*Problem:*

For all non-negative real numbers with sum , prove that

.

*Solution:*

Assume without loss of generality that

. Moreover, denote by the .

Then we get that

. From the hypothesis we deduce also that .

Let us now transform the factors of the inequality in terms of .

Thus we have that:

and .

Define by the Left Hand Side of the Inequality, that is

.

We will now prove that . Denote the Left Hand Side by . We must prove that .

We know that

.

We claim that the second factor of is greater than zero. Indeed, this is true as since we get that:

.

Multiplying these Inequalities we have that or . Adding up the we get that

.

Thus we have prove that

.

But from the last inequality we deduce that

due to the maximized value of .

This completes the first scale of the proof. Now we only need to prove that if

then ,

which is obviously true since it is of the form , *Q.E.D.*

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