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for every n, exists a number that appears n times in sequence f(m)=m+S(m)

Source: Rioplatense Olympiad 2001 level 3 P6

September 6, 2018
sum of digitsnumber theorySequence

Problem Statement

For m=1,2,3,...m = 1, 2, 3, ... denote S(m)S(m) the sum of the digits of mm, and let f(m)=m+S(m)f(m)=m+S(m). Show that for each positive integer nn, there exists a number that appears exactly nn times in the sequence f(1),f(2),...,f(m),...f(1),f(2),...,f(m),...
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