Problem:
Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is . What is the smallest possible value of ?
Answer Choices:
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Solution:
Let the two sequences be and , and assume without loss of generality that . The definitions of the sequences imply that , so . Because and are relatively prime, divides and divides . It follows that . The minimum value of results from choosing , and , in which case .
The problems on this page are the property of the MAA's American Mathematics Competitions