This master thesis is composed of three chapters with the titles Solving Mathematical Problems, Inequalities and Arithmetic. The first chapter begins with a description of the term mathematical problem and several different classifications of a mathematical task. The problem-based teaching method is described and the guidelines for performing this type of lessons are presented. Also, the method of solving a mathematical problem, presented by George P´olya in his book “How to solve it”, is briefly described. At the end of the chapter, the heuristic teaching method is explained, and the heuristics used to solve mathematical problems are listed and described. In the second and third chapters, mathematical problems in the field of inequality and arithmetic are solved. Before solving the mentioned problems, it is necessary to recall several of the most important and well-known theorems in these fields. Among them are the Cauchy-Schwarz inequality and inequalities between arithmetic, geometric, quadratic and harmonic means of numbers. Furthermore, after summarizing the basic concepts related to the divisibility of numbers, the divison theorem and Euclidean algorithm, the fundamental theorem of arithmetic was proved, the concept of congruences was introduced and the Chinese remainder theorem was proved. Thereby, certain techniques have been mastered, which enable mathematical tasks that may be difficult and demanding to solve, to be solved in a much simpler way.