120 Awesome Algebra Problems

This book, employing a problem-solving approach, reviews useful algebraic techniques that many of us don’t think of initially. Therefore, this book should be of interest to those who prepare for Olympiad competitions, to mathematics researchers including student researchers, as well as to those who enjoy recreationally reading a good collection of problems solved by ingenious methods. This book should be useful both as a book to

practice skills as well as a reference work when one is working in a particular area and wants to review advanced techniques.

 

The problems presented in the book cover a dozen fundamental areas including: matrices and determinants, polynomials, functional equations, sequences and series, inequalities, systems of, and individual, equations in both real numbers and integers, maxima-minima problems, equations involving exponents and logarithms, and quadratics.

 

The problems explore use of fundamental tools including, Vieta’s formulas, the AM-GM (Arithmetic Mean-Geometric Mean) inequality, Holder’s inequality, Cauchy-Schwartz, as well as the perhaps less familiar, Titu’s lemma, Chebychev’s inequality, and Muirhead’s inequality. Additional algebraic tools frequently used in solutions are Schur’s Theorem, the Chinese Remainder Theorem, Cayley-Hamilton, completion of squares, telescoping, and partial fraction decompositions. The problem solutions also explore exploitation of symmetry, skillful use of substitutions, and the clever consideration of cases.

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