Problem 4. For an infinite sequence of numbers $x_{1}, x_{2}, x_{3}, \ldots$, for all natural $n \geq 4$, the relation $x_{n}=x_{n-1} \cdot x_{n-3}$ holds. It is known that $x_{1}=1, x_{2}=1, x_{3}=-1$. Find $x_{2022}$.
# Answer: 1.

Solution. All members of the sequence have a modulus of 1. Mark the positive ones with a plus and the negative ones with a minus. Write out the beginning of the sequence:

$$
[++---+-]++---+-++\ldots
$$

It is clear that three consecutive members completely determine how the sequence will look further. Thus, if we see ++- again, as at the beginning, the sequence will start to repeat. Therefore, the period is seven, so the 2022nd position holds the same value as the sixth $(2022=6(\bmod 7))$.