Fall 2017, problem 54

Recall that a sum of two sets $A$ and $B$ is defined as $$ A + B = \left\{ a + b | a \in A, b \in B \right\}. $$ Can one find three sets $A$, $B$, and $C$ such that $A \cup B \cup C = \mathbb{Z}$, the sets are pairwise disjoint, and the sets $A+B$, $A+C$, and $B+C$ are pairwise disjoint? Can one do this if you replace $\mathbb{Z}$ with $\mathbb{Q}$?

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