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The problems in this book were selected from the Romanian Team Selection Tests from 1975 to 1986. They are' presented chronologically within each chapter. 1975 was the first author’s last year to take team selection tests, as he was a high school senior. The second author, much younger, took these tests at the end of the period covered in the book.
We learned a lot of mathematics from the problems encountered in various Romanian national contests (some of which are included) and this is something that stayed with us for our entire careers. The problems are old, some of them more than forty years, but still of use to today’s students, especially if they are involved in mathematical competitions. We felt that we had to do a lot more than just present a list ofLproblems accompanied by their solutions. Thus we gave more than one solution when possible, commented on the solutions in order to make them clearer or to get a connection to other problems, and even digressed to other relevant results. We trust that all of this made the exposition more interesting, at least for the beginners in Olympiad mathematics. Here and there we left unanswered questions related to some of the problems presented we hope the reader will accept and even enjoy these challenges. Of course, nowadays,“all answers are on the internet”, but we feel the urge to repeat the good old advice: try first to solve the problem on your own, and only after that seek the answers.
You will definitely see the different proportion between algebra and geometry on one side, and combinatorics and number theory on the other, if one compares to the problems from present days contests. This was not our choice, but rather the reality of those times. And you will find that the two authors of this book kindly invite you to read and learn something from it, so that you Will do well in mathematical contests, and, more importantly, in your future STEM journeys. Suggestions, comments, and remarks are highly welcomed.
We cannot end without thanking the authors of these problems (and apologize for not knowing all their names). Many thanks to Richard Stong for significant improvements of the text.
A booklet containing Romanian Team Selection Test problems from 1975 to 1983 along with sketches of their solutions was distributed at the final round of the 1984 competition. We used parts of this booklet (and some other materials, mentioned in the text) in developing the present book, and are grateful to all of those who contributed to these materials.
Preface
Algebra
Number Theory and Combinatorics
Number Theory
Combinatorics
Geometry and Trigonometry
Solutions to Algebra Problems
Solutions to Number Theory and Combinatorics Problems
Number Theory
Combinatorics
Solutions to Geometry and Trigonometry Problems