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

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
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 =
Background image of page 1
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

Ask a homework question - tutors are online