<s>[INST] Find the value of equation? Suppose 
\begin{align*}
  a+b+c &= 20\\
  \frac 1a + \frac 1b +  \frac 1c &= 30
\end{align*} 
Then find the value of 
$$
\frac ab + \frac ba + \frac ac + \frac ca + \frac bc + \frac cb 
$$
How i can apply A.M. $\ge$ G.M $\ge$ H.M.  in this?
How i can achieve this?
 [/INST] $$\frac ab + \frac ba + \frac ac + \frac ca + \frac bc + \frac cb $$ can be simplified as
$$ \frac{a^2c+b^2c+a^2b+bc^2+ab^2+ac^2}{abc} $$
which is
$$ \frac{\left[ac(a+c)+bc(b+c)+ab(a+b)\right]}{abc}$$
and that is
$$ \frac{\left[ac(20-b)+bc(20-a)+ab(20-c)\right]}{abc}$$
$$ = \frac{\left[20(ab+bc+ca)-3abc\right]}{abc}  $$
But 
$$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=30$$ implies
$$ ab+bc+ca=30abc$$
So you can do further simplification to get $597$
</s>