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Coefficients of Polynomial

Source: 2017 Greece National Olympiad Problem 4

May 2, 2017
polynomialalgebra

Problem Statement

Let uu be the positive root of the equation x2+x−4=0x^2+x-4=0. The polynomial P(x)=anxn+an−1xn−1+...+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0 where nn is positive integer has non-negative integer coefficients and P(u)=2017P(u)=2017. 1) Prove that a0+a1+...+an≡1mod  2a_0+a_1+...+a_n\equiv 1\mod 2. 2) Find the minimum possible value of a0+a1+...+ana_0+a_1+...+a_n.
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