Primer for Mathematics Competitions.
Title:
Primer for Mathematics Competitions.
Author:
Zawaira, Alexander.
ISBN:
9780191561702
Personal Author:
Physical Description:
1 online resource (363 pages)
Contents:
Contents -- Preface -- 1 Geometry -- 1.1 Brief reminder of basic geometry -- 1.1.1 Geometry of straight lines -- 1.1.2 Geometry of polygons -- 1.1.3 Geometry of the fundamental polygon - the triangle -- 1.1.4 Geometry of circles and circular arcs -- 1.2 Advanced geometry of the triangle -- 1.3 Advanced circle geometry -- 1.4 Problems -- 1.5 Solutions -- 2 Algebraic inequalities and mathematical induction -- 2.1 The method of induction -- 2.2 Elementary inequalities -- 2.3 Harder inequalities -- 2.4 The discriminant of a quadratic expression -- 2.5 The modulus function -- 2.6 Problems -- 2.7 Solutions -- 3 Diophantine equations -- 3.1 Introduction -- 3.2 Division algorithm and greatest common divisor -- 3.3 Euclidean algorithm -- 3.4 Linear Diophantine equations -- 3.4.1 Finding a particular solution of ax + by = c -- 3.4.2 Finding the general solution of ax + by = c -- 3.5 Euclidean reduction, or 'divide and conquer' -- 3.6 Some simple nonlinear Diophantine equations -- 3.7 Problems -- 3.8 Solutions -- 4 Number theory -- 4.1 Divisibility, primes and factorization -- 4.2 Tests for divisibility -- 4.3 The congruence notation: finding remainders -- 4.4 Residue classes -- 4.5 Two useful theorems -- 4.6 The number of zeros at the end of n! -- 4.7 The Unique Factorization Theorem -- 4.8 The Chinese Remainder Theorem -- 4.9 Problems -- 4.10 Solutions -- 5 Trigonometry -- 5.1 Angles and their measurement -- 5.2 Trigonometric functions of acute angles -- 5.3 Trigonometric functions of general angles -- 5.4 Graphs of sine and cosine functions -- 5.5 Trigonometric identities -- 5.5.1 The Pythagorean set of identities -- 5.5.2 Addition formulas -- 5.5.3 Double angle formulas -- 5.5.4 Product formulas -- 5.5.5 Sum formulas -- 5.6 Trigonometric equations -- 5.7 Problems -- 5.8 Solutions -- 6 Sequences and Series -- 6.1 General sequences.
6.2 The summation notation -- 6.3 Arithmetic Progressions -- 6.4 Geometric Progressions -- 6.5 Sum to infinity of a Geometric Progression -- 6.6 Formulas for sums of squares and cubes -- 6.7 Problems -- 6.8 Solutions -- 7 Binomial Theorem -- 7.1 Pascal's triangle -- 7.2 A formula for the coefficients -- 7.3 Some properties of Pascal's triangle -- 7.4 Problems -- 7.5 Solutions -- 8 Combinatorics (counting techniques) -- 8.1 The fundamental principle of enumeration -- 8.2 Factorial arithmetic -- 8.3 Partitions and permutations of a set -- 8.3.1 Definition of terms -- 8.3.2 The general partitioning formula -- 8.3.3 The general permutation formula -- 8.3.4 Circular permutations -- 8.4 Combinations -- 8.5 Derangements -- 8.6 The exclusion-inclusion principle -- 8.7 The pigeon-hole principle -- 8.8 Problems -- 8.9 Solutions -- 9 Miscellaneous problems and solutions -- 9.1 Problems -- 9.2 Solutions -- Further Training Resources -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Z.
Abstract:
A comprehensive resource containing an entertaining selection of problems in mathematics. Including numerous exercises, illustrations, hints, and solutions, it is aimed at students of mathematics looking for an introduction to problem solving in mathematics, as well as Mathematical Olympiad competitors and other recreational mathematicians. - ;The importance of mathematics competitions has been widely recognised for three reasons: they help to develop imaginative capacity and thinking skills whose value far transcends mathematics; they constitute the most effective way of discovering and nurturing mathematical talent; and they provide a means to combat the prevalent false image of mathematics held by high school students, as either a fearsomely difficult or a dull and uncreative subject. This book provides a comprehensive. training resource for competitions from local and provincial to national Olympiad level, containing hundreds of diagrams, and graced by many light-hearted cartoons. It features a large collection of what mathematicians call "beautiful" problems - non-routine, provocative, fascinating, and. challenging problems, often with elegant solutions. It features careful, systematic exposition of a selection of the most important topics encountered in mathematics competitions, assuming little prior knowledge. Geometry, trigonometry, mathematical induction, inequalities, Diophantine equations, number theory, sequences and series, the binomial theorem, and combinatorics - are all developed in a gentle but lively manner, liberally illustrated with examples, and consistently motivated by. attractive "appetiser" problems, whose solution appears after the relevant theory has been expounded. Each chapter is presented as a "toolchest" of instruments designed for cracking the problems collected at the end of the chapter. Other topics, such as
algebra, co-ordinate geometry, functional equations and probability, are introduced and elucidated in the posing and solving of the large collection of miscellaneous problems in the final toolchest. An unusual feature of this book is the attention paid throughout to the history of mathematics - the origins of the ideas, the terminology and some of the problems, and the celebration of mathematics as a multicultural, cooperative human achievement. As a bonus the aspiring "mathlete" may encounter, in the most enjoyable way possible, many of the topics that form the core of the standard school curriculum. -.
Local Note:
Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2017. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.
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