IL-GILLIS 2011 Problem 5

We have two lists of numbers: One initially containing 1,6,11,...,96, and the other initially containing 4,9,14,...,99. In every turn, we erase two numbers from one of the lists, and write $\frac{1}{3}$ of their sum (not necessarily an integer) in the other list. We continue this process until there are no possible moves. [list=a] [] Prove that at the end of the process, there is exactly one number in each list. [] Prove that those two numbers are not equal. [/list]