ILL 1984 Problem 45

Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$ .