Find the value....

Kind of a cheap solution, but plug in x=1 and you get 1/3 + 1/3 + 1/3. I find this approach helpful for competition math: when you don't have a lot of time, instead of proving something for all x, try plugging in an easy number that fits the requirements of x and you'll get the answer.

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

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

$\Rightarrow \frac{x^a}{x^a+x^b+x^c}$ + $\frac{x^b}{x^b+x^a+x^c}$ + $\frac{x^c}{x^c+x^b +x^a}$

$\Rightarrow \frac {x^a+x^b+x^c}{x^a+x^b+x^c}$

$\Rightarrow 1$