KöMaL 2011 A532

Prove that there exist a real number $c$>0 such that whenever each of the numbers $a_0,a_1,\ldots,a_n$ are 1 or -1 and the polynomial ($x$-1)$^k$ divides the polynomial $a_0+a_1x+\ldots+a_nx^n$ then $k$<$c^. ln^2$($n$+1).