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Injective function with sum of reciprocal integer

Source: Romania TST 1 P3, 2013

April 5, 2013
functioninductioninequalitiesalgebra unsolvedalgebra

Problem Statement

Determine all injective functions defined on the set of positive integers into itself satisfying the following condition: If SS is a finite set of positive integers such that ∑s∈S1s\sum\limits_{s\in S}\frac{1}{s} is an integer, then ∑s∈S1f(s)\sum\limits_{s\in S}\frac{1}{f\left( s\right) } is also an integer.
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