We have tried Facebook competitions in the past and they have not been very successful. At the end of the day you need to bring people to your Facebook page and unless you are very lucky and your page just happens to start trending then it is very hard to get followers. What we have found is that you need to use normal / old methods of attaching people outside of social media. Once you have the interest you can then get them to your Facebook page so you can collect more information from them and have a long term link to them.
So for instance if they saw you are an exhibition or on a advent then they would follow that through to your Facebook page and hopefully they would like your page or enter your competition. Social media does not in itself being people to your page. Of course if you start trending or when you get a large number of followers then social media does start working as a way of advertising and spreading the word.
Be very careful of Facebook adverts as we found they did nothing. A lot of people are very dubious as to how many people actually click on Facebook adverts. There are also a lot of companies promising you X number of likes for say £100 but in reality them are just paying for people in say China with 100’s of Facebook accounts to follow you. None of them will be real customers or interested parries. We know this from experience when our likes went up by 100 in the space of 2 minutes just as we were getting to a deadline for a campaign we paid for. When looking at the followers they were all completely different demographics to our buyers and all from far flung places.
I think you would be better off advertising through say leaflets, exhibitions, old style methods and then connecting/talking to your potential customers through Facebook.