1997 USAMO Problems/Problem 5
Prove that, for all positive real numbers
Because the inequality is homogenous (i.e. can be replaced with without changing the inequality other than by a factor of for some ), without loss of generality, let .
Lemma: Proof: Rearranging gives , which is a simple consequence of and
Thus, by :
Rearranging the AM-HM inequality, we get . Letting , , and , we get By AM-GM on , , and , we have So, .
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