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Omni-MATH / 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 ≥9, then we can subtract 9 from it, and add 1 t…

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problem

Let SS be the set of 1010-tuples of non-negative integers that have sum 20192019. For any tuple in SS, if one of the numbers in the tuple is ≥9\geq 9, then we can subtract 99 from it, and add 11 to the remaining numbers in the tuple. Call thus one operation. If for A,B∈SA,B\in S we can get from AA to BB in finitely many operations, then denote A→BA\rightarrow B. (1) Find the smallest integer kk, such that if the minimum number in A,B∈SA,B\in S respectively are both ≥k\geq k, then A→BA\rightarrow B implies B→AB\rightarrow A. (2) For the kk obtained in (1), how many tuples can we pick from SS, such that any two of these tuples A,BA,B that are distinct, $A\not\rightarrow B$.
Plain-text mathematical notation (without MathML)
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 ≥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∈S we can get from A to B in finitely many operations, then denote A→B.

(1) Find the smallest integer k, such that if the minimum number in A,B∈S respectively are both ≥k, then A→B implies B→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, $A\not\rightarrow B$.
Original LaTeX notation
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, $A\not\rightarrow B$.

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Source and history

task_id
f3136877-2cc1-553e-9412-a7ddb869f2dd
task_key
test--f3136877-2cc1-553e-9412-a7ddb869f2dd
task_revision_id
1
config / split
/ test

Official source

initial import