First Round, January 2019

Starting with a positive integer, a fragment of that number is any positive number obtained by removing one or more digits from the beginning and/or end of that number. For example: the numbers $2$, $1$, $9$, $20$, $19$, and $201$ are the fragments of $2019$. What is the smallest positive integer $n$ such that the following holds: there is a fragment of $n$ such that when you add this fragment to $n$ itself, you get $2019$?