iMac (2009-2011ish) chimes, lights up a grey screen then goes dark, turns off 10 seconds later

I've thought of some things that may help other people in this situation.

First, if you have a keyboard with a light that lights up when Caps Lock is on, you can use that to determine whether the computer is turned on and responsive. When the computer turns off, the light will go out; and If the computer is hung or powered off, then pressing Caps Lock will not change the state of the light. This is how I determined that the computer was turning itself off.

Second, I remembered that booting while I held down the Option key would bring up the menu to select a boot disk. My iMac's display still goes dark after half a second, but when I shine a flashlight on it I can make out the boot disk icon ... and the computer does not turn itself off. But as soon as I press the Return key to start booting from that disk, though, it turns itself off a few seconds later.

That turns one confusing problem into two simple ones: (1) the computer remains on and responsive even though the display is dark (this probably means it only needs a new inverter board), and (2) it turns itself off as soon as it tries to boot from the internal hard drive (this probably means it needs a new drive or a new OS install). I can fix these.

I hope this helps someone else!