*Problem:*

If are non-negative real numbers with sum equal to then find the maximum value of the expression

*Solution:*

First we may, without loss of generality, assume that Now, we shall prove that

It holds that

Therefore, we only need to prove that

For we get

or equivalently the above Inequality reduces to the But this one holds due to the AM-GM Inequality since we have

Now, for on the other side, we only need to show that

or

But, again, from the AM-GM Inequality we acquire

So, finally, we are left to find the maximum value of the expression

We have according to the AM-GM Inequality

Thus, the maximum value of the expression is equal to and it is obtained if and only if *Q.E.D.*