VN-TST 2015 Problem 1

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 $c_0+c_1+c_2+\cdots+c_n$ .