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Large real polynomial with real roots

Source: Germany VAIMO 2015 - #1

July 11, 2015
algebrapolynomialreal numberRootInteger

Problem Statement

Find the least positive integer nn, such that there is a polynomial P(x)=a2nx2n+a2n−1x2n−1+⋯+a1x+a0 P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 with real coefficients that satisfies both of the following properties: - For i=0,1,…,2ni=0,1,\dots,2n it is 2014≤ai≤20152014 \leq a_i \leq 2015. - There is a real number ξ\xi with P(ξ)=0P(\xi)=0.
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