Hard Core bad news: no boost, no weapons.
Hard Core good news: no one can shoot you or reverse your controls.
This is pretty straightforward racing. You should be able to get in front of the pack during your first lap if you follow these tips:
1. Time your start. Hit the gas (Right Trigger) just as the count hits "Start" and you get a little boost that can take you to the front.
2. Make good use of your power slide (A button) to set yourself up for corners.
3. Don't use the brake, just let off the gas.
4. To win the Tournament you do not have to win every race, you just need to have the highest combined point total at the end. The Tournament consists of 5 race tracks. The position that you finish in each race will award you points in the Tournament total. (10 for 1st, 8 for 2nd, etc.) If you get 1st place in three races and second in the other two, you'll win.