Titu Andreescu Bogdan Enescu Mathematical Olympiad Treasures Second Edition Titu Andreescu School of Natural Sciences and Mathematics University of Texas at Dallas Richardson, TX 75080 USA titu.andreescu@utdallas.edu Bogdan Enescu Department of Mathematics “BP Hasdeu” National College Buzau 120218 Romania bogdanenescu@buzau.ro ISBN 978-0-8176-8252-1 e-ISBN 978-0-8176-8253-8 DOI 10.1007/978-0-8176-8253-8 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2011938426 Mathematics Subject Classification (2010): 00A05, 00A07, 05-XX, 11-XX, 51-XX, 97U40 © Springer Science+Business Media, LLC 2004, 2011 All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper Springer is part of Springer Science+Business Media (www.birkhauser-science.com) Preface Mathematical Olympiads have a tradition longer than one hundred years The first mathematical competitions were organized in Eastern Europe (Hungary and Romania) by the end of the 19th century In 1959 the first International Mathematical Olympiad was held in Romania Seven countries, with a total of 52 students, attended that contest In 2010, the IMO was held in Kazakhstan The number of participating countries was 97, and the number of students 517 Obviously, the number of young students interested in mathematics and mathematical competitions is nowadays greater than ever It is sufficient to visit some mathematical forums on the net to see that there are tens of thousands registered users and millions of posts When we were thinking about writing this book, we asked ourselves to whom it will be addressed Should it be the beginner student, who is making the first steps in discovering the beauty of mathematical problems, or, maybe, the more advanced reader, already trained in competitions Or, why not, the teacher who wants to use a good set of problems in helping his/her students prepare for mathematical contests We have decided to take the hard way and have in mind all these potential readers Thus, we have selected Olympiad problems of various levels of difficulty Some are rather easy, but definitely not exercises; some are quite difficult, being a challenge even for Olympiad experts Most of the problems come from various mathematical competitions (the International Mathematical Olympiad, The Tournament of the Towns, national Olympiads, regional Olympiads) Some problems were created by the authors and some are folklore The problems are grouped in three chapters: Algebra, Geometry and Trigonometry, and Number Theory and Combinatorics This is the way problems are classified at the International Mathematical Olympiad In each chapter, the problems are clustered by topic into self-contained sections Each section begins with elementary facts, followed by a number of carefully selected problems and an extensive discussion of their solutions At the end of each section the reader will find a number of proposed problems, whose complete solutions are presented in the second part of the book v vi Preface We encourage the beginning reader to carefully examine the problems solved at the beginning of each section and try to solve the proposed problems before examining the solutions provided at the end of the book As for the advanced reader, our advice is to try finding alternative solutions and generalizations of the proposed problems In the second edition of the book, we added two new sections in Chaps and 3, and more than 60 new problems with complete solutions University of Texas at Dallas “B.P Hasdeu” National College Titu Andreescu Bogdan Enescu Contents Part I Problems Algebra 1.1 An Algebraic Identity 1.2 Cauchy–Schwarz Revisited 1.3 Easy Ways Through Absolute Values 1.4 Parameters 1.5 Take the Conjugate! 1.6 Inequalities with Convex Functions 1.7 Induction at Work 1.8 Roots and Coefficients 1.9 The Rearrangements Inequality 3 11 14 17 20 24 27 31 Geometry and Trigonometry 2.1 Geometric Inequalities 2.2 An Interesting Locus 2.3 Cyclic Quads 2.4 Equiangular Polygons 2.5 More on Equilateral Triangles 2.6 The “Carpets” Theorem 2.7 Quadrilaterals with an Inscribed Circle 2.8 Dr Trig Learns Complex Numbers 37 37 40 45 50 54 58 62 66 Number Theory and Combinatorics 3.1 Arrays of Numbers 3.2 Functions Defined on Sets of Points 3.3 Count Twice! 3.4 Sequences of Integers 3.5 Equations with Infinitely Many Solutions 3.6 Equations with No Solutions 3.7 Powers of 3.8 Progressions 71 71 74 77 81 85 88 90 93 vii viii Contents 3.9 The Marriage Lemma Part II 96 Solutions Algebra 4.1 An Algebraic Identity 4.2 Cauchy–Schwarz Revisited 4.3 Easy Ways Through Absolute Values 4.4 Parameters 4.5 Take the Conjugate! 4.6 Inequalities with Convex Functions 4.7 Induction at Work 4.8 Roots and Coefficients 4.9 The Rearrangements Inequality 101 101 109 116 120 124 130 134 138 144 Geometry and Trigonometry 5.1 Geometric Inequalities 5.2 An Interesting Locus 5.3 Cyclic Quads 5.4 Equiangular Polygons 5.5 More on Equilateral Triangles 5.6 The “Carpets” Theorem 5.7 Quadrilaterals with an Inscribed Circle 5.8 Dr Trig Learns Complex Numbers 149 149 156 163 171 176 181 185 193 Number Theory and Combinatorics 6.1 Arrays of Numbers 6.2 Functions Defined on Sets of Points 6.3 Count Twice! 6.4 Sequences of Integers 6.5 Equations with Infinitely Many Solutions 6.6 Equations with No Solutions 6.7 Powers of 6.8 Progressions 6.9 The Marriage Lemma 197 197 201 205 213 219 224 229 237 244 Glossary 249 Index of Notations 251 Index to the Problems 253 Part I Problems 6.8 Progressions 239 or k≤ N a − + r1 r1 It follows that the number of terms of the first progression belonging to the set {1, 2, , N } equals a N − +1 r1 r1 Similarly, we deduce that the number of terms of the progression with common difference ri belonging to the set {1, 2, , N } equals N a − +1 ri ri Since the progressions form a partition of the set of positive integers, we must have n i=1 N a − + = N ri ri Using the inequality x ≤ x < x + 1, we obtain n N≤ i=1 N a − +1 n2 be an integer We define Bk = + m k , for k = 1, 2, , n Observe that for k ≥ Bk > + k m For k ≤ n we have k k! k(k − 1) + ··· + + m k!mk 2!m k n(n − 1) · · · (n − k + 1) n(n − 1) ≤1+ + + ··· + m k!mk 2!m Bk = + 6.8 Progressions 241 =1+ n(n − 1) n(n − 1) · · · (n − k + 1) k + + ··· + m m 2!m k!mk−1