Tuyển tập đề thi vô địch bất đẳng thức thế giới P1

Preface This work blends together classic inequality results with brand new problems, some of which devised only a few days ago.. Finally, but not in the end, we would like to extend our

Preface

This work blends together classic inequality results with brand new problems, some of which devised only a few days ago What could be special about it when so many inequality problem books have already been written? We strongly believe that even if the topic we plunge into is so general and popular our book is very different

Of course, it is quite easy to say this, so we will give some supporting arguments This book contains a large variety of problems involving inequalities, most of them difficult, questions that became famous in competitions because of their beauty and difficulty And, even more importantly, throughout the text we employ our own solutions and propose a large number of new original problems There are memorable problems

in this book and memorable solutions as well This is why this work will clearly appeal to students who are used to use Cauchy-Schwarz as a verb and want to further improve their algebraic skills and techniques They will find here tough problems, new results, and even problems that could lead to research The student who is not

as keen in this field will also be exposed to a wide variety of moderate and easy problems, ideas, techniques, and all the ingredients leading to a good preparation for mathematical contests Some of the problems we chose to present are known, but we have included them here with new solutions which show the diversity of ideas pertaining to inequalities Anyone will find here a challenge to prove his or her skills If

we have not convinced you, then please take a look at the last problems and hopefully you will agree with us

Finally, but not in the end, we would like to extend our deepest appreciation

to the proposers of the problems featured in this book and to apologize for not giving all complete sources, even though we have given our best Also, we would like

to thank Marian Tetiva, Dung Tran Nam, Constantin Tanadsescu, Calin Popa and Valentin Vornicu for the beautiful problems they have given us and for the precious comments, to Cristian Baba, George Lascu and Calin Popa, for typesetting and for the many pertinent observations they have provided

The authors

Contents Preface

CHAPTER 1

Problems

1 Prove that the inequality

Tournament of the Towns, 1993

5 Find the maximum value of the expression z° + y? + 2° — 3xyz where x? + y? + z* =1and z,y,z are real numbers

6 Let a,b,c, x,y,z be positive real numbers such that «+ y+ z= 1 Prove that

Old and New Inequalities 9 When does equality hold?

JBMO 2002 Shortlist

10 [ loan Tomescu | Let x,y, z > 0 Prove that

+z (1 + 32)(x + 8y)(y + 9z)(z + 6)

When do we have equality?

1

Sma:

Gazeta Matematica

11 [ Mihai Piticari, Dan Popescu | Prove that

5(a? + b2 + e2) < 6(aŸ + bỶ + c3) + 1,

for all a,b,c > O witha+6+c=1

12 [| Mircea Lascu | Let 71, %2, ,%, € R,n > 2 anda > 0 such that x +

2 T Prove that x; € [ *) , for all

14 For positive real numbers a,b,c such that abc < 1, prove that

+-+—=>a+b+c

b ec oa

15 [ Vasile Cirtoaje, Mircea Lascu | Let a,b,c,2,y,z be positive real numbers

such thata+a>b+y>c+zanda+b+c=2+y+2z Prove that ay+bzr > ac+2z

16 | Vasile Cirtoaje, Mircea Lascu | Let a,b,c be positive real numbers so that

abc = 1 Prove that

18 Prove that ifn > 3 and x%1,%2, ,%, > 0 have product 1, then

24 Let a,b,c > 0 such that at + 64 + ct < 2(a?b? + bc? + c?a”) Prove that

aŠ + bŠ + e? < 2(ab + be + ca)

Kvant, 1988

Old and New Inequalities 11

25 Let n > 2 and %1, ,2, be positive real numbers satisfying

28 D Olteanu | Let a,b,c be positive real numbers Prove that

RP—k+e @Ồ -ae+da2 @-adth = a+b+e

Proposed for the Balkan Mathematical Olympiad

31 | Adrian Zahariuc | Consider the pairwise distinct integers 71, %2, ,2n,

n > 0 Prove that

ei taste ta? > xr + 2283 + +: + ®ay#i + 2n — 3.

32 [| Murray Klamkin ] Find the maximum value of the expression #7 + #23 +

-+ + 9212, +2722, when 21, %2, ,U%n_—1,£n > 0 add up to 1 and n > 2

Crux Mathematicorum

33 Find the maximum value of the constant c such that for any

L1,02, -,%n,°*- > O for which xp4) > 4%, +4 + -+ 2% for any k, the inequality

Vi + fiz to + Vin <cVui Fant Fy

also holds for any n

36 Find the maximum value of the expression

a”(b+ec+đd) +~b?(e+d+a) +c(d+a+b) + dđ”(a+b+ e)

where a, b,c,d are real numbers whose sum of squares is 1

37 | Walther Janous | Let x,y, z be positive real numbers Prove that

rt Veter) 0+Vp)+ 2025) 2+ Veraery) ~

Crux Mathematicorum

38 Suppose that aj < ag < < Gy are real numbers for some integer n > 2 Prove that

Old and New Inequalities 13

AO Let @1,@2, ,@, > 1 be positive integers Prove that at least one of the numbers %/a2, %%/a3, , °»-¥/@n, *%/a1 is less than or equal to V3

Adapted after a well-known problem

41 | Mircea Lascu, Marian Tetiva | Let x,y,z be positive real numbers which

satisfy the condition

42 | Manlio Marangelli | Prove that for any positive real numbers 2, y, z,

3(z2 + y?z + 272) (ay? + yz? + za") > xuz(œ + + z)

43 [| Gabriel Dospinescu |] Prove that if a,b,c are real numbers such that

max{a, b,c} — min{a, b,c} < 1, then

1+a@?4+b% +3 + 6abe > 3a7b + 3bŠc + 3c2a

44 | Gabriel Dospinescu | Prove that for any positive real numbers a, b, c we have

48 [ Gabriel Dospinescu ] Prove that if /x + /y+./z = 1, then

53 [ Titu Andreescu |] Let n > 3 and a1,a2, ,@, be real numbers such that

ay +ag+ +4@, >nand a?+az+ +a2 > n* Prove that mar{a,,a2, ,An} > 2

Old and New Inequalities 15

56 Prove that if a,b,c > 0 have product 1, then

(a+ b)(6b+c)\(e+a) > 4(a+b+c-1)

MOSP, 2001

57 Prove that for any a,b,c > 0,

(a7 + b` + c?)(a+b— e)(b+e— a)(e+a— b) < abe(ab + be + ca)

58 [| D.P.Mavlo | Let a,b,c > 0 Prove that

œ”+ 0” +c” + dbc > min {5.5 +3}

Kvant, 1993

61 Prove that for any real numbers a, b,c we have the inequality

3 (1+42)?(1+ð2)?(ø— e)?(b— e)® > (1+a?)(1+ð2)(1+ c?)(a — b)”(b — e)?(c— a)?

AMM

62 | Titu Andreescu, Mircea Lascu | Let a, x,y,z be positive real numbers such

that xyz = 1 and a > 1 Prove that

64 | Laurentiu Panaitopol | Let a1,a2, ,a@, be pairwise distinct positive inte- gers Prove that

a(Vầ + Vab) b(Vần+ Và) c(VAb+ ve +7

66 [ Titu Andreescu, Gabriel Dospineseu | Let a, 6, c,d be real numbers such that

(1 + a?)(1 + 67)(1 +c?)(1 + d?) = 16 Prove that

Old and New Inequalities 17

71 | Marian Tetiva | Prove that for any positive real numbers a, b,c,

a®—b bì—c3 -a? (a — b)? + (b-c)* + (c—a)?

a2 + b2 + e° + 2abe+ 3 > (1+ ø)(1+b)(1+ e)

75 [ Titu Andreescu, Zuming Feng | Let a,b,c be positive real numbers Prove

that

(Qa+b+c)? (2b+a+c)? (2c+a+b)? 3

2a2 + (b+c)? 26?+(at+c)? 2c? + (a+b)? ~~

USAMO, 2003

76 Prove that for any positive real numbers x,y and any positive integers m,n,

(n—1)(m—1)(a™*"ty™*") + (mtn-1 (ary +ary™) > mnlartr ty pyr" ag),

Austrian-Polish Competition, 1995

77 Let a,b,c, d,e be positive real numbers such that abcde = 1 Prove that

ltab+abed 1+6bc+bcde 1+cd+cdea 1+de+deab l+ca+ cabe

Crux Mathematicorum

78 [ Titu Andreescu | Prove that for any a,b,c, € (0, 3) the following inequality

holds

sỉn ø - sin(œ — Ö) - sin(œ —e) sinb-sin(6 — e) - sin(b — ø)_ sinc- sin(e — a) - sin(e — b) >0

sin( + c) sin(e + ø}) sin(a + b) —

TST 2003, USA

79 Prove that if a,b,c are positive real numbers then,

Va4 + b1 + c+ + Veh? + Pet 2a? > Va3b+ Bet Gat Vab? + be? + ca

KMO Summer Program Test, 2001

80 | Gabriel Dospinescu, Mircea Lascu | For a given n > 2 find the smallest constant k, with the property: if a1, ,@, > 0 have product 1, then

a1 Q2 + a203 fives $ — AanG1 F< ky

(a? +a2)(az +a) (a3 +.a3)(a? + a2) (a2 + a1)(a? +an) —

84 | Vasile Cirtoaje, Gheorghe Eckstein | Consider positive real numbers

#1,Z2, ,„ such that #+zs #„ = l Prove that

85 | Titu Andreescu | Prove that for any nonnegative real numbers a, b,c such

that a? + 6? + c? + abc = 4 we have 0 < ab+ be + ca — abc < 2

USAMO, 2001

Old and New Inequalities 19

86 | Titu Andreescu | Prove that for any positive real numbers a, b, c the following

inequality holds

OFFS | Save < maz{ (Va — v5)?, (vb Ve)”, (Ve - Va}

TST 2000, USA

87 | Kiran Kedlaya | Let a,b,c be positive real numbers Prove that

a + Vab+ Ÿabc _ af a+b atbte

88 Find the greatest constant & such that for any positive integer n which is not

a square, |(1 + /n) sin(zn)| > È

Vietnamese IMO Training Camp, 1995

89 | Dung Tran Nam ] Let #,,z > 0 such that (x + y+ 2)? = 32zyz Find the

ei ty* +24 minimum and maximum of ——————_

(ct+y+z)

Vietnam, 2004

90 | George Tsintifas ] Prove that for any a,b,c,d > 0,

(a+ 6)?(b+ e)3(e+ đ)®(d+ a)? > 16a?b?c2d?(œ + b+ e+ đì!

Crux Mathematicorum

91 [ Titu Andreescu, Gabriel Dospinescu | Find the maxinum value oÊ the ex-

pression

(ab)” | bc)” | (ca)”

where a,b,c are nonnegative real numbers which add up to 1 and n is some positive

94 | Vasile Cirtoaje | Let a,b,c be positive real numbers Prove that

(of) (OE i)e(oe£a) (ced ra(oed 1) (ba) ea

95 [ Gabriel Dospinescu |] Let n be an integer greater than 2 Find the greatest real number m, and the least real number Ä⁄„ such that for any positive real numbers

97 | Vasile Cirtoaje | For any a, b,c,d > 0 prove that

2(aŠ + 1)(b3 + 1)(eŸ + 1)(đ + 1) > (1+ abeđ)(1 + a”)(1 + 02)(1+ e2)(1+ d?)

Gazeta Matematica

98 Prove that for any real numbers a, b,c,

(z+b)*+(b+e)*+(e+a)°®> sa) (a1 + b* + €)

100 | Dung Tran Nam | Find the minimum value of the expression ¬ + 3 + :

where ø, ô,e are positive real numbers such that 2lab + 2be + 8ca < 12

Old and New Inequalities

102 Let a,b,c be positive real numbers Prove that

where a,, is the least among the numbers a1, a2, ,@n

104 [ Turkevici |] Prove that for all positive real numbers z, y, z, t,

ai+yttzt+tt+2eyzt> oe ytyest PP 4Prr testy’

106 Prove that 1Í aI,ds, , an, ÐỊ, ., 6, are real numbers between 1001 and

2002, inclusively, such that øŸ + gã + - + a2 = b‡ + bã + - + b2, then we have the

(a7 + b”)(b2 + c?)(c? + a2) > 8(a?bŸ + b?e? + ca”)

108 [ Vasile Cirtoaje | TÍ ø, ,é, đ are positive real numbers such that abcd = 1, then

110 | Gabriel Dospinescu | Let a1, a2, ,@,, be real numbers and let S be a non-empty subset of {1,2, ,2} Prove that

» < » (a; + +4a;)?

¿c8 1<i<j<n

'TST 2004, Romania

111 [| Dung Tran Nam ] Let 71, 22 , Z2004 be real numbers in the interval [—1, 1]

such that x? + x3 + + #3004 = 0 Find the maximal value of the 7, +22 + -+ 22904

112 | Gabriel Dospinescu, Calin Popa | Prove that if n > 2 and a1,a2, ,Gn are real numbers with product 1, then

(n—1)(a? +a? + -+a")+nayaz Gn > (a, +ag+-+-+an)(ap | tant +-+-+ar-t),

Miklos Schweitzer Competition

117 Prove that for any %1,%2, ,2%, > 0 with product 1,

» (œ — z;)” > Soe? —n

=1 1<i<j<n

A generalization of Turkevici’s inequality

Old and New Inequalities 23

118 | Gabriel Dospinescu | Find the minimum value of the expression

l-da la IlTa2 — l1—a2

120 | Vasile Cirtoaje, Mircea Lascu | Let ø, , e, #,,z be positive real numbers such that

121 [ Gabriel Dospinescu | For a given n > 2, find the minimal value of the

constant k,, such that if 71,272, ,2%, > 0 have product 1, then

122 [ Vasile Cirtoaje, Gabriel Dospinescu | For a given n > 2, find the maximal

value of the constant k„ such that for any 21, 22, ,2%n > 0 for which z?+23+ -+

x? = 1 we have the inequality

(l—2a1)(1—22) (l—ay) > knti re 2p

CHAPTER 2

Solutions

25

1 Prove that the inequality

VEO OP + VRE oP + JPET ae > SX

holds for arbitrary real numbers a, Ö, e

Komal

First solution:

Applying Minkowsky’s Inequality to the left-hand side we have

V4 +(1—ð)?+02 +(1— e?+V@+(1—a)® > Jlatb+o24 (3—-a—b_-o

Denoting a+b+c=<2 we get

8\" 9 _ 9 (a+b+c)?+(3-a-—b-c) =2(x-3) +5 > 3

and the conclusion follows

Second solution:

We have the inequalities

Ja? + (1-6)? + fb? + (1-0)? + Ve? + (1-a)? >

5 lal += 8) Wl + ite, lel + [tal

(—ad—b1—-o < 7% —-a (i _b(1_o

By the AM-GM Inequality,

b Vabe < Vabe < ——

and

Old and New Inequalities 27 Summing up, we obtain

Vabe + V{1— a)(1T— b)(1— e) < Vb-We+ W1—b-WT1—e< 1,

by the Cauchy-Schwarz Inequality

Third solution:

Let a = sin?z,b = sin? y,c = sin? z, where z,y,z € (0, 3): The inequality becomes

sinz:siny:sinz+cosx:-cosy:cosz < Ì and it follows from the inequalities

sinz-siny:sinz+cosx-cosy-cosz < sinx-siny +cosz-cosy = cos(x —y) <1

3 | Mircea Lascu | Let a,b,c be positive real numbers such that abc = 1 Prove

that

b+e Ta + ete > Ja+Vb+ Vet 3 eta a+b

Gazeta Matematica

Let t= xy + yz + zx Let us observe that

(2? +y? + 23 — 32yz)? = (@ ty +2)? — xy — yz — za)? = (L+2#)(1-— t)Ẻ

and thus the maximum value is 1

6 Let a,b,c, x,y,z be positive real numbers such that «+ y+ z= 1 Prove that

aœ + bụ + cz +2\/(xy + 0z + z#)(ab + be + ca) <a+b+e

Ukraine, 2001 First solution:

We will use the Cauchy-Schwarz Inequality twice First, we can write ax+by+

cz < Va2 + b2 + cÈ - +2 + 2 + z2 and then we apply again the Cauchy-Schwarz

Inequality to obtain:

ax+by+cz + 2/(ry + yz + z#)(ab + be + ca) <

Vie Sia? + V2ab- V2” zụ <