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Let $n \ge 3$ be a prime number and $a_{1} < a_{2} < \cdots < a_{n}$ be integers.
Prove that $a_{1}, \cdots,a_{n}$ is an arithmetic progression if and only if there exists a partition of $\{0, 1, 2, \cdots \}$ into sets $A_{1},A_{2},\cdots,A_{n}$ such that
$a_{1} + A_{1} = a_{2} + A_{2} = \cdots = a_{n} + A_{n}$
where $x + A$ denotes the set $\{x + a \vert a \in A \}$.