Question 42
Let $n$ ($n \geq 1$) be an integer. Consider the equation
$$
2 \cdot \left\lfloor \frac{1}{2x} \right\rfloor - n + 1 = (n+1)(1 - n x)
$$
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 $m \leq y$.)
$$
2 \cdot \left\lfloor \frac{1}{2x} \right\rfloor - n + 1 = (n+1)(1 - n x)
$$
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 $m \leq y$.)