Canon A85 power on problem

I think we can rule out a problem with the cards, which leaves the camera. maybe one of the cf pins is bent?? look into the slot with a light and check for any pins out of line, if you find any, see if you gently push it back into place. I saw the exact same problem, having the same fix.

try also connecting AC through a cable and the camera to the computer, see if it makes any change.

Whats your warranty status? use it if you still got it, if not, get a repair estimate.

Hope it works out