if p+q+r = 0 then p+q = -r , q+r = -p and r+p = -q
$ \frac{1}{x^p+x^{-q}+1}+\frac{1}{x^q+x^{-r}+1}+\frac{1}{x^r+x^{-p}+1}$ = $ \frac{x^q}{x^{p+q}+1+x^{q}}+\frac{1}{x^q+x^{-r}+1}+\frac{1}{x^r+x^{-p}+1}$
= $ \frac{x^q}{x^{-r}+1+x^{q}} + \frac{1}{x^q+x^{-r}+1} + \frac{1}{x^r+x^{-p}+1}$ = $ \frac{x^q+1}{x^{q}+x^{-r}+1}+\frac{1}{x^r+x^{-p}+1}$ = $ \frac {x^{q+r}+x^r}{x^{q+r}+1+x^{r}}+\frac{1}{x^r+x^{-p}+1}$
= $ \frac{x^r+x^{-p}}{x^r+x^{-p}+1}+\frac{1}{x^r+x^{-p}+1}$= $ \frac{x^r+x^{-p}+1}{x^r+x^{-p}+1}$ = 1 Ans
