IMO Problem 4

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 the beginning of the game, player A chooses (and announces) a positive integer $N$ . In every round, player A picks some positive integer $m \leq N$ and provides $\frac {1}{2^m}$ units of ink from the pot. Player B then picks an integer $k$ and blackens the interval from $\frac {k}{2^m}$ to $\frac {k+1}{2^m}$ (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 $[0, 1]$ s is not completely blackened. Decide whether there exists a strategy for player A to win in a finite number of moves.

This problem is from the IMO 2013 SLThis problem is from the IMO.This problem is part of this set .