In debates, Christopher Hitchens often offers the following challenge: "Name one moral action performed by a believer that could not have been done by a nonbeliever." Leaving aside the fact that this challenge is based on a misunderstanding of the theist objection (he thinks the objection is that without God, we couldn't know right from wrong, when the actual objection is that there wouldn't be any right or wrong), the challenge itself is completely invalid when proposed by an atheist because, logically, it could never be answered to the satisfaction of an atheist, even if valid answers exist.
Why is this the case? Think about what type of moral action Hitchens is asking for. Why might an atheist be unable to do something moral? The answer can't be a physical inability. Nobody is stopping atheists from doing moral things. So why might an atheist not be able to do something moral that a theist could do? The only possibility is that there might be certain acts that only theists would recognize as being moral. Atheists, not recognizing those acts as being good, would not attempt to do them as moral acts.
And now we come to the logical problem. By the definition of the answer required, we must come up with something that is moral, but that atheists do not recognize as being moral. But if atheists do not recognize it as moral, Hitchens won't recognize it as moral. Therefore, any true example (if it exists) of a moral good that meets the criterion of not being recognized by atheists will, by definition, necessarily be rejected by Hitchens, and any answer that fails to meet the criterion will be easily refuted by him. It is unanswerable.
Since it is logically impossible to give an answer that will satisfy Hitchens, he may as well ask us to draw him a square circle and then declare himself the winner when we fail. In the end, his challenge is nothing but a rhetorical trick, and it should be exposed and dismissed as such. Hitchens should never get away with even asking it, let alone demanding we give him an "acceptable" answer in order to defend theism.
As it happens, there is an answer to Hitchens's question--one that seemed obvious to me immediately--and it illustrates perfectly the problem with the challenge. The highest moral good a person can do is to worship the living, true, sovereign God--to love Him with all one's heart, soul, mind, and strength. Not only will no atheist ever do this, no atheist can do this. But of course, since they do not recognize worship as a real, valid moral good, no atheist would accept this response to Hitchens's challenge. They necessarily reject it precisely because it correctly answers the challenge; because it succeeds, it fails. Any correct answer that exists will necessarily fail. Only an invalid question could lead to a paradox like this.
(UPDATE: To avoid confusion, I want to clarify that the defeating answer to the challenge is not worship. The answer is this: The question cannot logically be answered and is therefore invalid. Although I do think worship is one moral thing an atheist can't do, in this post worship is merely an illustration to show why the question is invalid.)