Problem: (2003 USAMO – #5) Let be positive real numbers. Prove that
Solution: Well first of all we notice that the inequality is homogeneous. Therefore, we can set to an arbitrary value. Note that using works particularly well.
Then the inequality becomes
Well, noticing that we have
as desired. QED.
Comment: This was a pretty cool problem, too. The most important part was splitting the fraction and getting rid of the quadratic term. After both the squared terms are gone, it’s pretty simple to finish off with only linear and constant terms.
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