Fix for the closing apps problem on unjailbroken iPhones
I have no idea if this is already known, but there is a quick fix for the closing apps problem if you are not near your computer and cannot reactivate your iTunes account and sync.
Downloaded apps close seconds after starting, bulit-in ones run just fine.
1.) Open Appstore
2.) Download and install any free app that you do not yet have installed. 3.)This will prompt you for your iTunes username and password. Enter them.
4.) Run the newly installed app
5.) Exit back to the home screen and problem solved. You should now be able to run the other downloaded apps as normal.
If that does not work, try resetting the settings first and then start at 1.)
hope that helps and wasn't too redundant.