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# DISCRETE MATH I MATH 574

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This 1 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 574 at University of South Carolina - Columbia taught by Ma Filaseta in Fall. Since its upload, it has received 20 views. For similar materials see /class/229527/math-574-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

MATH 574 NOTES 4 THE PIGEONHOLE PRINCIPLE Examples 1 A person has 20 socks in a drawer 10 of which are black and 10 of which are brown How many socks must he pull out to assure he has two socks the same color How many socks must he pull out to assure he has two socks the opposite color 2 Given ve lattice points in the plane show that the line segment containing some two of them has midpoint a lattice point Note that the same is not true if only four lattice points are given 3 Show that in a group of two or more people who shake hands no one shakes hands with his or herself and no one shakes hands with the same person twice some two of them shook the same number of hands 4 Suppose that the various line segments joining pairs of 6 noncollinear points in the plane how many are there are each colored either red or blue Show that there must be a triangle formed with all of its edges the same color 5 Given any n 1 integers from the set 1 2 271 show that there are at least two of them say a and b such that a divides b 6 Show that for every real number a there are integers a and b with b 7 0 such that 1 1 lt 1 a b b2 Discuss good rational approximations in general 7 Given a sequence of n2 1 distinct real numbers there exists a subsequence of length n 1 which is either increasing or decreasing Example The sequence 7 5 6 1 10 2 9 3 8 of 8 num bers does not have such a subsequence of length 4 but if 4 or 0 or 11 is added to the end it does

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