CN-TST 2019 Problem 8

Let $S$ be the set of $10$ -tuples of non-negative integers that have sum $2019$ . For any tuple in $S$ , if one of the numbers in the tuple is $\geq 9$ , then we can subtract $9$ from it, and add $1$ to the remaining numbers in the tuple. Call thus one operation. If for $A,B\in S$ we can get from $A$ to $B$ in finitely many operations, then denote $A\rightarrow B$ .

(1) Find the smallest integer $k$ , such that if the minimum number in $A,B\in S$ respectively are both $\geq k$ , then $A\rightarrow B$ implies $B\rightarrow A$ .

(2) For the $k$ obtained in (1), how many tuples can we pick from $S$ , such that any two of these tuples $A,B$ that are distinct, $A\not\rightarrow B$ .