Herrer M.L. Let $S(n)$ denote the sum of the digits of the number $n$ (in decimal notation). Do there exist three distinct natural numbers $m, n$, and $p$ such that $m+S(m)=n+S(n)=p+S(p)$?
Here is one of such triples:

$m=9999999999999, S(m)=117 ; \quad n=10000000000098, S(n)=18 ; \quad p=10000000000107, S(p)=9$.

For these numbers $m+S(m)=n+S(n)=p+S(p)=10000000000116$.

## Answer

They exist.