ClassW04ApproxAlgs/ProofsGroupsExercises3

1. * There are 10 cities in Fatland. Two airlines control all of the flights between the cities. Each pair of cities is connected by exactly one one flight (the flight connects the cities in both directions). Prove that one airline can provide two traveling cycles with each cycle passing through an odd number of cities and with no common cities shared by the two cycles.

2. Given an initial sequence a₁, a₂, …, a_n of real numbers, we perform a series of steps. At each step, we replace the current sequence x₁,x₂,…,x_n with the sequence |x₁-a|,|x₂-a|,…,|x_n-a| for some number a that can vary with each step.

(a) Prove that no matter what sequence we start with, there is some way to do steps to reach the sequence 0,0,…,0.

(b) * Determine (with proof) the maximum, over all sequences of length n, of the minimum number of steps required to reach 0,0,…,0.

3. In a set of objects, each is either red or blue, and each is either round or square. There is at least one red object, at least one blue object, at least one round object, and at least one square object. Prove that there exist two objects that are different both in color and in shape.

4. A real number is assigned to each vertex of a finite connected graph so that the number on any vertex is the arithmetic mean (average) of the numbers on the adjacent vertices. Prove that all vertices' numbers are equal.

5. We are given an infinite set of rectangles in the plane, where each rectangle has vertices of the form (0,0), (0,n), (0,m) (n,m) for positive integers m and n (m and n vary from rectangle to rectangle).

(a) Prove that there exist two rectangles in the set such that one contains the other.

(b) * Prove or disprove: there must exist an infinite sequence R₁,R₂,…,R_n of rectangles in the set such that R₁ contains R₂, R₂ contains R₃, and so on. (R_i contains R_i+1 for each i).