IMO ShortList 2009_Combinatorics

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rakeen
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IMO ShortList 2009_Combinatorics

Unread post by rakeen » Sun May 22, 2011 10:49 am

since everyone is posting problems in this sub-forum why should I get back! Here's a problem I looked at yesterday and found interesting.
Though I haven't found nothing about it. This problem was proposed by Newzeeland. Hopefully you guys present many solutions in a simple
way+understandable way. :)

Consider 2009 cards, each having one gold side and one black side, lying in parallel on a long
table. Initially all cards show their gold sides. Two players, standing by the same long side of
the table, play a game with alternating moves. Each move consists of choosing a block of 50
consecutive cards, the leftmost of which is showing gold, and turning them all over, so those
which showed gold now show black and vice versa. 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? ۝ ۞ ۩
r@k€€/|/

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