ClassW04ApproxAlgs/ProofsGroupsExercises4

1. Elizabeth has a secret hiding place where she presently has 35 coins, 3 baseball cards, and 39 pieces of candy. Her younger brother John has his own secret hiding place, but that does not prevent him from taking things from Elizabeth's (her place isn't as secret as she thinks). John always takes two items at a time, each of a different kind so that the number won't decrease too quickly, and he always adds one item of the third kind. On day, some time later, Elizabeth is shocked to find that all the items in her hiding place were of the same kind. Was it all candy?

2. The vertices of a cycle with n vertices are labelled with integers. A "flip" operation takes three consecutive vertices with labels (x,y,z) where y<0 and replaces the labels, respectively, with (x+y, -y, and z+y). Prove or disprove: it is possible to label the vertices so that the sum of the labels is positive, and then to do an infinite sequence of flip operations.

Example: (2,-1,0) -> (1,1,-1) -> (0,0,1) (and then you are stuck)

3. The positive integers are to be partioned into several subsets A₁,A₂,…,A_n such that, for i=1,2,...,n, if x∈ A_i then 2x¬∈ A_i. What is the minimum value of n?

4. Prove that any three real numbers x,y,z satisfy the inequality |x|+|y|+|z|-|x+y|-|y+x|-|z+x|+|x+y+z| ≥ 0.