2011 USAMO Problems/Problem 1
Let , , be positive real numbers such that . Prove that
Since it is natural to consider a change of variables: with the inverse mapping given by: With this change of variables, the constraint becomes while the left side of the inequality we need to prove is now
Therefore it remains to prove that
We note that the product of the three (positive) terms is 1/8, therefore by AM-GM their mean is at least 1/2, and thus their sum is at least 3/2 and we are done.