Problem 2. How many solutions in integers does the equation

$$
\frac{1}{2022}=\frac{1}{x}+\frac{1}{y} ?
$$
Answer: 53.

Solution. By eliminating the denominators, we obtain the equation

\[
(x-2022)(y-2022)=2022^{2}.
\]

Since \(2022^{2}=2^{2} \cdot 3^{2} \cdot 337^{2}\), the number \(x-2022\) can have the factors \(2^{0}, 2^{1}, 2^{2}\) - a total of three options. Similarly for the other factors. The number \(x-2022\) thus has \(3 \cdot 3 \cdot 3=27\) possible natural values, which means 54 integer values. However, one of these corresponds to \(x=0\), which does not satisfy the original equation. Therefore, there are 53 roots (for each suitable value of \(x\), we have \(y=\frac{2022^{2}}{x-2022}+2022\) - an integer).