I liked A Beautiful Mind... and if I were voting,I would have given it best adapted screenplay. Because not only did I fall for "the lie" once, I fell for it twice. To me, that's good story telling.
But I would have given Peter Jackson Best Director because he earned it. He did make the best movie of the year. And Lord of the Rings should have been best picture. Well, Hollywood can congratulate themselves... they stayed about "crass commercialism".
I have neither read nor seen "A Beautiful Mind" but I think LotR deserved Best Adapted Screenplay. The screenwriters took Tolkien's work and translated it to the screen without losing anything in translation. It was beautiful.