CIIM 2012 Problem 6

Let $n \geq 2$ and $p(x) = x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$ a polynomial with real coefficients. Show that if there exists a positive integer $k$ such that $(x-1)^{k+1}$ divides $p(x)$ then $$\sum_{j=0}^{n-1}|a_j| > 1 +\frac{2k^2}{n}.$$