ELMO SL 2014 N6

Show that the numerator of $$\frac{2^{p-1}}{p+1} - \left(\sum_{k = 0}^{p-1}\frac{\binom{p-1}{k}}{(1-kp)^2}\right)$$ is a multiple of $p^3$ for any odd prime $p$ .

Proposed by Yang Liu