This is a very intrigueing idea due to Russian problem proposer – Fedor Petrov

We know by AM-GM that ,

Introduce a parameter and get the following:

Then read the last inequality from the other point: for any positive $a,b$ there exist positive such that

Here

Or, we may write:

How may this observation help? Put

Then for appropriate we have:

So, we get the Cauchy-Schwarz Inequality. A lot of other inequalities also may be proved by this idea

**Problem**

Prove that for any four nonnegative reals $a,b,c,d$ the following inequality holds-

*Source:Proposed at 239 Lyceum Traditional Olympiad **(Author : Fedor Petrov)*

We have

And for any positive $A$ and $B$ there exist appropriate $x$ and $y$ ,for which equality holds

Let in terms of the problem. For some positive $x,y$ we have –

By AM-GM

and we are done!

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