Intel 20" Imac has random shut-down problems- need help!

Hello, I am a new member. I have a 20" intel core 2 duo iMac, and when I got it, there seemed to be no problems. Over the last year, though, it started to randomly shut off and not turn back on unless I pull the power cord and let it sit for at least 15 minutes. It seems to have gotten worse. Yes, it is usable, but I am part of a game community, and I design vehicles for this game. I play the game often to make sure the vehicle is performing to my liking, but it is now risky due to this random shutdown syndrome. It seems to have no definite pattern except when I play games like this or anything that demands the Graphics Card. I am not able to get a new imac (just spent the money on a brand new trumpet for band), so what can I do to fix this? I took it to an apple repair store, and they probed it, but couldn't get it to fail. I took it back here, and now it acts up again. I just don't see what is causing this problem. It also seems that every forum I have checked on has a different problem stated- like "shuts off every 3 days", or something like that. There is no pattern to mine, as I said. I am completely stumped, what can I do to fix this?