Please post the make/model of your computer: is it a desktop or a laptop?
--- Since you didn't say anything about a battery I suppose it's a desktop but I can't assume that
Okay, it sounds to me like the computer runs ok once it starts
--- Are there any issues using it then?
--- How many beeps do you hear when the computer does start up ok normally: how many when it doesn't?
When it doesn't start, what happens exactly?
--- Any error messages? If so please quote them exactly
--- Does it happen on a daily basis?
--- Are there some days it doesn't happen?
--- Or on a Shut down, starting back up during the same day?
Try starting up in Safe Mode with Networking
--- Can you do so over a period of a few days or so to give it enough time so you'll know whether or not running in Safe Mode with Networking matters?
I can't tell whether or not this would make a difference at this time, but the next time you have to shut down by pulling the plug, wait a few seconds, hold the power button in for about 10 seconds to fully discharge the capacitors. Wait at least 5 minutes. Plug the cord back into the power source and see what happens when you start back up