TITLE: How do authors make their problems/exercises for their math books?
QUESTION [31 upvotes]: 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?

REPLY [3 votes]: I personally am a (Euclidean) geometry enthusiast. To write problems, I pretty much go onto some geometry software like Geogebra or Geometers Sketchpad (I use the first because its free from their website) and I fool around, try to find some cool things. Once I notice something interesting, I investigate. Are these four points concyclic? Are these three points colinear? Are these angles the same? Etc. I fool around to determine if they could be true, and if they could then I think about I would go about proving this finding rigorously. I also use my previous knowledge, both in how to prove a statement (thus completing the problem so it's able to be given to others), but more importantly on what to fool around with. For example, yesterday I remembered this problem that I had read a while ago: 

Let ABCD be a trapezoid with AB || CD, let points X and Y be on AB and CD such that XA/XB = YD/YC. There are points P and Q on line XY such that < DPC = < DAB and < AQB = < ADC. Prove that A, P, Q, and D lie on a circle. 

Then I fooled around a bit, drew the diagram, and added points and lines semirandomly. I ended up with this problem (continuing the notations of the previous question), and I ended up finding a nice solution: 

Let PD and AQ intersect at M, and let PC and BQ intersect at N. Prove that MN is parallel to CD. 

Now you have a brand new problem that builds on a previous problem!