Problem 1683 (difficulty: 7/10)

Show that if the complex numbers \(\displaystyle a_1,\ldots,a_n\) are all different and \(\displaystyle p(z)=(z-a_1)\cdot\ldots\cdot(z-a_n)\), then

\(\displaystyle \sum_{j=1}^n \frac{p''(a_j)}{(p'(a_j))^3} = 0. \)

Give me another random problem!