MathDB

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

2021

in one of the

k

colors so that there are exactly

a_1

numbers of the first color,

a_2

numbers of the second color,

\ldots

, and

a_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

?
Proposed by Arsenii Nikolaiev

Problem Statement