ma017-2010-hw2 - m chips between them You may take one chip...

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MA017 2010 HOMEWORK 2 Due 2010-10-12 Tue. (1) Write n ones on the board, and repeat the process “erase any two numbers a,b and write a + b 4 ” until one number x remains. Show that x 1 /n . (2) Given finitely many points in the plane, any three of which form a triangle of area at most one, show that all of them lie inside some triangle of area four, including its boundary. (3) Starting with the number 7 2008 , cross out the first digit and add it to the remaining number (e.g., 325 7→ 28). You repeat this until a ten-digit number x remains. Show that x has two equal digits. (4) Find the largest possible number of elements in a finite set S of real numbers having the property: given any three elements in S , there are two whose sum is in S . (5) (a) Start with n piles holding
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Unformatted text preview: m chips between them. You may take one chip from each of two piles and place them in a third pile. Is is always possible to transfer all chips to the same pile? (You may assume n,m ≥ 3.) (b) Start with three piles of chips, distributed in some manner. Your score s is initially zero. You may transfer one chip from any pile with x chips onto any other pile with y chips, and your score changes by s 7→ s + y-x + 1. The game ends when the chips return to their original distribution; what is the maximum possible score attainable at this point? (6) Show that for any six disjoint circles of radius one in the plane, the distance between the centers of some pair of them is at least 2 √ 3. 1...
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