MathDB

35

Part of PEN O Problems

Problems(1)

O 35

Source:

5/25/2007
Let n≥3 n \ge 3 be a prime number and a1<a2<⋯<an a_{1} < a_{2} < \cdots < a_{n} be integers. Prove that a1,⋯ ,an a_{1}, \cdots,a_{n} is an arithmetic progression if and only if there exists a partition of {0,1,2,⋯ } \{0, 1, 2, \cdots \} into sets A1,A2,⋯ ,An A_{1},A_{2},\cdots,A_{n} such that a_{1} \plus{} A_{1} \equal{} a_{2} \plus{} A_{2} \equal{} \cdots \equal{} a_{n} \plus{} A_{n}, where x \plus{} A denotes the set \{x \plus{} a \vert a \in A \}.
arithmetic sequence