Few days ago, a friend of mine gave me this problem :
Let $X \subset \mathbb{Z}$ and suppose that $a_1, a_2, \cdots, a_n$ are integers such that $X+a_1, \cdots, X+a_n$ is a partition of $\mathbb{Z}$. Prove that $X+p=X$ for some nonzero integer $p$.
I found a solution after sometime (my solution is by using indicator function $v_i = 1$ for $i \in X$ and $0$ otherwise).
But I'm more interested on the source of problem since I'm pretty sure I've seen this problem before, maybe in a blog post or mathematics competition. It's bugging me since I can't remember where did I read this problem before.
I've tried googling it, with no luck.
I'm more interested on the source, can someone tell me where is this problem from?