1. We will call a word consisting only of the letters "a", "b", "c" diverse if it does not contain two identical subwords in a row. (For example, the word "abcac" is diverse, while the word "abcbc" is not.) Does there exist a diverse word of 15 letters?
1. Example of a diverse word: abcabacabcacbc.

There exists an infinite diverse word (Thue's theorem). Its construction is a rather difficult task.