2015 IMO Shortlist N4

Suppose that $a_{0}, a_{1}, \ldots$ and $b_{0}, b_{1}, \ldots$ are two sequences of positive integers satisfying $a_{0}, b_{0} \geq 2$ and $$a_{n+1}=\operatorname{gcd}\left(a_{n}, b_{n}\right)+1, \quad b_{n+1}=\operatorname{lcm}\left(a_{n}, b_{n}\right)-1$$ for all $n \geq 0$. Prove that the sequence $\left(a_{n}\right)$ is eventually periodic; in other words, there exist integers $N \geq 0$ and $t>0$ such that $a_{n+t}=a_{n}$ for all $n \geq N$. (France)

假设 $a_{0}, a_{1}, \ldots$ 和 $b_{0}, b_{1}, \ldots$ 是满足 $a_{0}, b_{0} \geq 2$ 和 $$a_{n+1}=\operatorname{gcd}\left(a_{n}, b_{n}\right)+1, \quad 的两个正整数序列b_{n+1}=\operatorname{lcm}\left(a_{n}, b_{n}\right)-1$$ 对于所有 $n \geq 0$。证明序列$\left(a_{n}\right)$最终是周期性的；换句话说，存在整数 $N \geq 0$ 和 $t>0$，使得所有 $n \geq N$ 的 $a_{n+t}=a_{n}$ 。 (法国)