Don't tell them you are a minor.
Something my grandfather taught me is "Just do it until someone says STOP".
What are they going to do, ask to see your birth certificate?
If they catch you for some reason, they may ask you to voluntarily return the prize. They will also probably strip the title from you.
I had to quit school at 15 years old so I could help pay bills. I have lied on every single job application I have filled out. They NEVER check the records.
I lied about my age to purchase liquor too. If I got busted, it would not be my problem. The responsibility would fall on the store for not ID-ing me properly. The police could only bust me for possession of alcohol as a minor. That is a pretty major offense. I doubt lying on a video game contest application would get you any legal repercussions.
Even if there where legal repercussions, you are a minor, so most of them will result in a warning, or really minimal punishment.
I am not saying it is cool to be dishonest. But in certain cases it dosent hurt anything to tell a white lie. Especially if it is going to help your career, moral, and outlook on life.
The reason they say 18 or over is probably because there is Gore and nudity in some of the contestants games. (which is stupid, my students always have better porn links than I do, They also play the goriest games.)
Also, dont take legal advice from the internet... this post is not legal advice.