10.2. Which is greater, $2021^{2021}$ or $2022^{2020}$?
Answer. $2021^{2021}>2022^{2020}$

Solution. $\quad$ Consider $\quad$ the ratio $\quad \frac{2022^{2020}}{2021^{2021}}=\frac{2022^{2021} \cdot 2022^{-1}}{2021^{2021}}=\left(\frac{2022}{2021}\right)^{2021} \cdot \frac{1}{2022}=$ $\left(\frac{2021+1}{2021}\right)^{2021} \cdot \frac{1}{2022}=\left(1+\frac{1}{2021}\right)^{2021} \cdot \frac{1}{2022}2022^{2020}$.

Remark. Here, the fact that $2 \leq\left(1+\frac{1}{n}\right)^{n}<3$ was used.