2012 IMO Shortlist C4

Players $A$ and $B$ play a game with $N \geq 2012$ coins and 2012 boxes arranged around a circle. Initially $A$ distributes the coins among the boxes so that there is at least 1 coin in each box. Then the two of them make moves in the order $B, A, B, A, \ldots$ by the following rules: - On every move of his $B$ passes 1 coin from every box to an adjacent box. - On every move of hers $A$ chooses several coins that were not involved in $B$ 's previous move and are in different boxes. She passes every chosen coin to an adjacent box. Player $A$ 's goal is to ensure at least 1 coin in each box after every move of hers, regardless of how $B$ plays and how many moves are made. Find the least $N$ that enables her to succeed.

玩家 $A$ 和 $B$ 用 $N \geq 2012$ 硬币和围绕圆圈排列的 2012 年盒子玩游戏。最初，$A$ 将硬币分配到各个盒子中，以便每个盒子中至少有 1 个硬币。然后他们两人按照以下规则按照 $B, A, B, A, \ldots$ 的顺序进行移动： - 他的 $B$ 每次移动都会将 1 个硬币从每个盒子传递到相邻的盒子。 - 在 $A$ 的每次移动中，都会选择 $B$ 之前的移动中未涉及的几个硬币，并且这些硬币位于不同的盒子中。她将每一枚选定的硬币传递到相邻的盒子中。玩家 $A$ 的目标是确保每次移动后每个盒子中至少有 1 个硬币，无论 $B$ 如何玩以及移动了多少次。找到能让她成功的最少$N$。