MS_Kekas wrote: You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots, a_k$. It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there is exactly $1$ number of the first color, $2$ numbers of the second color, $\ldots$, and $k$ numbers of the $k$-th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$, such that the total number of numbers colored in the same color as $x$ is exactly $x$. What are the possible values of $k$? Where are $a_1,a_2,...,a_k$ being used?

carefully wrote: MS_Kekas wrote: You are given a positive integer $k$ and not necessarily distinct positive integers $a_1, a_2 , a_3 , \ldots, a_k$. It turned out that for any coloring of all positive integers from $1$ to $2021$ in one of the $k$ colors so that there is exactly $1$ number of the first color, $2$ numbers of the second color, $\ldots$, and $k$ numbers of the $k$-th color, there is always a number $x \in \{1, 2, \ldots, 2021\}$, such that the total number of numbers colored in the same color as $x$ is exactly $x$. What are the possible values of $k$? Where are $a_1,a_2,...,a_k$ being used? Sorry, corrected