For $a>0$, simplify $1/(1+a^{m-n}+a^{m-p})+1/(1+a^{n-m}+a^{n-p})+1/(1+a^{p-m}+a^{p-n})$.

Multiply the first fraction's top and bottom by $a^{-m}$, the second's by $a^{-n}$, the third's by $a^{-p}$. Each becomes $a^{-m}/S,a^{-n}/S,a^{-p}/S$ with $S=a^{-m}+a^{-n}+a^{-p}$, so the sum is $S/S=1$.