Problem

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Tags: algebra



Let $\alpha$ be the positive root of the equation $x^2+x=5$. Let $n$ be a positive integer number, and let $c_0,c_1,\ldots,c_n\in \mathbb{N}$ be such that $ c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015. $ a. Prove that $c_0+c_1+c_2+\cdots+c_n\equiv 2 \pmod{3}$. b. Find the minimum value of the sum $c_0+c_1+c_2+\cdots+c_n$.