Uh... nobody has mentioned the good ole Ka-Bar? It's been carried by the military for decades. I got mine for $50, looks like Amazon has them for a little more right now.
If you want a smaller knife, check out the Ka-Bar Becker Necker. Wrap it with paracord and you're good to go. - $40
On the "stays sharp" point - realize that if it holds an edge well, it'll also be harder to sharpen. Most quality knives in your budget would be just fine though. Same point for serrated knives, you can't sharpen them in the woods.
If you want a smaller knife, check out the Ka-Bar Becker Necker. Wrap it with paracord and you're good to go. - $40
On the "stays sharp" point - realize that if it holds an edge well, it'll also be harder to sharpen. Most quality knives in your budget would be just fine though. Same point for serrated knives, you can't sharpen them in the woods.