With colleagues at work I ran the tipping competition from 1998 till the end of 2015. Unless people are dirt poor, have the entry fee as a flat fee paid up front.
My colleagues had to finish off the tipping comp run by a fellow colleague who had a heart attack mid-season 1997 and he used to let people pay on a per week basis. Problem is that those who were well out of the placings in the last 4-6 weeks would enter their tips but would not pay. as such they were chasing people for money upwards of 2-3 weeks after the competition had concluded. So generally speaking to avoid aggravation, make it payment up front to join. No pay, no start. Trust me it will save you a headache.
My other suggestion, run the comp online. There are a number of websites which offer free tipping comps. Find one which meets your needs rule wise and use it. trust me having to sort through 30-40 sheet of paper with peoples tips, add them up, update a table etc take a bit of time. Let a dedicated server do it for you.