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positive real number (IrMO 1989)

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January 17, 2014
algebra unsolvedalgebra

Problem Statement

Let aa be a positive real number and let
b=a+a2+13+a−a2+13b= \sqrt[3] {a+ \sqrt {a^{2}+1}} + \sqrt[3] {a- \sqrt {a^{2}+1}}.
Prove that bb is a positive integer if, and only if, aa is a positive integer of the form 12n(n2+3)\frac{1}{2} n(n^{2}+3), for some positive integer nn.
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