Shuts down during boot up

My daughter's imac shut down today and won't complete it's boot up. First off, you have to pull the power cord and plug it back in for it to respond to the power button. If you don't unplug/plug the cord, it won't do anything when you push the power button. So, once it starts, it makes the chime, the fan comes on full blast, the Apple logo comes up and the progress bar starts to crawl to the right. The status bar gets about half way and the computer just turns completely off.

-Tried plugging it directly into a different outlet. Tried unplugging it, holding down the power button for a while, then plugging back in. Unplugged all the externals. Plugged in the original keyboard and mouse. None of this made any difference.

-I held the options key when it started and got to the screen that shows the main drive, recovery drive and wireless login. It stayed on the whole time I put in the wifi password and selected the recovery disk. After selecting the recovery disk, the logo and progress bar came on screen, but again, the computer shutdown when the progress bar got about half way.

I've read the posts about opening the computer and looking at the diagnostic led’s and all that, but this is the model with the screen that's essentially taped into the frame. I'd rather not mess up that seal unless I have no other option.

Is there anything else I can try before I have to open it up? Has anyone had this same problem that could suggest an actual fix?

Thanks.