Elementary Combinatorial Geometry

The present book is a collection of all problems proposed at Balkan Mathematical Olympiads (BMO), also containing some of the problems discussed during these contests by the jury. It is not only a collection of elementary problems but a page of history - the history of BMO, one of the first reginoal mathematical competition for highschool students.As it is already accepted, International Mathematical Olympiads (IMO), which was created in 1959, had an important contribution to development of mathematical education of young students all around the world. They extended the area of problems used in mathematical education of young people and also improved their quality. Many mathematicians realized how important is to use some parts of their research problems as elementary educational problems.All problems are presented with complete solutions. Many problems have several alternative solutions and we also present some extensions. The reader can follow the real increasing of the quality of problems as an illustration of the improvement of mathematical preparation of students. An additional preparatory addendum, containing notions and classical useful results has been added to the end of the book. Cuprins PrefacePart I Problems and SolutionsThe 1st BMOThe 2nd BMOThe 3rd BMOThe 4th BMOThe 5th BMOThe 6th BMOThe 7th BMOThe 8th BMOThe 9th BMOThe 10th BMOThe 11th BMOThe 12th BMOThe 13th BMOThe 14th BMOThe 15th BMOThe 16th BMOThe 17th BMOThe 18th BMOThe 19th BMOThe 20th BMOThe 21st BMOThe 22nd BMOThe 23rdBMOPart II Supplementary ProblemsGeometryAlgebra and Number TheoryAppendixThe Euler LineThe Nine Point CircleEuler Triangle FormulaLeibniz's RelationThe Einstein CriterionYoung's InequalityGlossaryIndex of notationsReferences