Inspiron 600m does not power up, flashing LED's (intermittent problem)

I have a Dell Inspiron 600m from November 2004.  For the last month, I've had an intermittent problem when I attempt to turn on the notebook.  I'll press the power button and the three LED's (cap lock, num lock, and scroll lock) will blink rapidly for about five seconds and the power led will light up like normal.  After this five second period, the computer shows no signs of life.  If I repeat this process, eventually the computer will power up like nothing was ever wrong.

I've tried powering up the computer on a freshly charged battery, and I've tried removing the battery and using just the AC power supply.  Neither makes any difference.  I've also tried re-seating the RAM chips, the DVD drive, and the hard drive.  So far nothing seems to make any difference.

I've read other posts about this problem, but I haven't seen any definitive conclusion as to what is causing this type of problem.  It it's the motherboard, then the repair will probably cost more than the notebook is worth.  But, if it's the power button, it should be fairly cheap and easy to replace this.  (I can't imagine what else would cause this.)

Does anyone know what would cause the three LED's (cap lock, num lock, and scroll lock) to blink rapidly for about five seconds when the power on button is pressed?  And, has anyone encountered this problem and reached a successful resolution?

Thanks,

- Ed