# 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.

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