Problem

Source: ELMO 2014 Shortlist N6, by Yang Liu

Tags: modular arithmetic, number theory proposed, number theory



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