My Mac had a failure as well, it's a 27" iMac from July 2011 I believe.
Someone is checking what the issue is right now. We first thought it was the graphics card. After having replaced it, the issue was still there. (You can read about the symptoms my Mac had in this thread.) Now we're testing the logic board.
It's interesting that you mentioned the RAM. Because not so long ago I upgraded my RAM. I bought it from Crucial, it's 32 GB of RAM. Crucial says that this works fine with the iMac I have, they tested it with it. But perhaps it was still a bit too much.
I have tested the computer with the original RAM again, it worked once but that was it. But maybe because I touched the RAM for too long (You know there's this whole magnetic thing, that you shouldn't touch it and shouldn't install it without special tools etc).
But I also believe, like already mentioned, that heat has something to do with it. I had this game that I play a lot, my iMac gets really hot when I play it. I've heard it shuts down itself if it's too hot, but it never did for me. So is it true that post-2012 iMacs don't have the heat issue?
Also the 32 GB of RAM might have contributed to the heat problem as well: More RAM means more memory usage, means more heat, etc.
I really should have bought Apple Care. But in the end you pay more, because you really NEED Apple Care. If not, your product will fail (by coincidence!) a few days or weeks after your warranty has run out. This is all programmed people. In America people probably don't know about this, but over here in Europe there's talk about this kind of stuff all the time. All products that we buy are meant to fail not long after their warranty has expired... it's too bad because earlier Macs were built like Tanks. Apple doesn't stand for quality anymore. It's still the best OS, but the name of the company has been tarnished by the poor quality. Actually even with Apple Care it will fail at some point, and maximum you can get is two years. Which isn't a long time... Best thing is to build your own Mac. Buy all best parts yourself and get it built by someone who knows his way around. Might even cost less actually, and will last forever. Trust me. I just turned 20 and have paid over 2000€ for this Mac only one and a half years ago, and now I'll have to pay again... it just isn't woth it.
Well I'll keep you updated anyway, we'll see if it's the Logic board or something else.