well, for 1 you need something like:
there exists a winner, such that for all y, if y beats x then x is the same person as y.
and for the second part... I dunno. I mean the consequent of the implication: "then there is no winner" is going to be the negation of part 1, I believe. So, you need to find a way to describe the antecedent, "If every competitor beats exactly one other competitor", and I think you're going to need the "x is the same person as y" predicate to describe the "exactly one" part.
Hmmm... what about:
"the negation of the above." ? Does that work?