Below are some problems that I have culled from Prof. Titu Andreescu’s encyclopaedic literature on mathematics olympiads.
Use Cauchy Schwarz inequality to prove the following:
Let x, y, . Prove that
Let a, b, x, y, z be positive real numbers. Prove that
Let a, b, . Prove that
Perhaps, applying the Cauchy Schwarz inequality directly may be cumbersome, or even impossible. In that case, use the following equivalent lemma for Cauchy Schwarz Inequality:
If a, b, x, y are real numbers and x, , then the following inequality holds:
(I have solved the above questions using lemma only!! Yet to check whether a direct application of Cauchy Schwarz inequality will work out 🙂 You can enlighten me on this)