Titu Andreescu Bogdan Enescu

MathematicalOlympiadTreasures

Second Edition

Titu AndreescuSchool of Natural Sciences and

MathematicsUniversity of Texas at DallasRichardson, TX 75080USAtitu.andreescu@utdallas.edu

Bogdan EnescuDepartment of MathematicsBP Hasdeu National CollegeBuzau 120218Romaniabogdanenescu@buzau.ro

ISBN 978-0-8176-8252-1 e-ISBN 978-0-8176-8253-8DOI 10.1007/978-0-8176-8253-8Springer 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, 2011All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission 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 inconnection 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 arenot identified as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.

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Preface

Mathematical Olympiads have a tradition longer than one hundred years. The firstmathematical competitions were organized in Eastern Europe (Hungary and Ro-mania) by the end of the 19th century. In 1959 the first International Mathemati-cal 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 ofparticipating countries was 97, and the number of students 517.

Obviously, the number of young students interested in mathematics and math-ematical competitions is nowadays greater than ever. It is sufficient to visit somemathematical forums on the net to see that there are tens of thousands registeredusers and millions of posts.

When we were thinking about writing this book, we asked ourselves to whom itwill be addressed. Should it be the beginner student, who is making the first stepsin discovering the beauty of mathematical problems, or, maybe, the more advancedreader, already trained in competitions. Or, why not, the teacher who wants to use agood 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 arerather easy, but definitely not exercises; some are quite difficult, being a challengeeven for Olympiad experts.

Most of the problems come from various mathematical competitions (the Interna-tional Mathematical Olympiad, The Tournament of the Towns, national Olympiads,regional Olympiads). Some problems were created by the authors and some arefolklore.

The problems are grouped in three chapters: Algebra, Geometry and Trigono-metry, and Number Theory and Combinatorics. This is the way problems are clas-sified 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 se-lected problems and an extensive discussion of their solutions. At the end of eachsection the reader will find a number of proposed problems, whose complete solu-tions are presented in the second part of the book.

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vi Preface

We encourage the beginning reader to carefully examine the problems solved atthe beginning of each section and try to solve the proposed problems before ex-amining 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 proposedproblems.

In the second edition of the book, we added two new sections in Chaps. 1 and 3,and more than 60 new problems with complete solutions.

Titu AndreescuBogdan Enescu

University of Texas at DallasB.P. Hasdeu National College

Contents

Part I Problems

1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.1 An Algebraic Identity . . . . . . . . . . . . . . . . . . . . . . . . 31.2 CauchySchwarz Revisited . . . . . . . . . . . . . . . . . . . . . 71.3 Easy Ways Through Absolute Values . . . . . . . . . . . . . . . . 111.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Take the Conjugate! . . . . . . . . . . . . . . . . . . . . . . . . . 171.6 Inequalities with Convex Functions . . . . . . . . . . . . . . . . . 201.7 Induction at Work . . . . . . . . . . . . . . . . . . . . . . . . . . 241.8 Roots and Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 271.9 The Rearrangements Inequality . . . . . . . . . . . . . . . . . . . 31

2 Geometry and Trigonometry . . . . . . . . . . . . . . . . . . . . . . 372.1 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 372.2 An Interesting Locus . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 Cyclic Quads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Equiangular Polygons . . . . . . . . . . . . . . . . . . . . . . . . 502.5 More on Equilateral Triangles . . . . . . . . . . . . . . . . . . . . 542.6 The Carpets Theorem . . . . . . . . . . . . . . . . . . . . . . . 582.7 Quadrilaterals with an Inscribed Circle . . . . . . . . . . . . . . . 622.8 Dr. Trig Learns Complex Numbers . . . . . . . . . . . . . . . . . 66

3 Number Theory and Combinatorics . . . . . . . . . . . . . . . . . . 713.1 Arrays of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 713.2 Functions Defined on Sets of Points . . . . . . . . . . . . . . . . . 743.3 Count Twice! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.4 Sequences of Integers . . . . . . . . . . . . . . . . . . . . . . . . 813.5 Equations with Infinitely Many Solutions . . . . . . . . . . . . . . 853.6 Equations with No Solutions . . . . . . . . . . . . . . . . . . . . 883.7 Powers of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 903.8 Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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viii Contents

3.9 The Marriage Lemma . . . . . . . . . . . . . . . . . . . . . . . . 96

Part II Solutions

4 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.1 An Algebraic Identity . . . . . . . . . . . . . . . . . . . . . . . . 1014.2 CauchySchwarz Revisited . . . . . . . . . . . . . . . . . . . . . 1094.3 Easy Ways Through Absolute Values . . . . . . . . . . . . . . . . 1164.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.5 Take the Conjugate! . . . . . . . . . . . . . . . . . . . . . . . . . 1244.6 Inequalities with Convex Functions . . . . . . . . . . . . . . . . . 1304.7 Induction at Work . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.8 Roots and Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 1384.9 The Rearrangements Inequality . . . . . . . . . . . . . . . . . . . 144

5 Geometry and Trigonometry . . . . . . . . . . . . . . . . . . . . . . 1495.1 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . 1495.2 An Interesting Locus . . . . . . . . . . . . . . . . . . . . . . . . . 1565.3 Cyclic Quads . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1635.4 Equiangular Polygons . . . . . . . . . . . . . . . . . . . . . . . . 1715.5 More on Equilateral Triangles . . . . . . . . . . . . . . . . . . . . 1765.6 The Carpets Theorem . . . . . . . . . . . . . . . . . . . . . . . 1815.7 Quadrilaterals with an Inscribed Circle . . . . . . . . . . . . . . . 1855.8 Dr. Trig Learns Complex Numbers . . . . . . . . . . . . . . . . . 193

6 Number Theory and Combinatorics . . . . . . . . . . . . . . . . . . 1976.1 Arrays of Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1976.2 Functions Defined on Sets of Points . . . . . . . . . . . . . . . . . 2016.3 Count Twice! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2056.4 Sequences of Integers . . . . . . . . . . . . . . . . . . . . . . . . 2136.5 Equations with Infinitely Many Solutions . . . . . . . . . . . . . . 2196.6 Equations with No Solutions . . . . . . . . . . . . . . . . . . . . 2246.7 Powers of 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2296.8 Progressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2376.9 The Marriage Lemma . . . . . . . . . . . . . . . . . . . . . . . . 244

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Index of Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

Index to the Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253

Part IProblems

Chapter 1Algebra

1.1 An Algebraic Identity

A very useful algebraic identity is derived by considering the following problem.

Problem 1.1 Factor a3 + b3 + c3 3abc.

Solution Let P denote the polynomial with roots a, b, c:

P(X) = X3 (a + b + c)X2 + (ab + bc + ca)X abc.Because a, b, c satisfy the equation P(x) = 0, we obtain

a3 (a + b + c)a2 + (ab + bc + ca)a abc = 0,b3 (a + b + c)b2 + (ab + bc + ca)b abc = 0,c3 (a + b + c)c2 + (ab + bc + ca)c abc = 0.

Adding up these three equalities yields

a3 + b3 + c3 (a + b + c)(a2 + b2 + c2)+ (ab + bc + ca)(a + b + c) 3abc = 0.Hence

a3 + b3 + c3 3abc = (a + b + c)(a2 + b2 + c2 ab bc ca