I'm suspecting either the logic board or the power supply.

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I just had the power supply replaced a couple of months ago for mine. It did not have the magic serial number but I had Applecare. It seems that it is not unusual. Don't you have applecare for it? So far I have had it pay for itself every time.

Or else you could order the part, I suppose...

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That was my mistake. I didn't get it with AppleCare mainly because at the time I just couldn't afford it on top of the cost of the iMac. I know that the power supply is about $200 so that's doable in time, but it's just not worth paying the price for the logic board. For that, I might as well just get myself a Core 2 Duo Mac mini and run with that until I can afford something else. :-/