[nongtum]

$$\begin{eqnarray}
\frac{1}{1+a^p+\frac{1}{a^q} }+\frac{1}{1+a^r+\frac{1}{a^p} }+\frac{1}{1+a^q+\frac{1}{a^r} }
&=& \frac{a^q}{a^q+a^{p+q}+1}+\frac{a^{p+q}}{a^{p+q+r}+a^{p+q}+a^q}+\frac{1}{1+a^q+a^{-r}}\\
&=& \frac{a^q}{a^q+a^{-r}+1}+\frac{a^{-r}}{1+a^{-r}+a^q}+\frac{1}{1+a^q+a^{-r}}\\
&=&1 \\
\end{eqnarray}$$