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Let n be a positive integer, and consider a sequence a_1, a_2, \ldots, a_n of positive integers. Extend it periodically to an infinite sequence a_1, a_2, \ldots by defining a_{n+i} = a_i for all i \geq 1. If 
  a_1 \leq a_2 \leq \cdots \leq a_n \leq a_1 + n
and 
  a_{a_i} \leq n + i - 1  \quad \quad \text{for}\ i = 1, 2, \ldots, n \text{,}
prove that 
  a_i + \cdots + a_n \leq n^2 \text{.}

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