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

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