First off, the contest requires 3 people to enter. After speaking to the man on the stage the first time you can then talk to the 3 judges on the ground floor to see what they would like.
Lucy, your 3rd party member, can be recruited in the arcade after you beat up BoJohn again, then go into the tunnel to meet her.
The judges want a futuristic costume, a heroic costume and an educational costume. You should have acquired all 3 by this point before you left the suburbs. You can enter the contest as many times as you like until you get it right.
Aside from the achievement, the reward for winning the contest is the unicorn costume, which has better healing properties than the liberty costume.
(The correct costumes are robot, liberty and knight).