AoPS - Problem

Letting $k=f(i),l=g(j)$, we have $$S=\sum_{k\in\{1,...,2016\}}\sum_{j\in\{1,...,2016\}}(f(i)-g(j))^{2559}=\sum_{k\in\{1,...,2016\}}\sum_{l\in\{1,...,2016\}}(k-l)^{2559}=\sum_{k=1}^{2016}\sum_{l=1}^{2016}(k-l)^{2559}=\sum_{1\leq k,l\leq 2016}((2017-k)-(2017-l))^{2559}=-\sum_{1\leq k,l\leq 2016}(k-l)^{2559}=-S$$because $f,g$ and $h(i)=2017-i$ are bijections on $\{1,...,2016\}$. Therefore, the sum is zero.