1. Consider the 2022 fractions

$$
\frac{0}{2022}, \frac{1}{2021}, \frac{2}{2020}, \ldots, \frac{2021}{1}
$$

in the form of the ratio of two non-negative integers, whose sum for each fraction is equal to 2022. How many of them take integer values?

(Jaroslav Zhouf)
SOLUTION. The denominators of the given fractions are natural numbers from 1 to 2022. The numerator $c$ of the fraction with a given denominator $j$ is determined by the equation $c+j=2022$, i.e., $c=2022-j$. Therefore, our fractions can be expressed, and we will immediately simplify this expression:

$$
\frac{2022-j}{j}=\frac{2022}{j}-1, \quad \text { where } \quad j=2022,2021, \ldots, 1
$$

(The values of the denominators $j$ are listed in descending order, as they are in the problem statement.)

According to the problem, our task is to determine how many of all the denominators $j$ divide the corresponding numerator $2022-j$. The simplification we performed makes our task easier: we need to find the number of $j$ from 1 to 2022 that are divisors of the number 2022.

Since the number 2022 has the prime factorization $2022=2 \cdot 3 \cdot 337$, its divisors are precisely the numbers 1, 2, 3, 6, 337, 674, 1011, and 2022, of which there are eight in total. These are the denominators $j$ of all the considered fractions with integer values. Their numerators $c=2022-j$ are, in order, the numbers 2021, 2020, 2019, 2016, 1685, 1348, 1011, and 0.

Conclusion. Exactly eight of the given fractions have integer values, specifically

$$
\frac{0}{2022}, \frac{1011}{1011}, \frac{1348}{674}, \frac{1685}{337}, \frac{2016}{6}, \frac{2019}{3}, \frac{2020}{2}, \frac{2021}{1}
$$

## NOTES.