If we put a prime number $p$ instead of $13$, what we get is called the Chevalley-Warning theorem. A quick proof of this theorem You can find here (Chapter 2): http://www.math.mcgill.ca/goren/SeminarOnCohomology/Ax.pdf It is very likely that the author had in mind a similar solution.

calculate $\sum_{(x_{1},x_{2}...x_{n})}f(x_{1},x_{2},...x_{n})^{12}$ we only need to prove that the time $x_{1}^{k_{1}}x_{2}^{k_{2}}...x_{n}^{k_{n}}$ appears (here $k_{1},k_{2}...k_{n}$ are arbitrary non-negative numbers satisfying $k_{1}+k_{2}+...k_{n}\le\ n-1$) is a multiple of $13$ and it's easy to prove it