MathDB

A word is a finite sequence of letters from some alphabet. A word is repetitive if it is a concatenation of at least two identical subwords (for example,

ababab

and

abcabc

are repetitive, but

ababa

and

aabb

are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.)
Romania (Dan Schwarz)

Problem Statement