We could not activate Battlefield 3 PC

Try repairing BF3 in Origin by right clicking Origin and selecting repair. That should replace any missing or corrupted files that may be causing your error. If that doesn't work I recommend speaking to a game advisor to see if there's anything happening on the back end with your account entitlements.

You can reach a game advisor here: https://help.ea.com/contact-us click on the orange button on the right to start a new case, follow the instructions and when you are presented with a contact method I recommend that you use the telephone option and the “call me” option so you don't have to stay on hold for a long time. The call back feature available to North America, Austria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Hungary, Ireland, Italy, Netherlands, Norway, Poland, Portugal, Russia, Spain, Sweden, Slovakia, Switzerland, and the UK.