IMO 1986 LL TUR73

Let (ai)i\inN be a strictly increasing sequence of positive real

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IMO 1986 LL TUR73

Origin: TUR

Problem

Let (ai)i\inN be a strictly increasing sequence of positive real numbers such that limi\to\inftyai = +\inftyand ai+1/ai \leq10 for each i. Prove that for every positive integer k there are infinitely many pairs (i, j) with 10k \leqai/aj \leq10k+1.