A_k is the number formed by the first k digits of sequence: 123456789101112 ...
Source: Rioplatense Olympiad 2016 level 3 P6
September 5, 2018
number theorysum of digitsSequenceDivisibility
Problem Statement
When the natural numbers are written one after another in an increasing way, you get an infinite succession of digits Denote the number formed by the first digits of this sequence . Prove that for all positive integer there is a positive integer which simultaneously verifies the following three conditions:
(i) divides ,
(ii) divides ,
(iii) divides the sum of the digits of .