2012 IMO Problems/Problem 3

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The liar’s guessing game is a game played between two players A and B. The rules of the game depend on two positive integers k and n which are known to both players.

At the start of the game the player A chooses integers x and N with 1≤x≤N. Player A keeps x secret, and truthfully tells N to the player B. The player B now tries to obtain information about x by asking player A questions as follows: each question consists of B specifying an arbitrary set S of positive integers (possibly one specified in some previous question), and asking A whether x belongs to S. Player B may ask as many questions as he wishes. After each question, player A must immediately answer it with yes or no, but is allowed to lie as many times as she wants; the only restriction is that, among any k+1 consecutive answers, at least one answer must be truthful.

After B has asked as many questions as he wants, he must specify a set X of at most n positive integers. If x∈X, then B wins; otherwise, he loses. Prove that:

(a) If n≥2k then B has a winning strategy.

(b) There exists a positive integer k0 such that for every k≥k0 there exists an integer n≥1.99^k for which B cannot guarantee a victory.