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Useful Tools in Lecture 3: The Extreme Principle

This lecture focuses on the application of the Extreme Principle in various mathematical problems, such as age averages, tangency of coins, paper removal, polynomial conditions, and triangle areas.

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Useful Tools in Lecture 3: The Extreme Principle

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  1. Lecture 3 (More Useful Tools) The Extreme Principle

  2. The Extreme Principle Consider a finite linearly ordered set (S, ). Then there exist a smallest element xS and a largest element yS, i.e. x  S  y. Notes: You may first need to define the order. A smallest element exists if S is an infinite but well ordered set, e.g. the natural numbers.

  3. Problem 1 Eight people sit around a lunch table. As it happens, each person’s age is the average of the two persons’ ages on his/her right and left. Show that all their ages are equal.

  4. Hint: Who’s the youngest?

  5. Problem 2 You are given a finite set of coins in the plane, all with different diameters. Show that one of the coins is tangent to at most 5 of the others.

  6. Hint: Consider the smallest circle.

  7. Problem 3 Fifteen sheets of paper of various sizes and shapes lie on a desktop covering it completely. The sheets may overlap and may even hang over the edge. Show that five of the sheets may be removed so that the remaining ten sheets cover at least 2/3 of the desktop.

  8. Hint: Which piece of paper would you remove first?

  9. Problem 4 Let p(x) be a real polynomial such that for all x, p(x) + p(x)  0. Does it follow that for all x, p(x)  0? [Example: p(x) = x2 + 1 satisfies both the condition and conclusion.]

  10. Hint: What is the degree of p(x) + p(x)? What is the degree of p(x)? Does p(x) achieve a minimum somewhere?

  11. Problem 5 Consider finitely many points in the plane such that, if we choose any three points A,B,C among them, the area of triangle ABC is always less than 1. Show that all these points lie within the interior or on the boundary of a triangle of area less than 4.

  12. Hint: Which triangle should you focus on?

  13. Thank You for Coming Wafik Lotfallah

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