The 14th United States of America Mathematics Olympiad
1985年第14届美国数学奥林匹克 
 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 ?
 Find all real roots of the quartic x^{4}  (2N + 1)x^{2}  x + N^{2} + N  1 = 0 correct to 4 decimal places, where N = 10^{10}.
 A tetrahedron has at most one edge longer than 1. What is the maximum total length of its edges ?
 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 .
 0 < a_{1} ≤ a_{2} ≤ a_{3} ≤ ... is an unbounded sequence of integers. Let b_{n} = m if a_{m} is the first member of the sequence to equal or exceed n. Given that a_{19} = 85, what is the maximum possible value of a_{1} + a_{2} + ... + a_{19} + b_{1} + b_{2} + ... + b_{85} ?

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