Trigonometry and Analytic Tools in Olympiad Geometry Problems, Part I

This is the first part of a series of papers aiming to show how trigonometry and analytic tools can help into tackling demanding Olympiad geometry problems. We present several novel techniques for tackling hard problems from various international and national mathematical competitions, and develop the appropriate theoretical background. Several insightful examples and practice problems are included for the reader to sharpen their problem solving skills. The number of asterisks next to each problem indicate its (subjective) difficulty level, ranging from (*) (early IMO problem) to (***) (hard IMO problem), or ocassionaly (****) (too hard for an IMO problem). Lastly, at the end of the paper we devote a section on providing some historical information on mathematicians whose names appeared in the paper.