Construct Before Prove

In this paper, I studied several elementary mathematical problems from scratch, whichinvolves various area, such as missile's trajectory in Cartesian coordinate, algebraic functionalanalysis and elementary geometry, etc., aims to provide a personal perspective for problem solving. The Problem B and E to G were selected from Putnam competition in 2024. I would like toshare a little bit experience and basic ideas on these classical and simple matters, encompassingsome of tricks and strategies (e.g., the application of the inversion techniques)which are notthe best in some sense, but actually are practical for solving such problems. This paper is aninstructive and educational material to investigate such methods of problem solving.