Is it ethical to post an exam question and answer to the internet? I am a private tutor for (mostly) high school maths and physics students. 

Recently a student came to me with an elegant little problem that he had been given on an exam. The problem was very short, and easy for him to remember, but he and others had found it difficult to solve. It required synthesizing the material he had studied in a way that required some creativity. (At least, at the high school level.)

I worked through the solution with him. 

I create content (videos and papers) that I publish on the internet. I would like to share this intruiging application of high school precalculus.

Here's the concern: Does the teacher or whoever else who created the problem own the problem? Am I unethically making the problem useless for other exams or homework by publishing it? What if I alter it so that the process is the same, but the numeric solution is different? Do I need the permission of the problem's author to use it at all?

To be clear, it's not a "word" problem, but simply a "solve this interesting equation" problem.
There is an alternative that avoids the question asked here entirely and would even be more valuable to your readers. 

Think about what it is about that equation (or other question) that makes it interesting. Abstract from that to a more general question that also covers the specific case. 

Then present a way of thinking about that class of problems that will lead someone to a successful solution. Explain the thinking process. Explain any connections to similar things. Etc. 

Then, you can publish that freely. 

Note that words, but not ideas, can be bound by copyright. Take the idea and expand it. Anyone is free to do that.