IMac Won't Start Unless Unplugged For Long Period

I discovered my iMac shut down and to start. I feared it had been fried in an electrical storm the day before, though the whole house is surge protected, and a printer on the same plug was unaffected. I tried unplugging briefly, still no luck. I tried hitting the start button ( on the iMac itself, not the keyboard) a few times over a couple of days with no luck, and unplugged it in prepartion to cart off to a repair shop, fearing the worst. A couple of days later, I plugged it in for one last try before taking it to the car and, bingo, it started! Everything was fine, everything in its place. 

A couple of weeks later, I once again notice the computer is turned off and will not restart. I uplug and replug - no luck. I unplug, let it sit, and try it again about 24 hours later.

Info:
iMac, Mac OS X (10.6.8)