RIOPLATENSE 2001 Problem 6

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