1. The fraction $\frac{1}{5}$ is written as an infinite binary fraction. How many ones are there among the first 2022 digits after the decimal point in such a representation? (12 points)
Solution. The smallest number of the form $2^{n}-1$ divisible by 5 is 15. Then

$$
\frac{1}{5}=\frac{3}{15}=\frac{3}{16-1}=\frac{3}{2^{4}-1}=\frac{3}{16} \cdot \frac{1}{1-2^{-4}}=\left(\frac{1}{16}+\frac{1}{8}\right)\left(1+2^{-4}+2^{-8}+2^{-12}+\cdots\right)=
$$

$\left(2^{-3}+2^{-4}\right)\left(1+2^{-4}+2^{-8}+2^{-12}+\cdots\right)=0,001100110011 \ldots=0,(0011) \quad$ in $\quad$ binary. Since $2022=4 \cdot 505+2$, among the first 2022 digits after the decimal point, there will be 1010 ones.

## Answer: 1010.