5. Which number is greater: $2023^{2023}$ or $2022^{2024} ?$

#
# Solution

Consider the ratio of numbers

$$
\frac{2023^{2023}}{2022^{2024}}=\frac{2023}{2022^{2}} \cdot\left(\frac{2023}{2022}\right)^{2022}=\frac{2023}{2022^{2}} \cdot\left(1+\frac{1}{2022}\right)^{2022}
$$

We will use the fact that

$$
21
$$

Then

$$
\frac{2023}{2022^{2}} \cdot\left(1+\frac{1}{2022}\right)^{2022}<\frac{2023 \cdot 3}{2022 \cdot 2022}<1
$$

Formula $(*)$ was accepted without proof, but any correct attempts to justify it were encouraged.

Answer: $\quad 2023^{2023}<2022^{2024}$.