# quiz3 - and X a ∈ A a = X b ∈ B b If | X | = 12 instead...

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Putnam Exam Seminar Quiz 3 Fall 2010 September 22 Problem 1. Given any thirteen real numbers, prove that there are two of them, say x and y , so that 0 < x - y 1 + xy < s 2 - 3 2 + 3 . Problem 2. Prove that if any ﬁve points are chosen on a sphere, then four of them lie on some closed hemisphere. Problem 3. Let M = { 1 , 2 , 3 ,..., 2048 } and let X M such that | X | = 15. Show that there are two disjoint subsets of X whose sum of elements is the same. That is, show that we can ﬁnd A,B X with A B =
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Unformatted text preview: and X a ∈ A a = X b ∈ B b . If | X | = 12 instead is this result still true? Problem 4. Given a set of n + 1 positive integers, none of which is greater than 2 n , prove that at least one member of this set must divide another member of this set. Problem 5. Given a set of n + 1 positive integers, none of which is greater than 2 n , prove that at least two of the integers are relatively prime....
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