2015 IMO Shortlist C4

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \leq n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. (Finland)

令 $n$ 为正整数。两个玩家 $A$ 和 $B$ 玩一个游戏，他们轮流选择正整数 $k \leq n$。游戏规则如下： (i) 玩家不能选择任一玩家在上一回合中选择过的号码。 (ii) 玩家不能选择与该玩家在任何先前回合中已选择的数字连续的数字。 (iii) 如果所有号码都被选中，则比赛为平局；否则不能再选择数字的玩家就输掉了游戏。玩家$A$ 进行第一回合。假设双方都使用最优策略，确定游戏的结果。 (芬兰)