Exponents these days! :/

Notice the first factor of $B$ is simply

$\displaystyle{\left(x^{m-n} \right)^{m^2 +mn +n^2} = x^{(m-n)(m^2 + mn +n^2)} = x^{m^3 - n^3}}$

Similarly for the other 2 factors of $B$ we get

$\displaystyle{B = x^{m^3 - n^3} \times x^{n^3 - l^3} \times x^{l^3 - m^3} = x^0 = 1}$

Notice for the first addend in $A$

$\dfrac{1}{x^{b-b} + x^{b-c} + x^{b-a} } = \dfrac{1}{x^b\left( S \right)}$

where $S = x^{-b} + x^{-c} + x^{-a}$ .

Similarly calculations for the other 2 addends of $A$ lead to

$A = \dfrac{1}{x^b S } + \dfrac{1}{x^a S } + \dfrac{1}{x^c S }$

$A = \dfrac{1}{ S} \left( \dfrac{1}{x^b} + \dfrac{1}{x^a} + \dfrac{1}{x^c} \right) = \dfrac{1}{ S} \left( x^{-b} + x^{-c} + x^{-a} \right) = 1$

hence $A + B = 2$ .

Simplify both of them separately. For $A$ ,

$\frac{1}{1+x^{b-c}+x^{b-a}}+\frac{1}{1+x^{c-a}+x^{c-b}}+\frac{1}{1+x^{a-b}+x^{a-c}}$

$=\frac{1}{\frac{x^{b}}{x^{b}}+\frac{x^{b}}{x^{c}}+\frac{x^{b}}{x^{a}}}+\frac{1}{\frac{x^{c}}{x^{c}}+\frac{x^{c}}{x^{a}}+\frac{x^{c}}{x^{b}}}+\frac{1}{\frac{x^{a}}{x^{a}}+\frac{x^{a}}{x^{b}}+\frac{x^{a}}{x^{c}}}$

$=\frac{1}{x^{b}\left(\frac{1}{x^{a}}+\frac{1}{x^{b}}+\frac{1}{x^{c}}\right)}+\frac{1}{x^{c}\left(\frac{1}{x^{a}}+\frac{1}{x^{b}}+\frac{1}{x^{c}}\right)}+\frac{1}{x^{a}\left(\frac{1}{x^{a}}+\frac{1}{x^{b}}+\frac{1}{x^{c}}\right)}$

$=\left(\frac{1}{x^{a}}+\frac{1}{x^{b}}+\frac{1}{x^{c}}\right)\left(\frac{1}{\frac{1}{x^{a}}+\frac{1}{x^{b}}+\frac{1}{x^{c}}}\right)$

$=\boxed{1}$

And for $B$ ,

$\left(\frac{x^{m}}{x^{n}}\right)^{m^{2}+mn+n^{2}}\times\left(\frac{x^{n}}{x^{l}}\right)^{n^{2}+nl+l^{2}}\times\left(\frac{x^{l}}{x^{m}}\right)^{l^{2}+lm+m^{2}}$

$=\left(\left(\frac{x^{m}}{x^{n}}\right)^{m^{2}}\times\left(\frac{x^{l}}{x^{m}}\right)^{m^{2}}\right)\times\left(\left(\frac{x^{m}}{x^{n}}\right)^{n^{2}}\times\left(\frac{x^{n}}{x^{l}}\right)^{n^{2}}\right)\times\left(\left(\frac{x^{n}}{x^{l}}\right)^{l^{2}}\times\left(\frac{x^{l}}{x^{m}}\right)^{l^{2}}\right)\times\left(\left(\frac{x^{m}}{x^{n}}\right)^{mn}\times\left(\frac{x^{n}}{x^{l}}\right)^{nl}\times\left(\frac{x^{l}}{x^{m}}\right)^{lm}\right)$

$=\frac{x^{lm^{2}}}{x^{nm^{2}}}\times\frac{x^{mn^{2}}}{x^{ln^{2}}}\times\frac{x^{nl^{2}}}{x^{ml^{2}}}\times\frac{x^{m^{2}n}}{x^{mn^{2}}}\times\frac{x^{n^{2}l}}{x^{nl^{2}}}\times\frac{x^{l^{2}m}}{x^{lm^{2}}}$

$=\boxed{1}$

Hence the answer is $\boxed{2}$ .