Here is a problem that appeared in Romanian Mathematical Olympiad (Junior Team Selection Test 2002).

If prove that

*First Solution:* Use Cauchy-Schwarz inequality to obtain:

Here we used the fact that (which follows from ), and (which follows from ). Taking square root of both sides, we obtain the desired inequality.

*Second Solution:* Since , there exists such that

Thus, the problem now reads as

or equivalently,

We can prove this inequality as follows

Done!

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