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5

Part of 2002 Putnam

Problems(2)

Putnam 2002 A5

Source:

3/12/2012
Define a sequence by a0=1a_0=1, together with the rules a2n+1=ana_{2n+1}=a_n and a2n+2=an+an+1a_{2n+2}=a_n+a_{n+1} for each integer n≥0n\ge0. Prove that every positive rational number appears in the set {an−1an:n≥1}={11,12,21,13,32,⋯ } \left\{ \tfrac {a_{n-1}}{a_n}: n \ge 1 \right\} = \left\{ \tfrac {1}{1}, \tfrac {1}{2}, \tfrac {2}{1}, \tfrac {1}{3}, \tfrac {3}{2}, \cdots \right\} .
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Putnam 2002 B5

Source:

3/12/2012
A palindrome in base bb is a positive integer whose base-bb digits read the same backwards and forwards; for example, 20022002 is a 44-digit palindrome in base 1010. Note that 200200 is not a palindrome in base 1010, but it is a 33-digit palindrome: 242242 in base 99, and 404404 in base 77. Prove that there is an integer which is a 33-digit palindrome in base bb for at least 20022002 different values of bb.
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