Problem

Source: Ukrainian Mathematical Olympiad 2021. Day 1, Problem 8.2

Tags: Strings, combinatorics



The Kotoras alphabet has exactly $11$ letters. It is known that for any positive integer $k$, there are exactly five $k$-letter bad words in the language of this tribe. Prove that it is possible to come up with a word in the Kotoras language with any number of letters that cannot be cut into two smaller words, of which at least one is considered bad. Proposed by Arseniy Nikolaiev