<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] Using Cauchy Schwarz, we can write $\displaystyle (a + b + c) \left(\frac{1}{a} +\frac{1}{b} + \frac{1}{c} \right) \geq 9$. Thus, we have, 
$$ \frac{a+b+c}{a} +\frac{a+b+c}{b} + \frac{a+b+c}{c} \geq 9 \implies 1+ \frac{b+c}{a} + 1 + \frac{a+c}{b} + 1 + \frac{a+b}{c} \geq 9$$
Hence, $$  \frac{a+b}{c}  + \frac{b+c}{a} +  \frac{c+a}{b} \geq 6 $$
</s>