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  • 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 interestinginequalities 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 Rolles 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 andr = 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. Polyas 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: Phm Vn Thun, L V

    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

    Phm Vn Thun, L V

    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 t3

    )

    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 hasfour 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, ..., ansatisfy a1 + a2 + + an = s, s < 2, define

    f (a1, a2, ..., an) = 1i< jn

    1

    1 (

    ai+a j2

    )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. Provethat

    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.

    -

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

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