I have no doubt this is a difficult one, because it does not happen all of the time.
HP Pavillion 500-301nd.
It runs to our satisfaction for years, but since some months it does not start all of the time.
On power on, normaly the first thing to be seen on the monitor is a HP logo (roundel wuth hp). I assume it is the BIOS that shows it. When that is seen all further booting is flawless. But often it is not shown, nothing happens (only ventilators are heard) and the only thing is to do a prolonged push on the power button to switch off.
Trying again somethimes helps, but most of the time, once it does not work it also does not when immediatly re-tried. Often in the morning everything is OK and then running until the evening without problems. Starting short after a shiutdown (even from a reboot action) often dos not work and after one or two re-trials, better leave the system off for some hours or until next morning.
I cleaned the system (removing dust) and re-seated the memory strip (which I did so clumsy that I needed a thread here to get that right).
I realize that there is a lot to guess here. Not reproducible always. Far before any logging or messages are available.