I did VEX last year when I was still a senior in high school. I believe that burnt IC is one of the H-bridge IC's for the integrated motor drivers on the Cortex.
The two ports with integrated drivers would be ports 1 and 10 (2-wire motor ports).
Check the motors in ports 1 and 10 to see if the wires have shorts. Normally the PTC in the motor would protect against things like this, but they don't protect against shorts outside the motor.
You could try to get the Cortex replaced under warranty, or desolder and replace the H-bridge driver yourself. There are two drivers per 2-wire port, one on top of the board and one below. See for more info here:
http://www.techspin.info/archives/856 (3/4ths of the way down).
The burnt part on the bottom of the PCB is this PMOS:
https://www.fairchildsemi.com/datasheets/FD/FDS4935BZ.pdf