Shuts off on startup!

This is my daughter's computer. I'm visiting this week and would love to solve the problem while I"m here. What happens is that when you power up the computer, it bongs, the screen turns white and then it shuts down.

If you hold the power button down for about 15 seconds, it starts normally. If you do a restart once it powers up, all is fine. If you shut it down and press/release the power button to start it, there is a bong, the screen lights up for a second, and then it dies.

So far, I've repaired permissons, reset the SMU and NVPRAM, run disk utilities, and fsck.

The computer is orignial, with all system (10.6.8) updates applied. I did find some nasty adware once I got it going, which I think I removed. The battery has been replaced, but other than that, everything is OEM. I did notice that the rubber feet are missing from the bottom of the case.

Ideas?????