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IMO ShortList 2008, Number Theory problem 3

Source: IMO ShortList 2008, Number Theory problem 3

July 9, 2009
greatest common divisornumber theorySequenceIMO Shortlist

Problem Statement

Let a0 a_0, a1 a_1, a2 a_2, … \ldots be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}. Prove that an≥2n a_n\ge 2^n for all n≥0 n\ge 0. Proposed by Morteza Saghafian, Iran
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