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,
Titu's Lemma: The sum of multiple fractions in the form where and are sequences of real numbers is greater than of equal to the square of the sum of all divided by the sum of all , where i is a whole number less than n+1. Titu's Lemma can be proved using the Cauchy-Schwarz Inequality after multiplying out the denominator of the RHS.
Consider the LHS of the proposed inequality. Split up the numerators and multiply both sides of the fraction by either a or 3a to make the LHS (Cyclic notation is a special notation in which all permutations of (a,b,c) is the summation command.)
Then use Titu's Lemma on all terms: owing to the fact that , which is actually equivalent to !
We proceed to prove that
(then the inequality in question is just the cyclic sum of both sides, since )
Indeed, by AP-GP, we have
Summing up, we have
which is equivalent to:
Dividing from both sides, the desired inequality is proved.
--Lightest 15:31, 7 May 2012 (EDT)
(by AM-GM) as desired.
We note that if we can prove that there exists a real number such that for all positive reals and , then we are done since both sides are a cyclic sum of , and . Note that if we multiply both and by a constant , the left and right sides will both multiply by (since the numerator on the left will multiply by and the denominator will multiply by . This means that if the inequality is satisfied for an ordered pair , then it is also satisfied for for any positive real . Thus, without loss of generality we can let ,and our inequality becomes Expanding this and rearranging we get . Since is a root of the left side, we can factor this as Note that for this to always be nonnegative, we must have be positive when and negative when . Thus, it must have a root at , so we plug in is a root and find that we must have . When , the expression factors as which obviously is nonnegative for all positive reals , so we are done.
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