IMO Shortlist 2013 problem C5

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Dodao/la: arhiva
21. rujna 2014.
Let r be a positive integer, and let a_0, a_1, \ldots be an infinite sequence of real numbers. Assume that for all nonnegative integers m and s there exists a positive integer n \in [m + 1, m + r] such that 
  a_m + a_{m+1} + \cdots + a_{m+s} = a_n + a_{n+1} + \cdots + a_{n+s} \text{.}

Prove that the sequence is periodic, i.e. there exists some p \geq 1 such that a_{n+p} = a_n for all n \geq 0.
Izvor: India