It is against Google TOS to ask for a link, so the best way to do this, as Dejan SEO have said is to cause a reason to link back to the competition page o the home page.
There are various methods of this, and the best has been mentioned - include a code snippet on your contest page, perhaps with a blog footer like so:
This post was for a competition hosted by Company Name, promoting Product. Visit their website for more details on this competition
(With links in obviously). This will trick a lot of people into thinking that it is necessary to include that footer following their post to keep their entry legitimate, and will obviously be gaining you the contextual link you're after, to multiple pages if you'd like.
The other method is via profile badges, or a specific badge for that competition. Encourage users to add your badges to their sites/blogs/individual posts, and make it as easy to do so as possible.
One final thing, make sure you give out badges to the finalists, again, linking to your pages. If they're winning something they'll feel obliged to use the badge!
Hope all of this helps EuropeanSEOguy!
