LED Quiz Buzzer Circuit With Logic Gates
A few notes on your current design:
 1. You cannot have floating inputs on logic gates (Your inverter inputs are floating)
 2. You cannot have multiple outputs connected together (outputs from your AND gates)

First you need to clearly define what it is you are trying to do. My guess is you want a circuit where when someone hits a button their light comes on. Once one light turns on everyone else's button is disabled. My guess is you will also need some way to reset this. This will probably require a latch.