midpoints of the line segments BC and EF, respectively. Prove that AN
    is perpendicular to NM.
  5. Determine the largest of all integers n with the property that n is divisible by all positive
    integers that are less than n^(1/3).
    Note: n^(1/3) is the cube root of n .
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