Hi infinity79! I want to give my input, hope it'll help.
We used to run a contest, till now actually, for all we know it's one way of link baiting
. We launched our contest (trivia contest) on Facebook, and promote it using other social media like Twitter. The answer to trivia can be found in one of the articles in our website, so they have to look over and search (for) that article containing the hint and the correct answer. We also tried this website: iWantPromo
a free promotion website. Try it, the information of their website is crawled fast by SE's bot, in just 24hours, expect your contest will appear in SERP.
I can't say anything for sure if you will face any legal problem when it comes in prizes. It depends on your country's governance I guess. I do a search and found a good source of info: What legal issues come up when running a contest or giveaway?