RU-ALLRUSSIAN 2016 Problem 115

Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$ . Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation \begin{align*} a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0 \end{align*} has not integer roots?