Inequalities: A Mathematical Olympiad Approach

217 Pages·2009·1.063 MB·English

by  Radmila Bulajich Manfrino, José Antonio Gómez Ortega, Rogelio Valdez Delgado

Radmila Bulajich Manfrino
José Antonio Gómez Ortega
Rogelio Valdez Delgado
Inequalities
A Mathematical Olympiad Approach
Birkhäuser
Basel · Boston · Berlin
Autors:
Radmila Bulajich Manfrino José Antonio Gómez Ortega
Rogelio Valdez Delgado Departamento de Matemàticas
Facultad de Ciencias Facultad de Ciencias, UNAM
Universidad Autónoma Estado de Morelos Universidad Nacional Autónoma de México
Av. Universidad 1001 Ciudad Universitaria
Col. Chamilpa 04510 México, D.F.
62209 Cuernavaca, Morelos México
México e-mail: [email protected]
e-mail: [email protected]
[email protected]
2000 Mathematical Subject Classification 00A07; 26Dxx, 51M16
Library of Congress Control Number: 2009929571
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ISBN 978-3-0346-0049-1 Birkhäuser Verlag, Basel – Boston – Berlin
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Introduction
This book is intended for the Mathematical Olympiad students who wish to pre-
pare for the study of inequalities, a topic now of frequent use at various levels
of mathematical competitions. In this volume we present both classic inequalities
and the more useful inequalities for confronting and solving optimization prob-
lems. An important part of this book deals with geometric inequalities and this
fact makes a big difference with respect to most of the books that deal with this
topic in the mathematical olympiad.
The book has been organized in four chapters which have each of them a
different character. Chapter 1 is dedicated to present basic inequalities. Most of
them are numerical inequalities generally lacking any geometric meaning. How-
ever, where it is possible to provide a geometric interpretation, we include it as
we go along. We emphasize the importance of some of these inequalities, such as
theinequalitybetweenthearithmetic meanandthe geometricmean,the Cauchy-
Schwarzinequality,the rearrangementinequality,the Jenseninequality,theMuir-
head theorem, among others. For all these, besides giving the proof, we present
several examples that show how to use them in mathematical olympiad prob-
lems. We also emphasize how the substitution strategy is used to deduce several
inequalities.
ThemaintopicinChapter2istheuseofgeometricinequalities.Thereweap-
plybasic numericalinequalities,asdescribedinChapter1,to geometricproblems
to provide examples of how they are used. We also work out inequalities which
have a strong geometric content, starting with basic facts, such as the triangle
inequality and the Euler inequality. We introduce examples where the symmetri-
cal properties of the variables help to solve some problems. Among these, we pay
special attention to the Ravi transformation and the correspondence between an
inequality in terms of the side lengths of a triangle a, b, c and the inequalities
that correspondto the terms s, r and R, the semiperimeter, the inradius and the
circumradius of a triangle, respectively. We also include several classic geometric
problems, indicating the methods used to solve them.
In Chapter 3 we present one hundred and twenty inequality problems that
have appeared in recent events, covering all levels, from the national and up to
the regional and international olympiad competitions.
vi Introduction
In Chapter 4 we provide solutions to each of the two hundred and ten exer-
cises in Chapters 1 and 2, and to the problems presented in Chapter 3. Most of
the solutions to exercises or problems that have appeared in international math-
ematical competitions were taken from the official solutions provided at the time
of the competitions. This is why we do not give individual credits for them.
Some ofthe exercisesandproblemsconcerninginequalities canbe solvedus-
ingdifferenttechniques,thereforeyouwillfindsomeexercisesrepeatedindifferent
sections. This indicates that the technique outlined in the corresponding section
can be used as a tool for solving the particular exercise.
The material presented in this book has been accumulated over the last fif-
teen years mainly during work sessions with the students that won the national
contest of the Mexican Mathematical Olympiad. These students were develop-
ing their skills and mathematical knowledge in preparation for the international
competitions in which Mexico participates.
WewouldliketothankRafaelMart´ınezEnr´ıquez,LeonardoIgnacioMart´ınez
Sandoval, David Mireles Morales, Jesu´s Rodr´ıguez Viorato and Pablo Sobero´n
Bravofor their carefulrevisionof the text and helpful comments for the improve-
ment of the writing and the mathematical content.
Contents
Introduction vii
1 Numerical Inequalities 1
1.1 Order in the real numbers . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The quadratic function ax2+2bx+c . . . . . . . . . . . . . . . . . 4
1.3 A fundamental inequality,
arithmetic mean-geometric mean . . . . . . . . . . . . . . . . . . . 7
1.4 A wonderful inequality:
The rearrangementinequality . . . . . . . . . . . . . . . . . . . . . 13
1.5 Convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.6 A helpful inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.7 The substitution strategy . . . . . . . . . . . . . . . . . . . . . . . 39
1.8 Muirhead’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2 Geometric Inequalities 51
2.1 Two basic inequalities . . . . . . . . . . . . . . . . . . . . . . . . . 51
2.2 Inequalities between the sides of a triangle . . . . . . . . . . . . . . 54
2.3 The use of inequalities in the geometry of the triangle . . . . . . . 59
2.4 Euler’s inequality and some applications . . . . . . . . . . . . . . . 66
2.5 Symmetric functions of a, b and c . . . . . . . . . . . . . . . . . . . 70
2.6 Inequalities with areas and perimeters . . . . . . . . . . . . . . . . 75
2.7 Erdo˝s-MordellTheorem . . . . . . . . . . . . . . . . . . . . . . . . 80
2.8 Optimization problems . . . . . . . . . . . . . . . . . . . . . . . . . 88
3 Recent Inequality Problems 101
4 Solutions to Exercises and Problems 117
4.1 Solutions to the exercises in Chapter 1 . . . . . . . . . . . . . . . . 117
4.2 Solutions to the exercises in Chapter 2 . . . . . . . . . . . . . . . . 140
4.3 Solutions to the problems in Chapter 3 . . . . . . . . . . . . . . . . 162
Notation 205
viii Contents
Bibliography 207
Index 209
Chapter 1
Numerical Inequalities
1.1 Order in the real numbers
A very important property of the real numbers is that they have an order. The
orderoftherealnumbersenablesustocomparetwonumbersandtodecidewhich
one of them is greater or whether they are equal. Let us assume that the real
numbers system contains a set P, which we will call the set of positive numbers,
and we will express in symbols x > 0 if x belongs to P. We will also assume the
following three properties.
Property 1.1.1. Every real number x has one and only one of the following prop-
erties:
(i) x=0,
(ii) x∈P (that is, x>0),
(iii) −x∈P (that is, −x>0).
Property1.1.2. Ifx,y ∈P,then x+y ∈P (in symbolsx>0,y >0⇒x+y >0).
Property 1.1.3. If x, y ∈P, then xy ∈P (in symbols x>0, y >0⇒xy >0).
Ifwetakethe“realline” asthegeometricrepresentationoftherealnumbers,
bythis wemeana directedline where the number “0”hasbeen locatedandserves
to divide the real line into two parts, the positive numbers being on the side
containingthenumberone“1”.Ingeneralthenumberoneissetontherighthand
side of 0. The number 1 is positive, because if it were negative, since it has the
property that 1·x=x for every x, we would have that any number x(cid:4)=0 would
satisfy x∈P and −x∈P, which contradicts property 1.1.1.
Now we can define the relation a is greater than b if a−b ∈ P (in symbols
a>b). Similarly,a is smaller than b if b−a∈P (in symbols a<b). Observethat
2 Numerical Inequalities
a<b is equivalent to b>a. We can also define that a is smaller than or equal to
b if a<b or a=b (using symbols a≤b).
We will denote by R the setof realnumbers and by R+ the setP ofpositive
real numbers.
Example 1.1.4. (i) If a<b and c is any number, then a+c<b+c.
(ii) If a<b and c>0, then ac<bc.
In fact, to prove (i) we see that a+c < b+c ⇔ (b+c)−(a+c) > 0 ⇔
b−a > 0 ⇔ a < b. To prove (ii), we proceed as follows: a < b ⇒ b−a > 0 and
since c>0, then (b−a)c>0, therefore bc−ac>0 and then ac<bc.
Exercise 1.1. Given two numbers a and b, exactly one of the following assertions
is satisfied, a=b, a>b or a<b.
Exercise 1.2. Prove the following assertions.
(i) a<0, b<0⇒ab>0.
(ii) a<0, b>0⇒ab<0.
(iii) a<b, b<c⇒a<c.
(iv) a<b, c<d⇒a+c<b+d.
(v) a<b⇒−b<−a.
1
(vi) a>0⇒ >0.
a
1
(vii) a<0⇒ <0.
a
a
(viii) a>0, b>0⇒ >0.
b
(ix) 0<a<b, 0<c<d⇒ac<bd.
(x) a>1⇒a2 >a.
(xi) 0<a<1⇒a2 <a.
Exercise 1.3. (i) If a>0, b>0 and a2 <b2, then a<b.
(ii) If b>0, we have that a >1 if and only if a>b.
b
The absolute value of a realnumber x, which is denoted by |x|, is defined as
(cid:2)
x if x≥0,
|x|=
−x ifx<0.
Geometrically,|x|isthedistanceofthenumberx(ontherealline)fromtheorigin
0. Also, |a−b| is the distance between the real numbers a and b on the real line.
1.1 Order in the real numbers 3
Exercise 1.4. For any real numbers x, a and b, the following hold.
(i) |x|≥0, and is equal to zero only when x=0.
(ii) |−x|=|x|.
(iii) |x|2 =x2.
(iv) |ab|=|a||b|.
(cid:3) (cid:3)
(cid:3)a(cid:3) |a|
(v) (cid:3) (cid:3)= , with b(cid:4)=0.
b |b|
Proposition1.1.5(Triangleinequality). The triangle inequality states that for any
pair of real numbers a and b,
|a+b|≤|a|+|b|.
Moreover, the equality holds if and only if ab≥0.
Proof. Both sides of the inequality are positive;then using Exercise1.3 it is suffi-
cient to verify that |a+b|2 ≤(|a|+|b|)2:
|a+b|2 =(a+b)2 =a2+2ab+b2 =|a|2+2ab+|b|2 ≤|a|2+2|ab|+|b|2
=|a|2+2|a||b|+|b|2 =(|a|+|b|)2.
In the previous relations we observe only one inequality, which is obvious since
ab≤|ab|.Notethat,whenab≥0,wecandeducethatab=|ab|=|a||b|,andthen
the equality holds. (cid:2)
The general form of the triangle inequality for real numbers x1, x2,...,xn,
is
|x1+x2+···+xn|≤|x1|+|x2|+···+|xn|.
Theequalityholdswhenallxi’shavethesamesign.Thiscanbeprovedinasimilar
way or by the use of induction. Another version of the last inequality, which is
used very often, is the following:
|±x1±x2±···±xn|≤|x1|+|x2|+···+|xn|.
Exercise 1.5. Let x, y, a, b be real numbers, prove that
(i) |x|≤b⇔−b≤x≤b,
(ii) ||a|−|b||≤|a−b|,
(iii) x2+xy+y2 ≥0,
(iv) x>0, y >0⇒x2−xy+y2 >0.
Exercise 1.6. For real numbers a, b, c, prove that
|a|+|b|+|c|−|a+b|−|b+c|−|c+a|+|a+b+c|≥0.