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P (k) >= \bin{2k - 1}{k}^2 , no of 4k digits numbers from 0,2 divisible by 2020

Source: 2020 Czech and Slovak Olympiad III A p6

November 24, 2020

inequalitiesBinomialnumber theorydivisible

Problem Statement

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)