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The 14th United States of America Mathematics Olympiad
1985年第14届美国数学奥林匹克
  1. Do there exist 1985 distinct positive integers such that the sum of their squares is a cube and the sum of their cubes is a square ?
  2. Find all real roots of the quartic x4 - (2N + 1)x2 - x + N2 + N - 1 = 0 correct to 4 decimal places, where N = 1010.
  3. A tetrahedron has at most one edge longer than 1. What is the maximum total length of its edges ?
  4. A graph has n > 2 points. Show that we can find two points A and B such that at least [n/2] - 1 of the remaining points are joined to either both or neither of A and B .
  5. 0 < a1 ≤ a2 ≤ a3 ≤ ... is an unbounded sequence of integers. Let bn = m if am is the first member of the sequence to equal or exceed n. Given that a19 = 85, what is the maximum possible value of a1 + a2 + ... + a19 + b1 + b2 + ... + b85 ?
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