Problem

Source: 2020 Czech and Slovak Olympiad III A p6

Tags: inequalities, Binomial, number theory, divisible



For each positive integer $k$, denote by $P (k)$ the number of all positive integers $4k$-digit numbers which can be composed of the digits $2, 0$ and which are divisible by the number $2 020$. Prove the inequality $$P (k) \ge \binom{2k - 1}{k}^2$$and determine all $k$ for which equality occurs. (Note: A positive integer cannot begin with a digit of $0$.) (Jaromir Simsa)