Before i start telling you what to do there are one or two things you should take a look at. Like, according to rules table names should be singular. so it would be 'competition' and not 'competitions' etc. . .
You are going to have to make a composite table between people and competitions (this is just another table between the two). Usually the easiest way to name it will be taking the two names of the two many to many relational tables and calling it that. So, for instance you could call it 'person competition'. then you will therefore have a one to many relationship from person to person competition, (ID will therefore become a FK in this new table). and then you will also have a one to many relationship from competitions to person competition and therefore this new table will also have competitionID in it.
This will solve your problem, as the new table will show the competitionID and the person(ID) that takes part in that particular competition.
I dont know what your case study says but unless it says a competition has only one sponsor and a sponsor can sponsor many competitions, then usually a competition can have many sponsors. So you would have a composite table betweeen these two aswell.
Looking at the date you posted this, i guess my reply is a bit late. . .