This is a problem from RMO 2015 Mumbai Region based on inequality.

**Problem: RMO 2015 Mumbai Region**

Let x, y, z be real numbers such that . Prove that

**Discussion**

Note that .

According to the given condition .

Therefore .

Adding 2(x+y+z) + 1 to both sides,

We wish to show

or

Replacing 2xyz by we have

(this is what we need to show).

Therefore we need to show

This is true by Cauchy Schwarz Inequality.

PROOF 2 (suggested by Arkabrata Das)

CHECK FILE: new doc 1720151229235503023

## Chatuspathi:

**Paper:**RMO 2015 (Mumbai Region)**What is this topic:**Inequality**What are some of the associated concepts:**Cauchy Schwarz Inequality**Where can learn these topics:**Cheenta**Book Suggestions:**Secrets in Inequalities

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