I can only say that my idea was completely different and I can not understand whether your argument eventually hits the aim.Ideas for problem #5:
Make the problem simpler: take B = identity map, so b =1, and take the constant a = 1, so we want to prove the map f taking x to x^2 + x, hits every point C with |C| ≤ 1/4, where |x^2| ≤ |x|^2
Perhaps it was not a good idea to post this problem for the Math Challenge. All my previous problems were easily solved and I decided to post something more interesting. And I posted an example from my article. I feel this was too much.