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# Understand the problem

### then is it true that $$a+b+c=3$$?

##### Source of the problem
TIFR GS 2017, Part 1 Problem 2

Inequality
Easy

# Introduction to Inequalities (New Mathematical Library)

Do you really need a hint? Try it first!

Try to use AM-GM-HM inequalities
We know $$A.M \geq H.M$$. Can you use that?
$$A.M \geq H.M$$ when are they equal?
Taking $$1,a,b,c$$ we know $$A.M \geq H.M$$ then   $$\frac{1+a+b+c}{4}\geq \frac{4}{1+\frac 1a+\frac 1b+\frac 1c} \Rightarrow (1+a+b+c)(1+\frac 1a+\frac 1b+\frac 1c)\geq 16$$.

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