TIFR GS 2017, Part 1 Problem 2

Inequality

Easy

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 \(\frac{1+a+b+c}{4}\geq \frac{4}{1+\frac 1a+\frac 1b+\frac 1c} \Rightarrow (1+a+b+c)(1+\frac 1a+\frac 1b+\frac 1c)\geq 16 \). #### Now they are equal when \(a=b=c=1\) i.e \(a+b+c=3\).

## Hence the statement is true.

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