Sample problems fromOlympiad Inequalities Book

This book is intended as a useful resource for high school and college stu-

dents who are training for national or international mathematical competi-

tions. But anybody who is interested in elementary mathematical inequali-

ties may find this book useful. This problem solving book is divided into

six chapters, containing more than fifty topics of interest to mathematical

olympiad contestants and coaches, demonstrating ideas and strategies in solv-

ing inequalities. The reader will find in the book clever applications of well

known results as well as powerful original methods, each is explained and

illustrated by carefully selected problems.

Chapter 1 Inequalities between means

This chapter starts with the fundamental fact x2 ≥ 0, upon which many interesting

inequalities are derived. It also serves to emphasize that powerful results can be obtained

by little means.

Chapter 2 Cauchy-Schwars

The classical result of Cauchy-Schwarz is revisited with examples illustrating sophis-

ticated, at time surprising, ways to apply the inequality. Other classical inequalities such

as those of Chebyshev and Holder are also discussed and shown how they might cooperate

with Cauchy-Schwarz inequality.

Chapter 3 Convexity

This chapter utilises calculus in solving inequalities. Based on simpe properties of

linear and convex functions, systematic methods are derived to tackle some advanced

problems. Also discussed is tangent line method, which gives a geometric interpretation

of bounds.

1

Chapter 4 Homogenous inequalities

Homogeneous inequalities constitutes a large class of inequality problems. This chap-

ter discusses various approaches to solving this class of inequalities, including the tech-

niques of homogenization, normalisation, the application of Rolle’s theorem to reduce

the number of variables, the use of limits and partitions, quadratic estimations, and estab-

lishing new bounds through isolated fudging. Especially in focus are powerful techniques

to solve inequalities by the change of variables p = a + b + c, q = ab + bc + ca and

r = abc and by transforming them to one of the following forms

(1) x(a − b)(a − c) + y(b − c)(b − a) + z(c − a)(c − b) ≥ 0,

(2) x(a − b)2 + y(b − c)2 + z(c − a)2 ≥ 0,

(3) M(a − b)2 + N(c − a)(c − b) ≥ 0.

All three, four variable symmetric polynomial inequalities can be solved using ideas in

this chapter.

Chapter 5 The method of Mixing Variables

The method of mixing variables has been used in various forms for decades - an ex-

ample is G. Polya’s delightful proof of the AM-GM inequalities. This chapter examines

this idea in depth with extension in different directions. The first three sections explain

why mixing variables work, give hints to find approriate variables to mix by taking equal-

ity cases into consideration. The most important results in this chapter are two theorems

which facilitates solutions for a large class of multi-variable inequalities.

Chapter 6 Further Topics and problems with solutions

The chapter starts with miscellaneous indenpendent topics touching upon various

aspects of solving inequalities. The discussion includes the interplay between trogono-

metric and algebraic substitution, absolute values, inequalities with special equality cases

and inequalities with ordered sequences.

Authors: Phạm Văn Thuận, Lê Vĩ

Hanoi University of Science, Vietnam

332 pages, LATEX typset, soft cover

Price: 12 (twelve) USD

It is available for sale in La Thanh Hotel where deputy leaders and contestants stay.

3

4

b

b

b

b

Phạm Văn Thuận, Lê Vĩ

Olympiad Inequalities

Introduction to the art of solving inequalities

bbb

f (x) = (x − a)(x − b)(x − c)

y = f (x)

b

b

O

a b c x

y

f

(

1 − t

3

)

f

(

1 + t

3

)

VIETNAM NATIONAL UNIVERSITY PRESS

5

The 11 out of 600 problems

Problem 1. Prove that if x, y, z are real numbers, then

7(x4 + y4 + z4) + 10(x3y + y3z + z3x) ≥ 0.

Problem 2. Prove that a, b, c are positve real numbers, then

a√

b2 + 14 bc + c2

+b

√

c2 + 14 ca + a2

+c

√

a2 + 14 ab + b2

≥ 2.

Problem 3. Let a, b, c, d be non-negative real numbers such that

a2 + b2 + c2 + d2 = 1.

Prove that

a + b + c + d ≥ a3 + b3 + c3 + d3 + ab + bc + cd + da + ac + bd.

Problem 4. Suppose that p, q, r, s are real numbers such that the following equation has

four roots (not neccessarily distinct)

x4 − px3 + qx2 − rx + s = 0.

Prove that (p2 − 2q)5/2 + 8ps ≥ 4(p2 − 2q)r.

Problem 5. Let n be a positive integers, n ≥ 2. Non-negative real numbers a1, a2, ..., an

satisfy a1 + a2 + · · ·+ an = s, s < 2, define

f (a1, a2, ..., an) = ∑1≤i< j≤n

1

1 −(

ai+a j

2

)2.

Prove that

1

2n(n − 1)/

[

1 −(

s

n

)2]

≤ f (a1, a2, ..., an) ≤n − 1

1 − s2/4+

1

2(n − 1)(n − 2).

Determine cases of equality.

Problem 6. Let a, b, c, d be non-negative real numbers such that a + b + c + d = 2.

Prove that

ab(a2 + b2 + c2) + bc(b2 + c2 + d2) + cd(c2 + d2 + a2) + da(d2 + a2 + b2) ≤ 2.

Problem 7. Prove that if x, y, z are postive real numbers then

x

y+

y

z+

z

x≥

(

x2 + y2 + z2

xy + yz + zx

)2/3

.

6

Problem 8. Let r, a, b, c be positive real numbers, put p = 2r − 3√

r + 2. Prove that

a

pa + rb + c+

b

pb + rc + a+

c

pc + ra + b≤

1

1 −√

r + r.

Problem 9. Prove that if a, b, c, d are non-negative real numbers then

1

a2 + b2 + c2+

1

b2 + c2 + d2+

1

c2 + d2 + a2+

1

d2 + a2 + b2≥

12

(a + b + c + d)2.

Problem 10. Let x, y, z be non-negative real numbers such that x2 + y2 + z2 = 1. Prove

that

∑cyclic

√

1 − xy.√

1 − yz ≥ 2.

Problem 11. Prove that if x, y, z ∈ R then

x(x + y)3 + y(y + z)3 + z(z + x)3 ≥8

27(x + y + z)4.

Do you think problem 10 can be solved using only Cauchy-Schwarz inequal-

ity? If not, have a look at this book. Do you believe that a solution for problem 9

is just a few line long with only simple reasoning? Problem 4 and 5 look intimi-

dating but we have strategies to deal with such types. Problem 3 is selected from

the section on symmetric polynomials which contains a large number of original

and nice inequalities. There is a very short and clever solution to problem 6. Can

you figure it out? Problem 11 is a refinement of a well-known inequality.

This book offers a lot more beautiful problems together with powerful meth-

ods and strategies.

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Do not miss this book when you are in Hanoi for IMO.