Hi
I had an A600 with very similar problems. Thing is if I left the computer for twenty minutes or so it would fire up and run just fine. I would then switch off and dead in the water. You could try taking the floppy drive off and checking to see if the cable undernieth is fitted properly, these can cause this kind of problem, particularly if the cable has been reconnected the wrong way. Capacitors can cause probs on the A600 and I would guess its either a dead chip or a capacitor. The machine was probably faulty when you got it so no way of really knowing.
Anyway I simply got another motherboard for mine and it works just fine now.
Check out the following three pages from my site which will show you the inner workings of the A600. Not much help, but great fun in the end..
http://www.scuzz.org.uk/amiga/amiga_scuzz93.htmhttp://www.scuzz.org.uk/amiga/amiga_scuzz94.htmhttp://www.scuzz.org.uk/amiga/amiga_scuzz101.htmFor the whole site check out...
http://www.scuzz.org.uk/amiga/a_amiga_inframe.htmscuzz
