I concur with matthewfrey, it's almost always the caps in these models. It's the work of two minutes to open it up and look.
There are shops that inexpensively fix this problem (the one I use charges $240 to go through the whole iMac, fix all the problems (including power supply replacement), and ship it back, with 12 months warranty.) Otherwise you're looking at a new logic board.
The caps inside the power supply can croak too. Sometimes you can see if you shine a flashlight through the holes in the PS.