B4. The equation $\sin x=\frac{x}{2021 \pi}$ has exactly $n$ solutions. Find $n$.
Solution: Notice that $x=0$ is a solution. As well, both functions are odd, so due to the symmetry if $x^{*}$ is a solution then $-x^{*}$ is also a solution.

Since $-1 \leq \sin x \leq 1$, all solutions are in the interval for which $-1 \leq \frac{x}{2021 \pi} \leq 1$, that is $[-2021 \pi, 2021 \pi]$.
The function $y=\sin x$ is $2 \pi$-periodic. There are 1010 periods and a half of a period on the segment $(0,2021 \pi]$. There are 2 intersections of the line $y=\frac{x}{2021 \pi}$ and the sine on each period except the first one $(0,2 \pi]$, where we have only one intersection. There are also two intersections on the last half of the period. Consequently, there are $1+2020=2021$ solutions on $(0,2021 \pi]$. Similarly, there are 2021 solutions on $[-2021 \pi, 0)$. In total, there are $n=1+2 \times 2021=4043$ solutions.
Answer: 4043 .