I installed Unity, intending to learn it before the contest. I made a new project as the first tutorial specifies, but immediately afterwards, it informed me that my trial had run out.

After less than five minutes.

Solutions, besides searching the warez sites for a serial?

Send email to david@otee.dk and tell him about it, then ask him for a new temporary serial number.

I emailed David yesterday and he provided a special serial for the contest length.
The download link he provided was not the most recent version though. Needed to get an update.

That's odd... My unity trial is working fine.