How do authors make their problems/exercises for their math books? [closed]

I want to be a math professor one day, but I'm wondering how to make my own original problems to give them to my students. I think that it is a responsibility of the professor to create original and useful problems for their students given that there is a lot of solutions online of classic books.

How do they do it?

To set the scene for my answer: since 2010 every year I have worked with and taught classes of "very-soon-to-be" students in a so called "Vorkurs Mathematik". This is a 4-week course at the very beginning of your first semester at university, so usually the students have just finished school ("Gymnasium" in Germany which I think would be high school in America?). In this course we quickly review everything they should have learned so far in school and then give some hints of what mathematics is really about (propositional calculus, basic set theory, basic linear algebra...) which usually isn't taught in school anymore. In addition in these past years I have worked with students of different fields (Mathematics, Physics, Engineering, Biology) who are already studying at university. For me three things are important (depending on the level you are teaching of course as @Martigan has pointed out):

1. Although there are lots of classic problems including solutions to be found on the internet, it is important to cover these (in detail) in your own class. For example:

   Prove that $\lim\limits_{n\to\infty} \frac 1n=0$.

   Proof: Given $\varepsilon >0$ choose $N\geq \frac{1}{\varepsilon}$. If $n>N$ then $$|\frac 1n-0|=|\frac 1n|=\frac 1n < \frac 1N<\varepsilon.$$

   This is a perfectly good answer one could find in any textbook and to you and me this is "the way" to do it. But for a student just starting to deal with problem like this, this solution has a major flaw as it does not point out your train of thought. Yes, $N\geq \frac{1}{\varepsilon}$ is a perfectly good choice as it obviously "works". And to you and me it should be obvious that one could have chosen $N\geq \frac{1}{\varepsilon} +1$ instead. But how did we get to the point of choosing $N$? To a new student this can be one hell of a problem if these basics are not covered, be it in a lecture or in a question which is then explicitly talked about. A nice approach was made by Christopher Wallace in this post.

   So in my opinion every professor should cover these classic problems at least on a beginners level. This doesn't mean that one should copy questions and solutions from a textbook, but rather to apply your own style to the solution e.g. solve the question yourself in a way that points out your train of thought.

2. If you only use these classic questions, it can get boring. So in addition to the official questions I had to cover in the seminars, I tried to come up with a fun or interesting or surprising application. One example from the Vorkurs, so aiming at soon-to-be-students:

   In german schools it is common to cover the reconstruction of a polynomial to given points. In those questions there are usually $n+1$ points given to reconstruct a polynomial of degree $n$ (sometimes exceptions are made in advanced classes which also covers family of curves). Based on this I often give them the following question, which is not part of the official program of the Vorkurs:

   Given a polynomial $f(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$ with $a_n\in \mathbb{N}, a_i\in\mathbb{N}_0$ for $i\in\{0,\dots, n-1\}$. You can ask (multiple times) for the value of any $x\in\mathbb{Z}$. Is it possible to reconstruct the polynomial? If yes, what is the minimum quantity of values you have to ask for?

   As the degree of this polynomial is unknown, they can't use their "normal way" of asking for $n+1$ values. But this question is solvable using only what they've learned in school and in the Vorkurs so far (you only need to know about representation of numbers to a different base). This is what I would consider an interesting and surprising application, as it takes on a known question but you have to think of a new way to solve it (and in my oppinion the answer is indeed surprising).

3. I find it very important to always be amazed and enthusiastic about your own subject. This might sound trivial as you wouldn't want to become a professor if you don't like the subject, but in teaching it is very important to convey your enthusiasm to your students. As I pointed out, your students have to deal with classic questions. So as their teacher you have to deal with these questions, too. And maybe you have to deal with these for the rest of your life, as the basics of mathematics are unlikely to change. For some it is very hard to keep their amazement for these "trivial questions", which I think leads to bad teaching. So I always try to keep in mind how amazed I was the first time I solved this kind of question and how good it felt to know exactly how I got to the solution.

Of course this doesn't need to hold if your students are more advanced as they should have figured this out themselves by then, and it also varies if you're teaching mathematicians or students which must take a maths course but are not really interested in the subject (in my own experience the biology students where often only focussed on how to pass the exam).

In short: take classic questions and apply your own style to them, thus making it at least part your (original?) question. Give interesting tidbits. And always be enthusiastic even about easy stuff.

Plagiarize,
Let no one else's work evade your eyes,
Remember why the good Lord made your eyes,
So don't shade your eyes,
But plagiarize, plagiarize, plagiarize...