IR-MO 2009 Problem 6

Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$ . Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$ .

Time allowed for this problem was 1 hour.