ISL 2011 N7

Let $p$ be an odd prime number. For every integer $a$, define the number $$S_{a}=\frac{a}{1}+\frac{a^{2}}{2}+\cdots+\frac{a^{p-1}}{p-1}$$ Let $m$ and $n$ be integers such that $$S_{3}+S_{4}-3 S_{2}=\frac{m}{n}$$ Prove that $p$ divides $m$.