How to get the word out about an academic competition?
If you are aiming to reach out to members of a specific field, then I'd suggest reaching out to the department chairs of universities that offer it. For example, if you are looking into something designed for communications majors, professors, etc., then perhaps look up colleges with communications programs. Then find the contact information for the department chair and send them an email. You can repeat this for as many schools as necessary.

If you are going for something a bit broader, you may want to call the administrative offices for the schools, or see if they have an email address that you can contact them through. 

This way, if the message comes across more like spam, then hopefully you've irked as few people as possible. If it's useful and the recipient is interested, then they will have the ability to disperse the information throughout their department or university. It's a lot of work, but sometimes that is what is necessary.