CONO-SUR 2022 Problem 3

Prove that for every positive integer $n$ there exists a positive integer $k$ , such that each of the numbers $k, k^2, \dots, k^n$ have at least one block of $2022$ in their decimal representation.

For example, the numbers 4202213 and 544202212022 have at least one block of $2022$ in their decimal representation.