hw06 - ( a +1) mice of which one is descended from the...

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In the Name of God Fall 2005 Discrete Mathematics Homework 6 Due: Aban 29 Problem 1. Inside a room of area 5, you place 9 rugs, each of area 1 and an arbitrary shape. Prove that there are two rugs which overlap by at least 1 9 . Problem 2. Twenty pairwise distinct positive integers are all less than 70. Prove that among their pairwise diFerences, there are four equal numbers. Problem 3. S is a set of n positive integers. None of the elements of S is divisible by n . Prove that there exists a subset of S such that the sum of its elements is divisible by n . Problem 4. Show that among ( ab + 1) mice, there is either a sequence of
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Unformatted text preview: ( a +1) mice of which one is descended from the preceding, or there are ( b +1) mice of which non descends from the other. Problem 5. Thirty-three rooks are placed on an 8 × 8 chessboard. Prove that you can choose ±ve of them which are not attacking each other. Problem 6. Let a , a 1 , · · · , a 100 and b , b 1 , · · · , b 100 be two permutations of the integers from 1 to 100. Prove that, among the products a 1 b 1 , · · · , a 100 b 100 , there are two with the same remainder upon division by 100. 1...
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