OME 2017 Problem 3

Let $p$ be an odd prime and $S_{q} = \frac{1}{2*3*4} + \frac{1}{5*6*7} + ... + \frac{1}{q(q+1)(q+2)}$ , where $q = \frac{3p-5}{2}$ . We write $\frac{1}{2}-2S_{q}$ in the form $\frac{m}{n}$ , where $m$ and $n$ are integers. Prove that $m \equiv n (mod p)$