I'll give you a hint. Contradictions do not exist. Whenever you think that you are facing a contradiction, check your premises. You will find that one of them is wrong.Taken literally, i.e. in the standard logical use, this is horribly wrong, even if the intended meaning is not. A ‘premiss’, in traditional logic, is one of a set of propositions that are used to support the conclusion of an argument. Being ‘wrong’ is not a term of traditional logic, but ‘false’ is clearly what is meant. It is certainly true that if two or more propositions involve or imply a contradiction, then at least one of them is false. But that does not imply, as Rand imagines, that contradictions do not exist. For a contradiction is simply defined as two premisses (or propositions) that contradict each other. If the premisses exist, so does the contradiction, just as a left and a right shoe make a pair.
What Rand probably meant was that contradictory premisses cannot be true. That is perfectly correct, and is the Principle of Contradiction itself. So she probably meant something quite simple and obvious, indeed a law of logic. Why she chose to put it that way is more difficult. Perhaps by ‘facing a contradiction’, she meant facing the state of affairs that is the truthmaker for a contradiction. And of course there could be no such truthmaker, even if there were truthmakers for other propositions (which I deny, of course).