SonofMan, there is an important distinction between things that are physically or biologically impossible and things that are logically impossible.
In the former category we can put such miracles as the virgin birth, raising the dead, parting the sea, or turning water into wine. By impossible we mean that those things do not happen in the natural course of affairs and there is no known mechanism by which they can happen. Nevertheless, they are not logically contradictory. We can, at least, imagine what they would look like.
In the latter category are square circles and the simultaneous existence of Pallas Athene’s perfect bullet and perfect shield (which is the same problem as “unstoppable force meets immovable object” or “rock God cannot lift”). In these cases, the problem is not one of physical difficulty that might be overcome. The problem lies in the definitions of the words themselves. As Pallas Athene points out, we can say these things and that makes them sound like real problems to consider, but if we think about it the words cannot actually refer to anything, even purely conceptually. They define themselves out of existence. A square is a shape with four equal sides that meet at right angles. A circle is a shape made up of all points equidistant from a common center. In Euclidean geometry, at least, nothing can meet both of those definitions simultaneously.
The problem with the other situation (bullet/shield, force/object, God/rock) is that at least one of the things is defined in relational terms. By existing, even in concept, it defines certain other things out of existing. If I have a bullet that, by definition, cannot be stopped by any defense, then in that universe there IS NO perfect defense. If I define the defense first, then there is no perfect bullet. Again, this is mostly a matter of words, though. In real life, one might make a bullet that penetrates all known defenses, only for someone else to later invent a better shield. Your physical bullet hasn’t popped out of existence, but it has been shown not to be the conceptual perfect bullet. Similarly, if God is characterized as a being that can exert potentially infinite force, it is no failing on His part to be unable to create a rock He can’t lift. The concept makes no sense in a universe with such a God in it. God can still create rocks of arbitrarily great mass; it’s just that He can also lift them and therefore they do not qualify as the unliftable rock.
As others have already pointed out, the Trinity does not constitute a logical contradiction because, contrary to your phrasing, we do not say “three people are simultaneously one person.” We say “three divine persons share the one divine nature.” It’s a slippery idea and there are no earthly examples, but it is not self-contradictory.
Usagi