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Ali Baba and the Forty Thieves

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In the Parallel Universes, Alibaba is the leader of Forty Thieves. One day, they thieves 100 golden coins. Forty Thieves want be divided 80 golden coins, but the greedy Alibaba just want to own all the treasure. So he said to the thieves that: I will divide the treasure to 2 groups. Then, I chosse a random groups and divide to 2 other groups. Untill there are 100 groups. At any time of this process, If you can find 40 groups which having the sum is 80 coins, you can take this. But if you cannot, you get nothing.

The thieves definitely can take 40 groups like that after a-th divide times whatever how Alibaba divide. What is the smallest of the value of a?

posted May 14 by anonymous

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