Preparing for Mathematics Olympiad PDF

Preparing for Mathematics Olympiad All views expressed here are my personal & are from my 3 year experience on search of good study material for young mathematicians. I have also given a rough grouping of some useful books. A. Why...

Preparing for Mathematics Olympiad All views expressed here are my personal & are from my 3 year experience on search of good study material for young mathematicians. I have also given a rough grouping of some useful books. A. Why to Prepare: To pursue Mathematics as a field of research later in life, starting with Undergraduate Courses at Chennai Mathematical Institute, Indian Statistical Institute or any university abroad. B. Traits Needed: Love & Devotion for beauty of mathematics C. When to start: One can appear for Regional Math Olympiad (RMO) from class VIII to Class XI (not for class XII from year 2014 onwards). I can’t comment when one should start as every child is special & different. NOTE REGARDING ALL BOOKS: I have read many, but not all of these books [as few are still in my “To Read” list] In case you can’t find any of these books in market (out of stock or its very costly) simply e-mail me at [email protected] and I will help you to get one or can suggest an equivalent book. D. School Mathematics v/s Olympiad Mathematics: “School mathematics” (or “IIT Mathematics” for class XI & XII students) or “Olympiad Mathematics” seem to be of different nature but at core, they are actually same thing i.e. “Mathematics”. Only our prospective is different in both cases. Here I can point out some similarities in both to convince you: 1. NCERT Mathematics Textbook for Class IX [NCF – 2005] is fantastic book to start for Olympiad mathematics as it touches nearly all topics (like geometry, polynomials, Number Theory (rational - irrational numbers), Introduction to mathematical modelling) which we study at advanced level for Olympiads. FOR EXAMPLE: I was spell bound by chapter - “Introduction to Euclid’s Axioms” and I ended up reading “Euclid’s Window by Leonard Mlodinow “ & “Fun & Fundamentals of Mathematics by Narlikar” 2. NCERT Mathematics Textbook for Class X [NCF – 2005] also consists of basics of “Number Theory” topics like “Euclid’s division algorithm”. Moreover the appendices on “Proofs in Mathematics” & “Mathematical Modelling” are worth reading even at later stages. FOR EXAMPLE: The discussion on ‘Proof by Contradiction’ is awesome. 3. NCERT Mathematics Textbook for Class XI[NCF – 2005] includes some of most fundamental & important topics of Olympiad mathematics like “Set Theory”, “Principle of Mathematical Induction”, “Summation of Series”, “Binomial Theorem” & “Permutation & Combinations”. Also appendices on “Infinite Series” & “Mathematical Modelling” are worth reading. 4. NCERT Mathematics Textbook for Class XII [NCF – 2005] major focus in on Calculus, but still its appendices on “Proofs in Mathematics” & “Mathematical Modelling” are worth reading. FOR EXAMPLE: In Appendix – 1 (Proofs in Mathematics), there are proofs for two beautiful theorems of Number theory (a) Prime Numbers are infinite & (b) is not prime. E. Suggested Readings to get an insight of “Beauty of mathematics” :  History of Development of Mathematics : 1. Men of Mathematics (Volume 1 & 2) by E. T. Bell 2. The Mathematical Experience by P. J. Davis & R. Hersh 3. Euclid’s Window : The story of geometry from parallel lines to hyper space by Leonard Mlodinow 4. Fermat’s Last Theorem by Simon Singh 5. Journey Through Genius by William Dunham 6. Notebooks of Srinivasa Ramanujan @ http://www.imsc.res.in/~rao/ramanujan/NotebookFirst.htm Must peep into the notebooks of Genius, many parts of which are still topic of research  Biographies or Autobiographies of some Great Mathematicians: 1. The Man who knew infinity by Robert Kanigel [biography of Srinivasa Ramanujan Iyenger] 2. The Man who loved only numbers by Paul Hoffmann [biography of Paul Erdȍs] 3. A Beautiful Mind by Sylvia Nasar [biography of John Nash] 4. Adventures of a Mathematician by Stanislaw M Ulam [autobiography] 5. A Mathematician's Apology by GH Hardy [autobiography] Gaurish Korpal Preparing for Mathematics Olympiad 1 F. Step by Step Learning: Here “Levels” have been marked by me as per maturity level of mind needed in my opinion to understand what’s written, in these books. Also order of books in each level again specifies increasing order of difficulty of book in that level. You can keep on switching between various levels and books as per your comfort level as every child is different. Level – 1 [a] (Expanding Horizons) Book Remarks Mathematical Circles (Russian Experience) by Fomin, Genkin, Itenberg Thought provoking [for VIII & IX class] A Mathematical Mosaic: Patterns & Problem-Solving by Ravi Vakil My favourite! Arithmetic and Algebra: Numbers and the beginnings of Algebra by Shirali Ideal for beginners. First Steps in Number Theory: A Primer on Divisibility by S.A. Shirali Discusses congruences in good detail Hands-on Geometry by Christopher M. Freeman Step-by-step guide to learn construction Non-Routine Problems in Mathematics by AMTI (Editor: V. K. Krishnan) Lovely book but has few wrong solutions The Cartoon Guide to Calculus by Larry Gonick An illustrative guide to elementary calculus What is Mathematics ? by Richard Courant and Herbert Robbins This book will be your friend for 4 years Level – 1 [b] (Introduction to Higher mathematics with “Little Mathematics Library”) These books include short introductory material (without much detail) on various topics which can help student to get an idea of different fields of mathematics, written especially for high school students preparing for Olympiad. All these can be downloaded legally from http://mirtitles.org/2012/09/06/little-mathematics-library-taking-stock/ Topic Book Lobachevskian Geometry by Smogorzhevsky General Remarkable Curves by Markushevich Dividing Line Segment in Given Ratio by Beskin 2D Constructions Geometrical Constructions using Compasses Only by Kostovskii Geometry The Ruler in Geometrical Constructions by Smogorzhevsky Geometry Proof in Geometry by Fetisov Proofs Induction in Geometry by Yaglom & Golovina Analytical Geometry Method of Coordinates by Smogorzhevsky Stereographic Projection by Rosenfeld & Sergeeva 3 D Geometry Images Of Geometric Solids by Beskin Fascinating Fractions by Beskin General The Method of Mathematical Induction by Sominskii Algebraic Equations of Arbitrary Degrees by Kurosh Complex Numbers Complex Numbers and Conformal Mappings by Markushevich Fundamental Theorem of Arithmetic by Kaluzhnin Algebra Number Theory Solving Equations In Integers by Gelfond Systems of Linear Inequalities by Solodovnikov Inequalities Inequalities by Korovkin Pascal’s Triangle by Uspenskii Combinatorics The Mote Carlo Method by Sobol Calculus of Rational Functions by Shilov General Plotting Graphs by Shilov Calculus Differentiation Explained by Boltyansky Differential Calculus Method Of Successive Approximations by Vilenkin Integral Calculus Areas and Logarithms by Markushevich Economics Elements Of Game Theory by Venttsel Miscellaneous Programming Posts Machine by Upensky Research Godel’s Incompleteness Theorem by Uspensky Gaurish Korpal Preparing for Mathematics Olympiad 2  Level – 2 [a] (Building basics: Learning Theory) CAUTION : Don’t stick to one topic & keep switching (as per your wisdom). Learning should be natural & effortless as most of textbooks are for undergraduate levels, so doesn’t worry if you don’t understand something in one go. Topic Text Book 1.The Foundations of Geometry by David Hilbert 2. Geometry Revisited by H. S. M. Coxeter and S. L.Greitzer 3. Triangles: Constructions & Inequalities by Subramanian & Murlidharan [AMTI] Geometry 4. The Elements of Coordinate Geometry by S. L. Loney 5. Projective Geometry by H. S. M. Coxeter 6. Geometric Transformations (Part – I,II,III) by I. M. Yaglom Trigonometry Plane Trigonometry (Part-1 & 2) by S. L. Loney 1. Higher Algebra by Hall & Knight General 2. Discrete Mathematics with Graph Theory by Goodaire & Parmenter 3. generatingfunctionology by H. S. Wilf Sequence & Series A Primer on Number Sequences by S.A. Shirali 1. An Introduction to Combinatorics by Alan Slomson Combinatorics 2. Combinatorics : Theory & Applications by V. Krishnamurthy Introduction to Linear Algebra by Gilbert Strang Linear Algebra OR Algebra An Introduction to Linear Algebra by V. Krishnamurthy, V. P. Mainra & J. L. Arora Inequalities – An Approach Through Problems by B. J. Venkatachala Inequalities OR Inequalities- A Mathematical Olympiad Approach by Manfrino, Ortega, Delgado General Elementary Number Theory by David Burton Number Diophantine An introduction to Diophantine Equations – A Problem Based Approach Theory Equations by Titu Andreescu, Ion Cucurezeanu & Dorin Andrica Probability Probability Theory (First Steps) by E. S. Wentzel Calculus (Vol. 1) by Tom M. Apostol Analysis OR Introduction to Calculus & Analysis (Vol. 1) by Richard Courant & Fritz John Functional Equations Functional Equations and How to Solve Them by Christopher G Small Iterations Adventures in Iterations (Vol. 1) by S. A. Shirali 1. Chaos by James Gleick Chaos 2. Videos on Chaos (9 Chapters) at http://www.chaos-math.org/en Level – 2 [b] (Optional Further Investigations - for specific topic lovers only) Topic Text Book 1. Geometry by Pogorelov Geometry 2. Introduction to Geometry by H. S. M. Coxeter 3. Non-Euclidean Geometry by H. S. M. Coseter 1. Algebra by Michael Artin General 2. Higher Algebra by A. Kurosh Inequalities Inequalities by G. H. Hardy, J. E. Littlewood & G. Polya Combinatorics A course in Combinatorics by J. H. van Lint & R. M. Wilson Algebra Number Theory An Introduction to the Theory of Numbers by G.H. Hardy Linear Algebra Linear Algebra by Hoffman Kenneth , Ray Kunze 1. The Theory of Probability by B. V. Gnedenko Probability 2. An Introduction to Probability Theory & its Applications by William Feller Calculus (Vol. 2) by Tom M. Apostol Analysis OR Introduction to Calculus & Analysis (Vol. 2) by Richard Courant & Fritz John Functional Equations Lectures on Functional Equations & their applications by J. Aczel Iterations Adventures in Iterations (Vol. 2) by S. A. Shirali Chaos Fractals, Chaos, Power Laws: Minutes from an Infinite Paradise by M. Schroeder Gaurish Korpal Preparing for Mathematics Olympiad 3 Level – 3 (Learning General Approach for Problem Solving) Book Remarks How to Solve it by G. Polya Classics on “Learning Problem Solving” Also solve “The Stanford Mathematics Mathematics and Plausible Reasoning (Vol 1 & 2) by G. Polya Problem Book” to practise the concepts Mathematical Discovery by G. Polya taught in these books. Techniques of problem solving by S. G. Krantz Covers a wide range of topics. Solving Mathematical Problems by Terence Tao Short & beautifully written book Problem Solving Through Recreational Mathematics by Averbach & Chein Innovative way of learning! Problem Solving Strategies by Arthur Engel Must read its first three chapters. The Art and Craft of Problem Solving by Paul Zeitz First four chapters are worth reading. (along with Student’s Manual & Instructor’s Manual) Many classical problems are there. How to Solve it - Modern Heuristics by Michalewicz & Fogel ---- Level – 4 (Problem Solving Ideas for specific Topics) All these books have a basic theme of “classification” of problems according to “useful ideas” Book Remarks Polynomials by E. J. Barbeau Good collection of Pell’s Equations by E. J. Barbeau Challenging problems Introduction to Functional Equations Theory & Problem Solving Strategies for Also includes introduction Mathematical Competitions & beyond by Costas Efthimiou to “Iteration” & “Chaos” Unique type of Functional Equations – A Problem Solving Approach by B. J. Venkatachala classification of problems Trigonometric Functions : Problem Solving Approach by Panchishkin & Shavgulidze Awesome Number Theory Structures, Examples, and Problems by Titu Andreescu and Dorin Andrica Good books for brushing A Path to Combinatorics for Undergraduates: Counting Strategies by Andreesu & Feng up of basics The Cauchy–Schwarz Master Class: An Introduction to art of Mathematical Inequalities Grand Finale of by J. M. Steele Inequalities Aspects of Combinatorics: A wide Ranging Introduction by Victor Bryant Assorted ideas... Straight Lines & Curves by Vasilyev & Gutenmacher Awesome Problems & Solutions in Euclidian Geometry by M. N. Aref & William Wernick ----- Level – 5 (Refining Problem Solving Art by Learning Tricks) Book Remarks Adventures in Problem Solving by S.A. Shirali Problems based upon interesting concepts. Also teaches how to get aid from computer. *The book Challenge & Thrill of pre College Mathematics by C. R. Pranesachar, B. J. Venkatachala, K. N. Ranganathan & V. Krishnamurthy * The supplementary book suggested here has many supplements this book as it consists some interesting topics in Geometry typos & some weird topics like Synthetic Division (pg 504), (like Erdȍs-Mordell Theorem) & Combinatorics (like Ramsey’s Theorem) linear equations in 4 variables (pg 438) which can be found which have been posed as exercise in this book. in classic texts like “Higher Algebra by Hall & Knight”. These focus on developing tricks into methods Mathematical Olympiad Challenges by T. Andreescu & R. Gelca and methods into mastery. Each section begins with a theme of problem solving with one or two Mathematical Olympiad Treasures by T. Andreescu & B. Enescu examples that are easy if we apply the theme, and then gives a whole bunch of problems that Problems from the book by Titu Andreescu & Gabriel Dospinescu need to be solved by variants of the basic theme. Contains really challenging problems divided into Mathematical Miniatures by Titu Andreescu & Svetoslav Savchev 50 ideas. Also a beautifully organised book! Problem - Solving Through Problems by L. C. Larson Don’t have solutions for all Problems proposed It consists of many difficult solved (& unsolved) Winning Solution by Edward Lozansky & Cecil Rousseau problems along with many new theorems. Gaurish Korpal Preparing for Mathematics Olympiad 4  Level - 6 (Practising for Confidence) Book Remark Gems – Junior by V. Seshan [AMTI] 331 RMO Level Problems Gems – Inter by S. R. Santhanam [AMTI] 148 INMO Level Problems Problem Primer for Olympiads by Pranesachar, Venkatachala & Yogananda 110 IMO Level Problems Good Subjective Questions Test of Mathematics at 10+2 Level by Indian Statistical institute [without solutions] Problems in Plane Geometry by I.F. Sharygin A good collection 101 Problems in Algebra by Titu Andreescu and Zuming Feng 102 Combinatorial Problems by Titu Andreescu and Zuming Feng Problems from the training 103 Trigonometry Problems by Titu Andreescu and Zuming Feng sessions of USA IMO Team 104 Number Theory Problems by Titu Andreescu, Zuming Feng and Dorin Andrica Selected Problems & Theorems in Elementary Mathematics by Shklyarsky, Yaglom Arithmetic & Algebra Problems Fifty Challenging problems in Probability by F. Mosteller Awesome Level – 7 (Practising for Perfection) Book Remark Carefully selected 130 The Wohascum County Problem Book by G. T. Gilbert, M. I. Krusemeyer, and L. C. Larson problems. The Math Problems Notebook by Valentin Boju and Louis Funar Ultimate Practise. A decent source of 360 Problems from Mathematical Contests by Titu Andreescu and Dorin Andrica practise IMO Compendium by Duˇsan Djuki´c, Vladimir Jankovi´c, Ivan Mati´c & Nikola Petrovi´ Past IMO till 2004 Problems from various Putnam & Beyond by by Titu Andreescu and Razvan Gelca nations & IMO Mother of all Problem The USSR Olympiad Problem Book by D.O. Shklarsky. N.N.Chentzov and l.M.Yaslom books. G. Strengthening of Brain by Recreational Mathematics: Book Remark Creativity of Ramanujan [For Primary & Middle School] by P. K. Srinivasan Fascinating insights of Ramanujan’s [Association of Mathematics Teachers of India (AMTI)] Notebooks (mainly “Magic Squares”) The Wonder World of Kaprekar Numbers by AMTI (Editor – R. Athmaraman) Generating Special number patterns Indianised versions of all classical Fun and Fundamentals of Mathematics by J. V. Narlikar and M. Narlikar puzzles from around the world. I loved reading this!! Mathematical Recreations & Essays by W. W. R. Ball & H. S. M. Coxeter Similar to the book by Narlikar The Moscow Puzzles: 359 Mathematical Recreations by Boris Kordemsky A USSR classic! Algebra Can be Fun by Ya. I. Perelman Another USSR Classic One of many awesome books by Puzzles to Puzzle You by Shakuntala Devi “Human Computer” Collection of Puzzles from “Scientific The Unexpected Hanging & other mathematical Diversions by M. Gardner American” journal Wheels, Life & Other Mathematical Amusements by M. Gardener These add new spirit to puzzle solving Entertaining Mathematical Puzzles by M. Gardener Really challenging problems with Challenging Mathematical Teasers by J. A. H. Hunter elementary solutions Good Old-Fashioned Challenging Puzzles by H. E. Dudeney Good classifications What is the name of this book? by R. M. Smullyan Awesome puzzles Mathematical Puzzles of Sam Loyd by Sam Loyd (edited by M. Gardener) Classic puzzles from puzzle master Dimensions (Videos made for inducing understanding of fourth dimension) Visit : www.dimensions-math.org Gaurish Korpal Preparing for Mathematics Olympiad 5 H. Suggested readings for those who still want more Serious Mathematics (for fun!): Book Remark Geometry and the Imagination by D. Hilbert A classic ! A treatise on Problems of Maxima and Minima solved by Algebra by Ramchundra Mind blowing book! (AMTI) The Penguin Dictionary of Curious & Interesting Numbers by David Wells A concise collection of various interesting terms generally not The Penguin Dictionary of Curious & Interesting Geometry by David Wells found in textbooks Proofs Without Words (Vol. I & II) by Roger B. Nelsen Just “see” simple proofs Mathematical Gems (Vol. I,II,III) by Ross Honsberger Higher level study of various topics Mathematical Diamonds by Ross Honsberger of mathematics Ingenuity in mathematics by Ross Honsberger 100 Great Problems of Elementary mathematics by Heinrich Dorrie Classical Problems The Book of Numbers by J. H. Conway and R. K. Guy Fabulous book The Art of Counting by Paul Erdȍs A collection of work of Paul Erdȍs Proofs from the book by M. Aigner and G. M. Ziegler Dedicated to Paul Erdȍs Echoes from Resonance - Number Theory by S. A. Shirali & C. S. Yogananda Articles from Resonance Journal Excursions in Calculus: An Interplay of the continuous and the discrete by Young ----- Geometric Etudes in Combinatorial Mathematics by Alexander Soifer ----- Art gallery Theorems & Algorithms by Rourke Research work The Drunkard’s Walk: How Randomness Rules our Lives by Leonard Mlodinow Enjoyable reading. Symmetry – A Journey into the Patterns of Nature by Marcus du Sautoy Beautifully written. The Code Book: How to make it, break it, hack it, crack it by Simon Singh Enjoyable reading. I. Web Resources: 1. Cut The Knot (free maths resources ): http://www.cut-the-knot.org 2. Questions from Past Mathematical Competitions can be downloaded from: http://artofproblemsolving.com 3. A collection of 4,100 Olympiad problems and about 1,700 other problems : http://mks.mff.cuni.cz/kalva/ 4. Shyam Sunder Gupta’s Recreational Mathematics resources : http://shyamsundergupta.com 5. The Prime Puzzles & Problems Connection by Carlos Rivera: http://www.primepuzzles.net/ 6. Math Forum Library: http://mathforum.org/library 7. Visual Calculus Portal: http://archives.math.utk.edu/visual.calculus/ 8. Wolfram Mathworld [a good source of reference] : http://mathworld.wolfram.com 9. Terence Tao’s Blog: http://terrytao.wordpress.com 10. Shailesh Shirali’s Blog: http://Joyofmathshirali.blogspot.in 11. Vipul Naik’s Website: http://www.vipulnaik.com J. Useful Mathematics Periodicals: Mathematics periodicals play a vital role, as they keep you updated and a chance to go through different prospective of good writers (e.g. I learnt “Principle of Inclusion & Exclusion” through an article written by B. Sury in “At Right Angles” Magazine) 1. At Right Angles : www.teachersofindia.org/en/periodicals/at-right-angles [A goldmine for high school students] 2. Bona Mathematica by Bhaskaracharya Pratishthana, 56/14, Vishnupant Damle Path, Erandavana, Pune 3. Mathematical Reflections : http://www.awesomemath.org/ 4. The Mathematics Student by Indian Mathematical Society, Department of Mathematics, University of Pune 5. The Mathematics Teacher by The Association of Mathematics Teachers of India, Chennai , India 6. Crux Mathematicorum : http://cms.math.ca/crux/ [older issues are available online] K. Useful Softwares: 1. Computer Algebra System: i. Mathematica (paid) ii. Sage (free) 2. Interactive Geometry Software: i. Geometer’s Sketchpad (paid) ii. GeoGebra (free) 3. Spread Sheet: i. Microsoft Office Excel (paid) ii. Libre Office Calc (free) – Gaurish Korpal 9th May 2014 Gaurish Korpal Preparing for Mathematics Olympiad 6