problem of the security light flashing and car not starting

Ignition switch is going bad http://www.cargurus.com/Cars/Discussion- d475_ds505015. The steps on the page in the link will get it going but eventually the ignition switch will have to be replaced.

Could be a bad starter solenoid or it's possible that the car is equipped with a chip key system. I've personally seen where a non chip key will operate the car but the security mil will laminate. After it locks out, the car will not start

My 2001 Grand Am does this to me and there is one of two options to start it; 1. Turn the key to the on position, wait 10 minutes, turn the key back to off and then start the car. If this doesn't work then try this; 2 Turn the key to the ACC position, wait 15 minutes, turn the key back to off and then start the car. My car works with the second option. Make sure the security light has stopped flashing before you try to start the car with either option.

"How to fix it" from Grand Am Owners Club - http://www.gaownersclub.com/forum/showthread.php?t=139899