Question 189
Given the set $A = \{1, 2, \dots, 4044\}$. One colors 2022 of its numbers white and the rest black. For each $i \in A$, define the weight of $i$ as the sum of the number of white numbers that are smaller than $i$ and the number of black numbers that are larger than $i$. For every positive integer $m$, find all positive integers $k$ such that there exists a way to color the numbers so that exactly $k$ numbers have weight $m$.