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,…,An such that, for i=1,2,...,n, if x∈ Ai then 2x¬∈ Ai. 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.