Should I write a "proof" to a professor I don't know?
The significance of your additional work is unclear. Have you talked about this with the professor that teaches your class? (s)he – or another prof in the department – should be involved if you want to take this further.

If the additional work is significant enough, you'll be perfectly within your rights to write and publish your own paper that presents the results. Trivial results usually aren't publishable on their own, but if it's a novel approach then it's certainly doable. The original work is old enough that you need not worry about "scooping" the original authors' results. Just make sure to check that a solution to the problem hasn't already been published elsewhere.

In any other case, there shouldn't be any harm in sending an email to the authors (whether or not they will respond is another matter). Don't send the solution in your first email! Just inform the authors that you think that you've solved the problem, and only send them your solution if they agree to let you co-author any publications that incorporate your results. Intellectual property rights are a tricky business, and you should always ask for the help of a professor when doing things like this. The paper's authors will also be much more likely to take notice if the first email is from a well known professor, rather than an unknown grad student.