While it may be somewhat difficult, it is possible to get the movie clip out of the video & attempt to match it to similar videos online automatically. I could technically create a bot/script for this. If you install a service like CloudFlare, it may help to prevent against this. Also, when you generate the clips, add some sort of watermark over the videos (this will also help w/ copyright potentially)

I'm not saying this is necessarily how the cheater is doing this, but it is technically possible.
There's no way for us to know how the cheater is "cheating" without letting someone technical have access to your site, look at the logs, IPs & requests, etc.

Also, make sure all entry forms require a captcha & there aren't any entry forms without.

Are you manually releasing a new video every week & then the first person to guess it wins?