Maraichev A. $^{2}$

Two players play the following game. There is a pile of stones. The first player, on each of his turns, takes 1 or 10 stones. The second player, on each of his turns, takes $m$ or $n$ stones. They take turns, with the first player starting. The player who cannot make a move loses. It is known that for any initial number of stones, the first player can always play in such a way as to win (regardless of how the second player plays). What can $m$ and $n$ be?
Let at least one of the numbers $m$ and $n$ be less than 9 (for example, $m$). With an initial pile of $m+1$ stones, the first player will be forced to take 1 stone on the first move ($m+1-1=0$). The first player can leave either $k-1$ stones or $k-10$. If in one case it resulted in $m$, and in the other $-n$, the number $|m-n|$ would equal 9. Contradiction. The number of stones in the pile will decrease, and in the end, the first player will win.

## Answer

$m, n \geq 9$ and $|m-n| \leq 9$.

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