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Unusual fractional part

Source: Romania JBMO TST 2024 Day 2 P2

July 31, 2024
number theoryfractional part

Problem Statement

For any positive integer nn{} define an={n/s(n)}a_n=\{n/s(n)\} where s(⋅)s(\cdot) denotes the sum of the digits and {⋅}\{\cdot\} denotes the fractional part. [*]Prove that there exist infinitely many positive integers nn such that an=1/2.a_n=1/2. [*]Determine the smallest positive integer nn such that an=1/6.a_n=1/6.
Marius Burtea
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