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 $(x+y, -y, z+y)$.
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: $(2,-1,0) \rightarrow (1,1,-1) \rightarrow (0,0,1)$ (and then you are stuck)