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TOT 240 1989 Autumn A S4 sum 1/d_i <0,9, infinite arithm, progressions

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March 12, 2021
algebrainequalitiesArithmetic Progression

Problem Statement

The set of natural numbers is represented as a union of pairwise disjoint subsets, whose elements form infinite arithmetic progressions with positive differences d1,d2,d3,...d_1,d_2,d_3,.... Is it possible that the sum 1d1+1d1+1d3+...\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... does not exceed 0.90.9? Consider the cases where (a) the total number of progressions is finite, and (b) the number of progressions is infinite. (In this case the condition that 1d1+1d1+1d3+...\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+... does not exceed 0.90.9 should be taken to mean that the sum of any finite number of terms does not exceed 0.9.)
(A. Tolpugo, Kiev)
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