IMO 2014 problem 1


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21. rujna 2014.
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Let a_0 < a_1 < a_2 < \cdots be an infinite sequence of positive integers. Prove that there exists a unique integer n \geq 1 such that 
  a_n < \frac{a_0 + a_1 + \cdots + a_n}{n} \leq a_{n+1} \text{.}
Izvor: International Mathematical Olympiad 2014, day 1