It sounds suspiciously like a hardware problem to me.

> Sometimes when I press the power button to turn on the computer, the fan
> turns, but the monitor doesn't get a signal and nothing boots. Pressing
> reboot fixes this problem.

When this occurs does the box beep? long, short, number of times?
If so the manual of you main board will tell you why it is not booting. 

It the box full of crap? dust, mouse pee, metal shavings?

If you have several memory in there you may want to try booting with
just one, then another, to rule out bad memory. It could be that memory
is not seated right. 
  



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