A Primer for Mathematics Competitions
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.
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2 Algebraic inequalities and mathematical induction
3 Diophantine equations
4 Number theory
6 Sequences and Series
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ABCD Appetizer Problem area of triangle arithmetic Binomial Theorem bisector Cauchy–Schwarz inequality centre circle circumcircle coefficients colour common congruence congruence arithmetic Consider cos2 cosine cosx cyclic quadrilateral denote diagram diameter Diophantine equations divided divisible equal equilateral Example factors figure formula geometric geometric series given gives Golden Ratio graph Hence inductive hypothesis inequality integers last digit Mathematical Induction Mathematical Olympiad mathematicians method multiple natural numbers notation Observe odd numbers ofthe pairs Pascal’s triangle permutations perpendicular pigeon-hole polygon positive integers possible Proof prove Pythagoras Pythagorean Pythagorean triple radian radius ratio real numbers rectangle remainder result right angles sequence shown sides sin2 sin6 sine sine rule sinx smallest Solution of Appetizer solve subsets tangent Toolchest total number triangle ABC trigonometric functions true units digit vertices whole number zero