2015 IMO Shortlist N1

Determine all positive integers $M$ for which the sequence $a_{0}, a_{1}, a_{2}, \ldots$, defined by $a_{0}=\frac{2 M+1}{2}$ and $a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor$ for $k=0,1,2, \ldots$, contains at least one integer term. (Luxembourg)

确定由 $a_{0}=\frac{2 M+1}{2}$ 和 $a_{k+1}=a_{k}\left\lfloor a_{k}\right\rfloor$ 定义的序列 $a_{0}, a_{1}, a_{2}, \ldots$ 至少包含一个整数项的所有正整数 $M$，其中 $k=0,1,2, \ldots$。 (卢森堡)