KöMaL 2010 A511

Show that for arbitrary positive integers $n$, $k$ there exists a polynomial $p$($x$), with degree at most $100\sqrt{nk}$ , such that $p(0) > \big(\big|p(1)\big|+\ldots+\big|p(n)\big|\big) + \big(\big|p(-1)\big|+\ldots+\big|p(-k)\big|\big)$.