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a_0+a_1+a_2+...+a_n is divisible by 2 , P(m) =2018, a_i >0, m=(-1+\sqrt{17})/2

Source: 1st Final Mathematical Cup 2019 FMC , juniors p2

October 6, 2020
Sumpolynomialalgebra

Problem Statement

Let m=−1+172m=\frac{-1+\sqrt{17}}{2}. Let the polynomial P(x)=anxn+an−1xn−1+...+a1x+a0P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0 is given, where nn is a positive integer, the coefficients a0,a1,a2,...,ana_0,a_1,a_2,...,a_n are positive integers and P(m)=2018P(m) =2018 . Prove that the sum a0+a1+a2+...+ana_0+a_1+a_2+...+a_n is divisible by 22 .
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