2020 IMO Shortlist C8

Players $A$ and $B$ play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player $A$ erases two numbers $x$ and $y$ from the blackboard, and then player $B$ writes one of the numbers $x+y$ and $|x-y|$ on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds: (1) one of the numbers on the blackboard is larger than the sum of all other numbers; (2) there are only zeros on the blackboard. Player $B$ must then give as many cookies to player $A$ as there are numbers on the blackboard. Player $A$ wants to get as many cookies as possible, whereas player $B$ wants to give as few as possible. Determine the number of cookies that $A$ receives if both players play optimally.

玩家 $A$ 和 $B$ 在黑板上玩游戏，黑板上最初包含 2020 个数字 1。在每一轮中，玩家 $A$ 从黑板上擦掉两个数字 $x$ 和 $y$，然后玩家 $B$ 在黑板上写下数字 $x+y$ 和 $|x-y|$ 之一。在某一轮结束时，如果出现以下情况之一，则游戏终止： (1) 黑板上的一个数字大于所有其他数字的总和； (2) 黑板上只有零。然后，玩家 $B$ 必须向玩家 $A$ 提供与黑板上的数字一样多的饼干。玩家$A$想要获得尽可能多的饼干，而玩家$B$想要给予尽可能少的饼干。确定如果两个玩家都发挥最佳状态，$A$ 收到的 cookie 数量。