Junior Balkan Mathematical Olympiad 2021 - Problem 1 - 1 July 2021

Problem 1. Let $n$ ( $n \ge 1$ ) be an integer. Consider the equation

$2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$ ,

where $x$ is the unknown real variable.

(a) Solve the equation for $n = 8$ .
(b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions.
(For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$ .)

proposed by Bulgaria