*Problem:*

If are positive real numbers satisfying the equality , then prove that

.

*Solution:*

Without loss of generality, assume that .

Since

and

,

using Chebyshev’s Inequality we get:

.

Moreover, From the AM-GM Inequality we have that

.

Therefore, it suffices to prove that

.

Set .

The above inequality can be written now as

, or .

From the AM-GM Inequality once again, we acquire that

.

And thus, it is enough to check that

,

which is equivalent to the obvious inequality

, *Q.E.D.*

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