As a recent IMO problem

Algebra Level 5

Assume the increasing sequence of integer numbers: \[a_0 \lt a_1 \lt ... \lt a_n, n \rightarrow \infty\] How many value of n satisfies the group of in-equalities: \[ a_n \lt \frac{a_0+a_1+...+a_n}{n} \lt a_{n+1}\]

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