2012 USAJMO Problems/Problem 3
Let , , be positive real numbers. Prove that
By the Cauchy-Schwarz inequality, so Since , Hence,
Again by the Cauchy-Schwarz inequality, so Since , Hence,
Split up the numerators and multiply both sides of the fraction by either a or 3a: Then use Titu's Lemma (like Cauchy): It suffices to prove that After some simplifying, it reduces to which is trivial by the Rearrangement Inequality.
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