IMO Shortlist 1967 problem 2


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Prove that
\frac{1}{3}n^2 + \frac{1}{2}n + \frac{1}{6} \geq (n!)^{\dfrac{2}{n}},
and let n \geq 1 be an integer. Prove that this inequality is only possible in the case n = 1.
Izvor: Međunarodna matematička olimpijada, shortlist 1967