Prediction is impossible with 100% accuracy, so the problem statement is nonsense. Logic will not help you here.
In a real game, take both boxes since nothing will change. Omega is a charlatan.
To explore the impossible even further, use a quantum random number server, like the one at http://qrng.anu.edu.au/, to make your choice. If several people do this, and Omega is always able to make the prediction, then it means our understanding of quantum mechanics is deeply wrong.
As idle curiosity, I wonder how Omega got the money.
Omega sounds like an accomplished con artist. I'm certain the money is dirty.