2011 USAJMO Problems/Problem 4

A word is defined as any finite string of letters. A word is a palindrome if it reads the same backwards as forwards. Let a sequence of words $W_0$ , $W_1$ , $W_2$ , $\dots$ be defined as follows: $W_0 = a$ , $W_1 = b$ , and for $n \ge 2$ , $W_n$ is the word formed by writing $W_{n - 2}$ followed by $W_{n - 1}$ . Prove that for any $n \ge 1$ , the word formed by writing $W_1$ , $W_2$ , $\dots$ , $W_n$ in succession is a palindrome.

For a word to be palindromic, the $nth$ letter must be equal to the $nth$ - $to$ - $last$ number for all letters in the word.

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2011 USAJMO Problems/Problem 4