I don't want books like "How to solve it". Instead I am looking for books that actually have problems in them. And preferably solutions too. Basically books with problems that help you develop mathematical problem solving ability.
For my background, I have graduated from high school and know high school math fairly well. But lot of the problems at high school level are reiterative and can be done by memorizing stuff. I am looking for problems that challenge you and at the same time make you a better problem solver.
The art and craft of problem solving by Paul Zeitz is a very good book and it meets your requirements. It has many head scratching problems which are easy to understand but hard to do.
The IMO compendium is also a good book, which includes the questions (and answers) from the previous IMO, as well as some additional questions. It also teaches some concepts.