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: It suffices to prove that After some simplifying, it reduces to which is trivial by the Rearrangement Inequality. -r31415
We proceed to prove that
Indeed, by AP-GP, we have
Summing up, we have
which is equivalent to:
Dividing from both sides, the desired inequality is proved.
Permuting the variables, we have all these three inequalities:
The inequality in question is just the sum of these, hence it is proved.
--Lightest 15:31, 7 May 2012 (EDT)
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