<s>[INST] Prove for all positive a,b,c that $\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$ Prove for all positive a,b,c $$\frac{a+b}{c}+\frac{b+c}{a}+\frac{c+a}{b} \geq6$$
My Try
I tried taking common denominator of the expression,
$\frac{a^2b+ab^2+b^2c+c^2b+ac^2+a^2c}{abc}$
How to proceed? Is there a way to write them as perfect squares to get the least value? a Hint is much appreciated. Thanks!
 [/INST] Notice that $$\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a} \ge 3 \left( \frac{a}{b} \frac{b}{c} \frac{c}{a}\right)^{1/3}=3~~~~ \mbox{by AM-GM}$$
and $$\frac{b}{a}+ \frac{c}{b}+ \frac{a}{c} \ge 3 \left( \frac{b}{a} \frac{c}{b} \frac{a}{c}\right)^{1/3}=3~~~~ \mbox{by AM-GM}.$$ Adding these two we get the required inequality.
In each case and overall equality occurs when $a=b=c$.
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