Difference between revisions of "2011 USAJMO Problems/Problem 4"
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By the principle of mathematical induction, the statement of the problem is proved. [[User:Lightest|Lightest]] 21:54, 1 April 2012 (EDT) | By the principle of mathematical induction, the statement of the problem is proved. [[User:Lightest|Lightest]] 21:54, 1 April 2012 (EDT) | ||
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+ | == Solution 2== |
Latest revision as of 20:56, 13 February 2024
Problem
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 ,
,
,
be defined as follows:
,
, and for
,
is the word formed by writing
followed by
. Prove that for any
, the word formed by writing
,
,
,
in succession is a palindrome.
Solution
Let be the reflection function on the set of words, namely
for all words
,
. Then the following property is evident (e.g. by mathematical induction):
, for any words
,
.
We use mathematical induction to prove the statement of the problem. First, ,
,
are palindromes. Second, suppose
, and that the words
(
,
,
,
) are all palindromes, i.e.
. Now, consider the word
:
By the principle of mathematical induction, the statement of the problem is proved. Lightest 21:54, 1 April 2012 (EDT)
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