The problems in this book are among the gems of mathematical competitions and showcase the beauty and elegance of mathematical concepts even at the school level. This work is more than just a collection of typical and original problems selected by a team of experienced teachers and university lecturers: all tasks are thematically organized and accompanied by detailed solutions. Each section begins with an introductory text that compiles relevant facts and strategies. Different approaches are explained through solution variants, demonstrating the use of mathematical tools. The intentional application of heuristic principles serves to develop general problem-solving skills. Whether for self-study, study groups, math circles, or training camps, the book offers rich training material to actively prepare oneself or others for mathematical competitions such as the Mathematics Olympiad, the Federal Mathematics Competition, or the Kangaroo Competition. The material can also be used in advanced mathematics classes to provide insights into various facets of mathematics, introduce solution techniques, and practice their application. The problems can generally be solved with knowledge from secondary education level II, but due to their sometimes significant difficulty, they can occasionally pose a challenge even for math enthusiasts. Finally, the book will also support young people transitioning from school to university in mathematically oriented degree programs. Some heuristic methods can indeed be applied in other disciplines and can even be useful in everyday life.