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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