Problem 796 (difficulty: 10/10)

Let \(\displaystyle p(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0\) be a polynomial with real coefficients and \(\displaystyle n\ge2\), and suppose that the polynomial \(\displaystyle (x-1)^{k+1}\) divides \(\displaystyle p(x)\) with some positive integer \(\displaystyle k\). Prove that

\(\displaystyle \sum_{\ell=0}^{n-1} |a_\ell| > 1+\frac{2k^2}{n}. \)

CIIM 4, Guanajuato, Mexico, 2012

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