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Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number k such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40$. The last player who can make a legal move wins.

(a) Does the game necessarily end?

(b) Does there exist a winning strategy for the starting player?

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    @Christian: it is traditional to include the year in at least one of the IMO questions2011-08-31

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There is a solution on page 780 of Dusan Djukic, Vladimir Jankovic, Ivan Matic, Nikola Petrovic, The IMO Compendium; A Collection of Problems Suggested for The International Mathematical Olympiads, 1959-2009, which I found on Google Books.

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It would be worth telling us what you have tried.

Some hints:

  • Are there are a finite number of possible positions (and if so what is an upper bound)?

  • Can there be a cycle of positions (consider the card with the smallest number turned over)?

  • Will the starting player have a winning strategy if initially there are only 41 cards? 42? etc?

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    @joriki: Thanks, corrected.2011-08-29