2006 IMO Shortlist A1

A sequence of real numbers $a_{0}, a_{1}, a_{2}, \ldots$ is defined by the formula $$a_{i+1}=\left\lfloor a_{i}\right\rfloor \cdot\left\langle a_{i}\right\rangle \quad \text { for } \quad i \geq 0 \text {; }$$ here $a_{0}$ is an arbitrary real number, $\left\lfloor a_{i}\right\rfloor$ denotes the greatest integer not exceeding $a_{i}$, and $\left\langle a_{i}\right\rangle=a_{i}-\left\lfloor a_{i}\right\rfloor$. Prove that $a_{i}=a_{i+2}$ for $i$ sufficiently large. (Estonia)

实数序列 $a_{0}, a_{1}, a_{2}, \ldots$ 由公式 a_{i+1}=\left\lfloor a_{i}\right\rfloor \cdot\left\langle a_{i}\right\rangle \quad \text { for } \quad i \geq 0 \text {; 定义这里$a_{0}$是任意实数，$\left\lfloor a_{i}\right\rfloor$表示不超过$a_{i}$的最大整数，$\left\langle a_{i}\right\rangle=a_{i}-\left\lfloor a_{i}\right\rfloor$。证明 $a_{i}=a_{i+2}$ 对于 $i$ 足够大。 (爱沙尼亚)