2008 Indonesia MO Problems/Problem 2
Prove that for every positive reals and ,
By the Cauchy-Schwarz Inequality, and , with equality happening in the earlier inequality when and equality happening in the latter inequality when . Because , By the AM-GM Inequality, we know that . For the equality case, , so . Additionally, by the AM-GM Inequality, . For the equality case, , so . Because , Therefore, since and and , we must have , with equality happening when .
Let Since this function is concave up, according to Jensen's inequality, we can get which means . In this problem, it turns into .The conclusion we try to find is that So we can see that . Take reciprocal for both sides we can get . Take RHS, . Now we have to prove that . which turns to . It is always correct according to inequality, it happens when . ~bluesoul
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