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For everyone having black bars on 1080p output, theres a checkbox for overscan, uncheck it and it should fix your issues. My 2012 13" MBA now mirrors perfect 1080p to my 37" TV.
 
It's not 1080p :(

You're wrong. Follow those instructions I posted, along with selecting "Best for Airplay," and as long as your TV is 1080p, your AppleTV is a version 3 with 1080p support, and you did as I typed, you're sending 1080p to the TV.
 

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