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If it's a common problem, maybe it has a common solution as well? You could try resetting the PRAM, but I imagine it's a hardware/design issue, in which case the appropriate course of action would be to take it in to the shop. However, if no solution has come up in your research, I don't know that there's a lot we can offer.
 
is this one of the very first macbooks? Mine had a problem with it getting too hot, the heatsink expands and touches the motherboard and shorts
 
make sure you've connected your power cable, if your macbooks a year+ old it could have a battery with low health (they deplete over time, like all batterys)

does it work with the magsafe connected?
 
Happened to me, the hard drive was corrupted. Try verbose mode or put in your OSX DVD and attempt to view/repair the partition.
 
Hey, I have the same problem keeps turning itself on then goes off at the startup screen (apple logo) and on and off again repeatedly until the battery goes dead! Any solutions to this as its prob not worth getting fixed in a shop as its old so I will try repair myself! Thanks for any info!
 
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