USAMO 2010 Problem 5

Let $q = \frac{3p-5}{2}$ where $p$ is an odd prime, and let $$S_q = \frac{1}{2 \cdot 3 \cdot 4} + \frac{1}{5 \cdot 6 \cdot 7} + \dots + \frac{1}{q(q+1)(q+2)}.$$ Prove that if $\frac{1}{p} - 2S_q = \frac{m}{n}$ for integers $m$ and $n$, then $m-n$ is divisible by $p$.