Using the numbers from 1 to 22 exactly once each, Antoine writes 11 fractions: for example, he can write the fractions $\frac{10}{2}, \frac{4}{3}, \frac{15}{5}, \frac{7}{6}, \frac{8}{9}, \frac{11}{19}, \frac{12}{14}, \frac{13}{17}, \frac{22}{21}, \frac{18}{16}$, and $\frac{20}{1}$.

Antoine wishes to have as many fractions as possible with integer values among the fractions written: in the previous example, he wrote three fractions with integer values: $\frac{10}{2}=5, \frac{15}{5}=3$, and $\frac{20}{1}=20$. What is the maximum number of fractions that can have integer values?
First, it is possible to make 10 integers with the fractions

$$
\left\{\frac{22}{11}, \frac{20}{10}, \frac{18}{9}, \frac{16}{8}, \frac{14}{7}, \frac{12}{6}, \frac{21}{3}, \frac{19}{1}, \frac{15}{5}, \frac{4}{2}, \frac{17}{13}\right\}
$$

Furthermore, it is impossible to have 11 integer fractions. Indeed, there are 3 prime numbers that have no multiple smaller than 22: 19, 17, and 13. Thus, the only way for one of these three numbers to form an integer fraction is with the number 1, which can only be used with one of these three numbers.

Comment from the graders: The exercise was generally very well done. However, some students, although they noticed the importance of 13, 17, and 19, did not think to associate two of them in the same fraction, and therefore only obtain 9 integer fractions, which is a pity. Other students were content with a configuration of 10 fractions without justifying that it is optimal. Finally, we remind you that $\frac{19}{8}$ is definitely not an integer.