Question 138
For each real number $a$ let $\lfloor a \rfloor$ be the largest integer not exceeding $a$. Find all positive real numbers satisfying
$$
x \cdot \lfloor x \rfloor + 2022 = \lfloor x^2 \rfloor.
$$
$$
x \cdot \lfloor x \rfloor + 2022 = \lfloor x^2 \rfloor.
$$