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45 people (a1, a2…….a45) meet and shake hands in a circular fashion. In other words there totally 45 handshakes involving the pairs (a1, a2), (a2, a3)……(a44, a45), (a45, a1). Then the size of the smallest set of people such that the rest have shaken hands with atleast one person in the set is.

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Middle earth is fictional land inhabited by Hobbits, Elves, Dwarves and men. The Hobbits and the elves are peaceful creatures who prefer slow, silent lives and appreciate nature and art. The dwarves and men engage in physical games. The game is as follows a tournoi is one where out of the two teams that play a match, the one that loses gets eliminated. The matches are played in different round, where every round half of the team get eliminated from tournament. If the there are 7 rounds played, in knock out turnoi, how many matches were played? Metal strip of width „x‟ cm. 2 metal strips are placed one over the other, then the combine length of 2 strips is „y‟. If „z‟ strips are placed in that manner. What is the final width of that arrangement?