JBMO 2021 Problem 1

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

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

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 $m \leq y$.)