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Difference between revisions of "2011 USAJMO Problems/Problem 4"

(Created page with 'A word is defined as a finite sequence of letters. A ''palindrome'' is a word that reads the same forwards and backwards. Consider the sequence of words W0,W1,W2... such that W…')
 
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A word is defined as a finite sequence of letters.  A ''palindrome'' is a word that reads the same forwards and backwards.  Consider the sequence of words W0,W1,W2... such that W0 = a, W1 = b, and for all n >= 2, Wn is the word formed by writing W(n-2) followed by W(n-1).  Prove that for all n >= 1, the word formed by writing W1,W2,...,Wn in sequence is a palindrome.
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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 <math>W_0</math>, <math>W_1</math>, <math>W_2</math>, <math>\dots</math> be defined as follows: <math>W_0 = a</math>, <math>W_1 = b</math>, and for <math>n \ge 2</math>, <math>W_n</math> is the word formed by writing <math>W_{n - 2}</math> followed by <math>W_{n - 1}</math>.  Prove that for any <math>n \ge 1</math>, the word formed by writing <math>W_1</math>, <math>W_2</math>, <math>\dots</math>, <math>W_n</math> in succession is a palindrome.

Revision as of 16:08, 2 May 2011

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.

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