2013 IMO Shortlist C8

Players $A$ and $B$ play a paintful game on the real line. Player $A$ has a pot of paint with four units of black ink. A quantity $p$ of this ink suffices to blacken a (closed) real interval of length $p$. In every round, player $A$ picks some positive integer $m$ and provides $1 / 2^{m}$ units of ink from the pot. Player $B$ then picks an integer $k$ and blackens the interval from $k / 2^{m}$ to $(k+1) / 2^{m}$ (some parts of this interval may have been blackened before). The goal of player $A$ is to reach a situation where the pot is empty and the interval $[0,1]$ is not completely blackened. Decide whether there exists a strategy for player $A$ to win in a finite number of moves. (Austria)

玩家$A$和$B$在实线上玩一场彩绘游戏。玩家$A$有一罐含有四个单位黑色墨水的油漆。这种墨水的数量 $p$ 足以使长度为 $p$ 的(闭合)实区间变黑。在每一轮中，玩家 $A$ 选择一些正整数 $m$ 并从罐中提供 $1 / 2^{m}$ 单位的墨水。然后玩家$B$选择一个整数$k$并将区间从$k / 2^{m}$变黑到$(k+1) / 2^{m}$(这个区间的某些部分可能之前已经被变黑了)。玩家$A$的目标是达到锅为空并且区间$[0,1]$没有完全变黑的情况。决定是否存在让玩家 $A$ 在有限步数内获胜的策略。 (奥地利)