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Part of 1998 Putnam

Problems(2)

1998 Putnam A4

Source:

4/19/2013
Let A1=0A_1=0 and A2=1A_2=1. For n>2n>2, the number AnA_n is defined by concatenating the decimal expansions of An−1A_{n-1} and An−2A_{n-2} from left to right. For example A3=A2A1=10A_3=A_2A_1=10, A4=A3A2=101A_4=A_3A_2=101, A5=A4A3=10110A_5=A_4A_3=10110, and so forth. Determine all nn such that 1111 divides AnA_n.
Putnamcollege contestsPutnam number theory
1998 Putnam B4

Source:

4/19/2013
Find necessary and sufficient conditions on positive integers mm and nn so that ∑i=0mn−1(−1)⌊i/m⌋+⌊i/n⌋=0.\sum_{i=0}^{mn-1}(-1)^{\lfloor i/m\rfloor+\lfloor i/n\rfloor}=0.
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