# Understand the problem

Let be positive real numbers such that .Find with proof that is the minimal value for which the following inequality holds:

##### Source of the problem

Albania IMO TST 2013

##### Topic

Inequalities

##### Difficulty Level

Medium

##### Suggested Book

Inequalities by BJ Venkatachala

# Start with hints

Do you really need a hint? Try it first!

Choose specific values of to show that no value of less than 3 works.

Choosing , the inequality becomes . Note that, as , the LHS grows as and the RHS grows as . For the LHS to be always greater than the RHS, it should be of a higher order. Hence, $x\ge 3$. Now show that the inequality is true for .

Note that the RHS can be rewritten as . This reminds us of the AM-GM inequality.

Indeed, .

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