That understanding is rather superficial. The DMCA has many nuances, among them being having a designated DMCA agent, sending out proper notices so people can post counter notices, honoring proper counter notices, not having active control over the content of postings. If you are a student, you should not attempt to use this as a means for enable cheating. You are pretty clueless if you think popular video sharing websites have no worry about being sued. The big ones like YouTube spend major money on policing infringement.
I am not your lawyer and you are not my client. Free advice here is without recourse and any reliance thereupon is at your sole risk. This is done without compensation as a free public service. I am licensed in IL, MO, TX and I am a Reg. Pat. Atty. so advice in any other jurisdiction is strictly general advice and should be confirmed with an attorney licensed in that jurisdiction.
Regards to the DMCA, while you can achieve a safe harbor for user generated content, as Attorney Burdick aptly notes, it can get more complicated than what you suggests here. For example, the law requires a proper recordation of your contact person or entity to be on file with the US Copyright Office. This is required for every domain even if they are owned by the same entity or person.
Further, you can easy run afoul of the statute if, for example, the content is edited, controlled or manipulated in any way by the ISP (you in this case).
You will obviously have a number of legal considerations to launch something like this (e.g., entity structure, trademark, terms, copyright and DMCA matters, among others), so now would be a good time to form a relationship with a good business/IP attorney.
You can start by calling around to several for a free phone consultation, get some insights then pick the best fit to work with.
DISCLAIMER: this is not intended to be specific legal advice and should not be relied upon as such. No attorney-client relationship is formed with the law firm of Natoli-Lapin, LLC on the basis of this posting.