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:

Problem 1:

Let x, y, . Prove that

Problem 2:

Let a, b, x, y, z be positive real numbers. Prove that

Problem 3:

Let a, b, . Prove that

.

Note: 

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:

Lemma:

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)

Nalin Pithwa