Show that $\dfrac{1}{1 + a^{y - x} + a^{z - x}} + \dfrac{1}{1 + a^{z - y} + a^{x - y}} + \dfrac{1}{1 + a^{x - z} + a^{y - z}}$ = 1.

Given,

$\dfrac{1}{1 + a^{y - x} + a^{z - x}} + \dfrac{1}{1 + a^{z - y} + a^{x - y}} + \dfrac{1}{1 + a^{x - z} + a^{y - z}}$ = 1.

Solving L.H.S. of the above equation,

$\Rightarrow \dfrac{1}{1 + a^{y - x} + a^{z - x}} + \dfrac{1}{1 + a^{z - y} + a^{x - y}} + \dfrac{1}{1 + a^{x - z} + a^{y - z}} \\[1em] \Rightarrow \dfrac{1}{1 + a^ya^{-x} + a^{z}.a^{-x}} + \dfrac{1}{1 + a^z.a^{-y} + a^x.a^{-y}} + \dfrac{1}{1 + a^x.a^{-z} + a^y.a^{-z}} \\[1em] \Rightarrow \dfrac{1}{1 + \dfrac{a^y}{a^x} + \dfrac{a^z}{a^x}} + \dfrac{1}{1 + \dfrac{a^z}{a^y} + \dfrac{a^x}{a^y}} + \dfrac{1}{1 + \dfrac{a^x}{a^z} + \dfrac{a^y}{a^z}} \\[1em] \Rightarrow \dfrac{1}{\dfrac{a^x + a^y + a^z}{a^x}} + \dfrac{1}{\dfrac{a^y + a^z + a^x}{a^y}} + \dfrac{1}{\dfrac{a^z + a^x + a^y}{a^z}} \\[1em] \Rightarrow \dfrac{a^x}{a^x + a^y + a^z} + \dfrac{a^y}{a^x + a^y + a^z} + \dfrac{a^z}{a^x + a^y + a^z} \\[1em] \Rightarrow \dfrac{a^x + a^y + a^z}{a^x + a^y + a^z} \\[1em] \Rightarrow 1.$

Hence, proved that,

$\dfrac{1}{1 + a^{y - x} + a^{z - x}} + \dfrac{1}{1 + a^{z - y} + a^{x - y}} + \dfrac{1}{1 + a^{x - z} + a^{y - z}}$ = 1.