2010 CMO 第 6 题

There is a deck of cards placed at every points $A_1, A_2, \ldots , A_n$ and $O$ , where $n \geq 3$ . We can do one of the following two operations at each step: $1)$ If there are more than 2 cards at some points $A_i$ , we can withdraw three cards from that deck and place one each at $A_{i-1}, A_{i+1}$ and $O$ . (Here $A_0=A_n$ and $A_{n+1}=A_1$ ); $2)$ If there are more than or equal to $n$ cards at point $O$ , we can withdraw $n$ cards from that deck and place one each at $A_1, A_2, \ldots , A_n$ .

Show that if the total number of cards is more than or equal to $n^2+3n+1$ , we can make the number of cards at every points more than or equal to $n+1$ after finitely many steps.

在每个点 $A_1、A_2、\ldots、A_n$ 和 $O$ 处放置一副纸牌，其中 $n \geq 3$ 。我们可以在每一步执行以下两个操作之一： $1)$ 如果在某些点 $A_i$ 有多于 2 张牌，我们可以从该牌组中取出三张牌，并在 $A_{i-1}、A_{i+1}$ 和 $O$ 各放置一张。 (这里 $A_0=A_n$ 和 $A_{n+1}=A_1$ )； $2)$ 如果在 $O$ 点有超过或等于 $n$ 张牌，我们可以从该牌组中取出 $n$ 张牌，并将一张牌放在 $A_1、A_2、\ldots、A_n$ 处。

证明如果牌总数大于或等于 $n^2+3n+1$ ，经过有限步后，我们可以使每个点的牌数大于或等于 $n+1$ 。