*Problem:*

Let be positive real numbers. Prove that

.

*1st solution:*

The left hand side can be rewritten as

.

We only need to prove that

,

which is true according to the AM-GM inequality, that is

,

which holds for all non-negative numbers.

*2nd solution:*

Bringing everything in the left hand side we get that

.

But this one holds because it is of the form

.

Equality occurs if and only if , *Q.E.D.*

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