APMO 2022 Problem 3

Find all positive integers $k<202$ for which there exists a positive integer $n$ such that

$$\left\{\frac{n}{202}\right\}+\left\{\frac{2 n}{202}\right\}+\cdots+\left\{\frac{k n}{202}\right\}=\frac{k}{2}$$

where $\{x\}$ denote the fractional part of $x$. Note: $\{x\}$ denotes the real number $k$ with $0 \leq k<1$ such that $x-k$ is an integer.