Facebook – How to Become a Fan of a Page

you have to use a "like box" instead of a "like button": http://developers.facebook.com/docs/reference/plugins/like-box

to sort out your 'Facebook Page ID' just log on to your fan page and check the url. the number at the end of the url is your page id.

If you are running a competition through your page, you will have the post regarding the competition on your page. The easiest way would be to post a "Congratulations" message with the profile picture of the winner. Many pages run contests through their page. Real fans who have 'Like'd your page will see it in their wall.