Welcome to Justanswer.com, my name is Dustin.I'm sorry to hear about the issue you are having, I will do my best to help you resolve it.
The geek squad at best buy is notorious for over charging, and for mis diagnosing issues. Im sorry to hear about the experience you had with them.
When you press the power button to turn on the machine. How many times does the power button blink on and off.. Can you count the blinks. ( and try a few times, is it always the same )
This may have been an adapter issue all along then. H ave you tried a new power adapter? http://amzn.to/2bPoTMn
Also, when they replaced your motherboard did they check or replace your power supply? http://www.cdsparts.com/p-3559-new-gateway-one-zx6800-zx6810-zx6900-zx6951-computer-power-supply-250-watt.aspx
As you see, im focusing on the power supply.. But it could also be the Ram. Because when best buy replaced the motherboard they would have kept the power adapter, power supply and ram from the previous machine. And one of those three things is causing this issue. Its doubtful the old motherboard needed to be replaced. ( did they keep the motherboard or give it back ? )
Dustin S.Senior Technician and Network Switching Systems Operator / Maintainer.
I seen you started a duplicate, and that duplicate has been closed so I cant see what you posted.
im willing to co ntinue working with you on this over the long haul. The last thing I want is for you to get bounced back and forth between tech support especially considering the ordeal you had with the Geek Squad at Best Buy.
So if your willing to keep in touch and communicate on this page, Im happy to work through this issue with you.
I strive to provide great service, and greatly appreciate a 5 star rating clicked and submitted when you are satisfied.. I always answer follow up questions.Dustin S.Senior Technician and Network Switching Systems Operator / Maintainer.