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Unformatted text preview: types of operation allowed: Type 1: Choose a nonempty box B j with 1 ≤ j ≤ 5 . Remove one coin from B j and add two coins to B j +1 . Type 2: Chooseanonemptybox B k with 1 ≤ k ≤ 4 . Removeonecoinfrom B k andexchange the contents of (possibly empty) boxes B k +1 and B k +2 . Determinewhetherthereisaﬁnitesequenceofsuchoperationsthatresultsinboxes B 1 ,B 2 ,B 3 ,B 4 ,B 5 being empty and box B 6 containing exactly 2010 2010 2010 coins. (Note that a b c = a ( b c ) .) Problem 6. Let a 1 ,a 2 ,a 3 ,... be a sequence of positive real numbers. Suppose that for some positive integer s , we have a n = max { a k + a nk  1 ≤ k ≤ n1 } forall n > s . Provethatthereexistpositiveintegers ` and N ,with ` ≤ s andsuchthat a n = a ` + a n` for all n ≥ N . Language: English Time: 4 hours and 30 minutes Each problem is worth 7 points Language: English Day: 2...
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 Fall '11
 T.S.
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