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sum 1/a_i <= 1+1/2 +1/3 +...+1/9, decimal representation

Source: Switzerland - Swiss TST 2001 p5

February 18, 2020
DigitsSumdecimal representationinequalitiesnumber theory

Problem Statement

Let a1<a2<...<ana_1 < a_2 < ... < a_n be a sequence of natural numbers such that for i<ji < j the decimal representation of aia_i does not occur as the leftmost digits of the decimal representation of aja_j . (For example, 137137 and 1372913729 cannot both occur in the sequence.) Prove that ∑i=1n1ai≤1+12+13+...+19\sum_{i=1}^n \frac{1}{a_i} \le 1+\frac12 +\frac13 +...+\frac19 .
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