Buffet Contest Canada 2015 Combinatorics-1

[rah4927]

Let $X$ be a subset of $\mathbb{Z}$. Denote $X + a = \{x + a|x\in X\}$. Show that if there exist non-negative integers $a_1, a_2, ..., a_n$ such that $X + a_1, X + a_2, ..., X + a_n$ form a partition of $\mathbb{Z}$, then there is a non-zero integer $N$ such that $X = X + N$.