Question 91
Let $a$ and $b$ be two real numbers. We define the sequences $\left(a_{n}\right)$ and $\left(b_{n}\right)$ by $a_{0}=a,\ b_{0}=b$ and, for every natural number $n$, $a_{n+1}=a_{n}+b_{n}$ and $b_{n+1}=a_{n} b_{n}$. Determine all pairs $(a, b)$ such that $a_{2022}=a_{0}$ and $b_{2022}=b_{0}$.