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Polynomial with integer roots.

Source: All russian olympiad 2016,Day2,grade 11,P5

May 5, 2016
number theoryInteger Polynomialalgebrapolynomial

Problem Statement

Let nn be a positive integer and let k0,k1,…,k2nk_0,k_1, \dots,k_{2n} be nonzero integers such that k0+k1+⋯+k2n≠0k_0+k_1 +\dots+k_{2n}\neq 0. Is it always possible to a permutation (a0,a1,…,a2n)(a_0,a_1,\dots,a_{2n}) of (k0,k1,…,k2n)(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?
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