*Solution ( by Vo Quoc Ba Can):*

WLOG, assume . Now we use a lemma:

We claim that

Proof: The lemma is equivalent with:

But, as so

& so the lemma is proved.

The inequality

.

This easily follows from AM-GM. Hence proved.

This solution is really genial for its simplicity and brilliancy…. and also the easiest solution (in my opinion) possible.

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