4802Pigeonhole

# 4802Pigeonhole - every subset of 3 attendees spoke a common...

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Math 4802 Fall 2006 Pigeonhole Principle Problems In-class Problems: 1. Suppose that 17 playing cards are chosen at random from a standard deck of 52 cards. Show that these cards must contain a ﬂush. 2. Five darts are thrown onto a square dartboard having side length 1. Show that two of the darts are within distance 2 / 2 of each other. 3. Suppose G is a graph with 2 n vertices and n 2 + 1 edges, n 2. Show that G contains a triangle (three vertices all of which are connected). 4. Suppose S is a subset of { 1 , 2 , . . . , 2 n } with | S | = n +1. Prove that there are two distinct numbers a, b S with a dividing b . 5. Given any set S of ten natural numbers between 1 and 99 inclusive, prove that there are two disjoint non-empty subsets X, Y of S with equal sums of their elements. 6. Prove that given any set S of real numbers with | S | = 7, there exist x, y S with 0 < x - y 1 + xy 1 3 . 1

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Homework Problems (due September 21): 1. Show that every polyhedron has two faces with the same number of edges. 2. Of the 1985 people in attendance at a recent international math con- ference, no one spoke more than 5 languages and at least 2 people in
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Unformatted text preview: every subset of 3 attendees spoke a common language. Prove that some language was spoken by at least 200 of the people at the meeting. 3. Prove that there exist integers a, b, c , not all zero and each of absolute value less than one million, such that | a + b √ 2 + c √ 3 | < 10-11 . 4. Given two relatively prime positive integers a, b and an integer m ≥ ( a-1)( b-1), prove that there exist nonnegative integers x, y such that m = ax + by . 5. A set S of integers is called “sum-free” if there do not exist x, y, z ∈ S with x + y = z . What is the maximum size of a sum-free subset of { 1 , 2 , 3 , . . . , 2 n + 1 } ? 6. Let B be a set of more than 2 n +1 n distinct points with coordinates of the form ( ± 1 , ± 1 , . . . , ± 1) in R n , with n ≥ 3. Show that there are three distinct points in B which are the vertices of an equilateral triangle. 2...
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4802Pigeonhole - every subset of 3 attendees spoke a common...

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