Question 168
Let $\alpha$ be a positive root of the equation $x^2 + x = 5$. Suppose that $n$ is a positive integer and $c_0, c_1, \dots, c_n$ are nonnegative integers satisfying the condition
$$
c_0 + c_1\alpha + c_2\alpha^2 + \dots + c_n\alpha^n = 2015.
$$
a) Prove that $c_0 + c_1 + \dots + c_n \equiv 2 \pmod{3}$.
b) Find the minimum of $c_0 + c_1 + \dots + c_n$.
$$
c_0 + c_1\alpha + c_2\alpha^2 + \dots + c_n\alpha^n = 2015.
$$
a) Prove that $c_0 + c_1 + \dots + c_n \equiv 2 \pmod{3}$.
b) Find the minimum of $c_0 + c_1 + \dots + c_n$.