2B. Determine the quotient of the numbers

$$
2021^{\sqrt{\log _{2021} 2022}} \text { and } 2022^{\sqrt{\log _{2022} 2021}}
$$
Solution. Let $2021^{\sqrt{\log _{2021} 2022}}=x \quad$ and $\quad 2022^{\sqrt{\log _{2022} 2021}}=y$. We have,

$$
\log x=\sqrt{\log _{2021} 2022} \cdot \log 2021=\sqrt{\frac{\log 2022}{\log 2021}} \cdot \log 2021=\sqrt{\log 2022 \cdot \log 2021}
$$

and

$$
\log y=\sqrt{\log _{2022} 2021} \cdot \log 2022=\sqrt{\frac{\log 2021}{\log 2022}} \cdot \log 2022=\sqrt{\log 2021 \cdot \log 2022}
$$

Therefore, $\log x=\log y$, so $x=y$, i.e., the desired quotient is $\frac{x}{y}=1$.