1691 Disequazioni Algebriche 2009-65_a_finale-Dispari (4).....

Problem Solving Old and New Problems for Mathematical Olympiads

1691 Algebraic Inequalities

3

Panagiote Ligouras

Problem Solving Old and New Problems for Mathematical Olympiads

Panagiote Ligouras

Via Col di Lana, 33

70011 Alberobello (Ba)


5

The book starts with a collection by Aassila M., Andreescu T.,

Barbeau E. J., Beckenbach E., Bourgade P., Bulajich Manfrino R.,

Cirtoaje V., Dorrei H., Dospinescu G., Enescu B., Engel A., Feng

Z., Gomez Ortega J.A., Hardy G.H., Klamkin M., Kolev E., Lascu

M., Larson L., Littlewood J.E., Mitrinovic D., Mushkarov O., Negut

A., Nikolov N., Panaitopol L., Pham Kim Hung, Pòlya G.,

Rabinowitz , reiman I., Shawyer B.L.R., Stergiou B., Soulami T.,

Todev R., Xiong Bin and of many other friends. Its intention is to

bring together, in the successive editions, all significant algebraic

inequalities. This will also be done by finding out the author, year in

which it was created and grade of difficulty on a scale of 1 to 5

(banal, easy, medium, requiring effort and difficult).

We haven’t classified the problems according to the techniques used

to solve them, or by their difficulty, in order to avoid limiting the

creativity and freedom of the person solving them.

In writing this book, the website www.mathlinks.ro has been very

useful. It is created and maintained by Dr Valentin Vornicu, who is

totally dedicated to all the kinds of mathematics that help in

preparing for national mathematics competitions and the

International Mathematics Olympics.

On the website’s forum, these mathematical inequalities can be

discussed much further. Alternative solutions can be suggested,

along with new problems that may help to develop the knowledge of

friends visiting the website. This also provides an opportunity to

underline the importance of mathematics as a discipline, for the

development of men and also of society. The problems that are

marked t=329751 or similar are present on the forum of the website

www.mathlinks.ro.


7

Preface ……………………………………………….…………... pg. 5

Contents ……………………………………………………………… 7

Part I

Problems with 1 variable ………………………..……………… 11

Problems with 2 variables ………………………..……………… 11





Problems with n > 6 variables …………………..……………… 11

Problems with find the maximum value ………..………………. 33

Problems with find the minimum value ………..………………. 33

Part II

Glossary …..……………………………………………………. 159

Notations ……………………………………………………. 165

Bibliography …………………………………………………….. 169

Further reading …..………………………………………….. 171

Index ……………………..…………………………………… 173


9

Part I


11

------------------------------- AI1 - 1 -------------------------------

Let a be real number ( a R ). Prove that:

1 1) 2, 0 ) 2, 0a a a b a a

a a! " # ! $ % &

------------------------------- AI1 - 2 -------------------------------

Let a be non-negative real number ( 0a " ). Prove that:

5 2 33 2a a a% ! " !

Titu Andreescu, 2004

------------------------------- AI1 - 3 -------------------------------


' ( ' (22 45 1 2 1a a a% ! " !

Panagiote Ligouras, 2009

------------------------------- AI1 - 3 -------------------------------


' ( ' (' (3 32 32 1 1 1a a a! " ! !

------------------------------- AI1 - 4 -------------------------------


' ( ' (' (' (3 33 4 24 1 1 1 1 , 0a a a a a! " ! ! ! "


------------------------------- AI1 - 5 -------------------------------


' ( ' (' (4 43 4 22 1 1 1a a a! " ! !


13

1 1 91 1 , 8 0

48a

a a

) *) *! ! " # #+ ,+ ,

%- .- .Gazeta Matematica, 2002

------------------------------- AI1 - 12 -------------------------------

Let a be real number ( a R ) such that 1 1a% & & . Prove that:

4 2 4 41 1 1 3a a a% ! % ! ! $The Mathscope, n.216.3

------------------------------- AI1 - 13 -------------------------------

Let a be positive real number ( 0a # ). Prove that:

3 42 4

2

1 11

21

a aa a

a

! !" % ! "

!CRUX, 35(4), p. 254, 2009

------------------------------- AI1 - 14 -------------------------------


3 6 2a a a a! ! $ !The Mathscope, n.332.5

------------------------------- AI1 - 15 -------------------------------


3 62 5 2 1a a a a! ! $ !Panagiote Ligouras, Mathlinks, 2009

------------------------------- AI1 - 16 -------------------------------

Let a be real number such that 2a " % . Prove that:

3 62 2 2 4a a a a! ! ! ! ! $ !Panagiote Ligouras, Mathlinks, 2009

------------------------------- AI1 - 17 -------------------------------


15

P r o b l e m s w i t h 2 v a r i a b l e s

------------------------------- AI2 - 21 -------------------------------

Let a, b be non-negative real numbers ( , 0a b " ). Prove that:

2a b ab! "

------------------------------- AI2 - 22 -------------------------------

Let a, b be real numbers ( ,a b R ). Prove that:

2 2 2a b ab! "

------------------------------- AI2 - 23 -------------------------------

Let , 0a b " . Prove that:

' (3 3a b ab a b! " !

------------------------------- AI2 - 24 -------------------------------


' (4 4 2 2a b ab a b! " !

------------------------------- AI2 - 25 -------------------------------


' ( ' (5 5 3 3 2 2a b ab a b a b a b! " ! " !

------------------------------- AI2 - 26 -------------------------------

Let ,a b R . Prove that:

' (2 4a b ab! "


17

------------------------------- AI2 - 33 -------------------------------

Let , 0a b # . Prove that:

2 2

2

a b a b

a b

! !"

!

------------------------------- AI2 - 34 -------------------------------


2 2

1 1 1

2

a b

a ba b

! ) *$ !+ ,

! - .

------------------------------- AI2 - 35 -------------------------------


4

ab a b

a b

!$

!

------------------------------- AI2 - 36 -------------------------------


4 1 1

a b a b$ !

!

------------------------------- AI2 - 37 -------------------------------

Let , 0a b # such that 1a b! / . Prove that:

2 21 1 25

2a ba b

) * ) *! ! ! "+ , + ,

- . - .

------------------------------- AI2 - 38 -------------------------------

Let , 0a b " and *n N . Prove that:

1 1n nn na b a b ab% %! " !


19

------------------------------- AI2 - 45 -------------------------------

Let , *a b R . Prove that:

2 2 2 2

3 3 3 3

2 3 2 3 4

2 3 2 3

a b b a

a ba b b a

! !! $

!! !The Mathscope, n 285.2

------------------------------- AI2 - 46 -------------------------------


4 2 2 4

1 a b

ab a b a b" !

! !Russia, 1995

------------------------------- AI2 - 47 -------------------------------

Let , 0a b # and 1k " . Prove that:

' ( ' ( ' (2 2 2 21

a kb b ka a b

a k ab bb ka a kb

! ! !! "

! % !! !

MathLinks, t=255035, 2009

------------------------------- AI2 - 48 -------------------------------

Let , 0, 1a b a b# ! / . Prove that:

1 1 2 2, 1, , 0

1 1 2 1a b a b

a b! " ! / #

% % %

------------------------------- AI2 - 49 -------------------------------

Let , 0, 2a b a b# ! / . Prove that:

2 22

1 1

b aa ba b

! $! !

------------------------------- AI2 - 50 -------------------------------

Let 2, 0, , 2, , 1, a b m N m r R rm ab r# " " / . Prove that:

' ( ' ( ' (1 1 2

1 1 1m m m

a b r! "

! ! !

Arkady Alt, CRUX, 34(2), n. 3319, 2008


21

------------------------------- AI2 - 57 -------------------------------


3 3 2 2a b a b ab! " ! ! %Norman Schaumberger, N.Y.S.M.T.J., n.153, 1984

------------------------------- AI2 - 58 -------------------------------


1b aa b! #Kàlmàn Szabò, T.M.G., n. 67.H, 1983

------------------------------- AI2 - 59 -------------------------------

Let ,a b R such that , 1a b & . Prove that:

1ab a b! # !PARABOLA, Q591, 1984

------------------------------- AI2 - 60 -------------------------------


' ( ' (2 2

1 1 1

11 1 aba b! "

!! !

------------------------------- AI2 - 61 -------------------------------

Let 0 , 1x y$ $ . Prove that:

' ( ' ( ' (' (2 22 21 1 1 1 1 5 1x y x y xy! ! ! ! % ! % " ! %

------------------------------- AI2 - 62 -------------------------------


' ( 3 331 1

2 , , 0a b

a b a ba b b a

) *! ! " ! #+ ,

- .Czech and Slovakia, 2000


23

------------------------------- AI2 - 69 -------------------------------

Let 1, , 0a b a b! $ # . Prove that:

3 3 2 2

1 1 120

a b a b ab! ! "

!Cao Minh Quang

------------------------------- AI2 - 70 -------------------------------

Let 1, , 0ab a b/ # . Prove that:

3 3

11 1

a b

b a! "

! !Le Thanh Hai

------------------------------- AI2 - 71 -------------------------------


2 2 2 2

38 8

a b ab a b

a b

) *% !! "+ ,+ , !- .

Yugoslavia, 1991

------------------------------- AI2 - 72 -------------------------------

Let3

, 1,2

x y0 1

2 34 5. Prove that:

2 2 3 2 3 2x y x y y x! " % ! %Moldova, 2001

------------------------------- AI2 - 73 -------------------------------

Let n be a positive integer, and let x and y be a positive real numbers such

that 1n nx y! / . Prove that

' (' (

2 2

4 41 1

1 1 1

1 1 1 1

n nk k

k kk k

x y

x y x y/ /

) * ) *! !# 6+ , + ,+ , + ,% % ! !- . - .7 7

IMO Shortlist 2007


25

P r o b l e m s w i t h 3 v a r i a b l e s

------------------------------- AI3 - 79 -------------------------------

Let a, b, and c be positive real numbers such that 2 2 2 1a b c! ! / . Prove

that:

2 2 2

3 3

21 1 1

a b c

a b c! ! "

% % %Sefket Arslanagic, CRUX, n. 2738

------------------------------- AI3 - 80 -------------------------------

Let a, b, and c be positive real numbers such that 0 , , 1a b c& & and

1ab bc ca! ! / . Prove that:

2 2 2

3 3

21 1 1

a b c

a b c! ! "

% % %

------------------------------- AI3 - 81 -------------------------------

Let a, b, and c be positive real numbers such that 0 , , 1a b c& & and

1ab bc ca! ! / . Prove that:

2 2 2

2 2 2

3 1 1 1

41 1 1

a b c a b c

a b ca b c

) *% % %! ! " ! !+ ,+ ,% % % - .

------------------------------- AI3 - 82 -------------------------------

Let a, b, and c be real numbers such that 1ab bc ca! ! / and

1 , , 1a b c% & & . Prove that:

' (' (2 2

2 2

9

21 1cyc

a b

a b

!"

% %7



27

------------------------------- AI3 - 88 -------------------------------

For positive real numbers a, b, and c with 1a b c! ! / , show that

1 1 12

1 1 1

a b c b c a

a b c a b c

! ! ! ) *! ! $ ! !+ ,% % % - .

You need not state when equality holds.14th Japanese MO, 2004

------------------------------- AI3 - 89 -------------------------------

a) If a, b and c are three real numbers, all different from 1, such that

1abc / , then prove that

' ( ' ( ' (

2 2 2

2 2 21

1 1 1

a b c

a b c! ! "

% % %.

b) Prove that equality is achieved for infinity many triples of rational

numbers a, b and c.

Walther Janous, IMO Shortlist, 2008 & EXCALIBUR, 13(3), 2008

------------------------------- AI3 - 90 -------------------------------

Let a, b, and c be positive real numbers such that , , 1a b c # . Prove that:

' ( ' ( ' (

4 4 4

2 2 248

1 1 1

a b c

b c a! ! "

% % %

EXCALIBUR, 13(4) & 13(5), 2008

------------------------------- AI3 - 91 -------------------------------

Let a, b, and c be positive real numbers such that , , 1a b c # and

ab bc ca abc! ! / . Prove that:

' ( ' ( ' (

4 4 4

2 2 2

4 4 4 4 4 454

1 1 1

c a a b b c

a b c

! % ! % ! %! ! "

% % %

Panagiote Ligouras, MathLinks, t=298586, 2009

------------------------------- AI3 - 92 -------------------------------

Let a, b, and c be non-negative real numbers such that 1a b c! ! / . Prove

that:

1 1 1 273

1 1 1 8ab bc ca$ ! ! $% % %

Sefket Arslanagic, CRUX, n. 2786


29

------------------------------- AI3 - 99 -------------------------------

Let a, b, c be non-negative real numbers, no two of which are zero. Prove

that

' (' ( ' (' ( ' (' (3 3 3

2 2 2 2 2 2 2 2 2 2 2 2

1

2 2 2 2 2 2

a b c

a b ca b a c b c b a c a c b! ! $

! !! ! ! ! ! !

Vasile Cirtoaje, MathLinks, 2005

------------------------------- AI3 - 100 -------------------------------

If a, b, c are non-negative real numbers, then

2 2 2

2 2 2 2 2 2 2 2 20

2 2 2

a bc b ca c ab

a b c a b c a b c

% % %! ! "

! ! ! ! ! !Nguyen Anh Tuan, MathLinks, 2005

------------------------------- AI3 - 101 -------------------------------

Let a, b, c be non-negative reals and let p, q be positive reals. Prove that

' (' (2 2 2 2

2 2 2

2

02cyc

a bc p a b c q

a b c

0 1% ! ! !2 3"2 3

! !2 34 5

7


------------------------------- AI3 - 102 -------------------------------

If a, b, c are non-negative real numbers, then

2 2 2

2 2 2 2 2 2 2 2 20

2 2 2

a bc b ca c ab

a b c a b c a b c

% % %! ! "

! ! ! ! ! !

Nguyen Anh Tuan, MathLinks, 2005

------------------------------- AI3 - 103 -------------------------------

Let a, b, c be non-negative real numbers, no two of which are zero. Prove

that

2 2 2 2 2 2

1 1 1 6

2 2 2a bc b ca c ab a b c ab bc ca! ! "

! ! ! ! ! ! ! !


31

------------------------------- AI3 - 109 -------------------------------

Let a, b, c be positive reals such that 3a b c! ! / . Prove that

2 2 2

2 2 21

2 2 2

a b c

b a c b a c! ! "

! ! !

------------------------------- AI3 - 110 -------------------------------

Let a, b, c be positive reals such that 3a b c! ! / . Prove that

2 2 2

3 3 31, 3

2 2 2

a b ca b c

b a c b a c! ! " ! ! /

! ! !

------------------------------- AI3 - 111 -------------------------------

Let a, b, c be non-negative real numbers, no two of which are zero. Then

2 2 2

2 2 2 2 2 21

2 2 2 2 2 2

a b c

b bc c c ca a a ab b! ! "

% ! % ! % !Vasile Cirtoaje

------------------------------- AI3 - 112 -------------------------------

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

3 3 3

2 2 2 2 2 2 32 2 2 2 2 2

a b c a b c

a ab b b bc c c ca a

! !! ! "

% ! % ! % !Nguyen Viet Anh

------------------------------- AI3 - 113 -------------------------------


12 2 2

a b c

a b b c c a! ! $

! ! !Gazeta Matematic , 2000

------------------------------- AI3 - 114 -------------------------------


1 1 1 3

2 2 2a b b c c a a b c! ! "

! ! ! ! !


33

------------------------------- AI3 - 121 -------------------------------

Show that for all positive reals a, b, c,

3 3 32 2 2

32 2 2

a b b c c a

a c b a c b

! ! !) * ) * ) *! ! "+ , + , + ,! ! !- . - . - .

MOP, 2004

------------------------------- AI3 - 122 -------------------------------


3

2 2 2 4

a b c

a b c a b c a b c! ! "

! ! ! ! ! !

------------------------------- AI3 - 123 -------------------------------


3

2 2 2 4

a b c

a b c a b c a b c! ! $

! ! ! ! ! !M&Y-9, 2004

------------------------------- AI3 - 124 -------------------------------


2 2 2 4

ab ca bc a b c

a b c a b c a b c

! !! ! $

! ! ! ! ! !M&Y-3, 2005

------------------------------- AI3 - 125 -------------------------------

Given three variables a, b and c satisfying the following equality 2 2 2

3 3 39a b c a b c! ! / and , ,a b c R! , prove that

1

2 2 2 4

ab bc ca

a b c a b c a b c! ! "

! ! ! ! ! !Shaastra Online Math Contest, 2009

------------------------------- AI3 - 126 -------------------------------


3

2 2 2 4

ab c ca b bc a a b c

a b c a b c a b c

! ! ! ! ! !! ! $

! ! ! ! ! !


35

------------------------------- AI3 - 133 -------------------------------


2 2 2 2 2 21

2 2 2 2 2

c ab b ac a bc a b c a b c

a b c a b c a b c b c c a a b

) *! ! ! ! !! ! $ ! ! !+ ,+ ,! ! ! ! ! ! ! ! !- .


------------------------------- AI3 - 134 -------------------------------


' ( ' ( ' ( ' (2 2 2

32 2 2

b c a c a b a b cabc a b c

a b c a b c a b c

! ! !! ! " ! !

! ! ! ! ! !MathLinks, t=259361, 2009

------------------------------- AI3 - 135 -------------------------------


' (' ( ' (' ( ' (' ( ' (9

2 2 2 8

a b c

b c a b c c a b c a a b c a b a b c! ! "

! ! ! ! ! ! ! ! ! ! !

------------------------------- AI3 - 136 -------------------------------


' (' (' (

' (' (' (

' (' (' (

3

2 2 2 4

a a b b b c c c a

b c a b c c a b c a a b c a b

! ! !! ! "

! ! ! ! ! ! ! ! !

------------------------------- AI3 - 137 -------------------------------


' ( ' (' (' (

' (2 2

2 2cyc

ab a c b cabc a b c

a b c a b c

! !" ! !

! ! ! !7MathLinks, t=259603, 2009

------------------------------- AI3 - 138 -------------------------------


3

2 2 2 4

a b a b c a c a c

b c a b c c a b c a a b c a b

! ! !6 ! 6 ! 6 "

! ! ! ! ! ! ! ! !D. Olteanu, Gazeta Matematic


37

------------------------------- AI3 - 145 -------------------------------

Let a, b, and c be positive real numbers such that 1abc / . Prove that

' (' (

' (' (

' (' (2 2 2

1 1 11

2 2 2

a a ab b b bc c c ca

a b b c c a

! ! ! ! ! !! ! "

! ! !


------------------------------- AI3 - 146 -------------------------------


' (' (' (

' (' (' (

' (' (' (2 2 2

1 1 2 1 1 2 1 1 24

2 2 2

a a ab a b b b bc b c c c ca c a

a b b c c a

! ! ! ! ! ! ! ! ! ! ! !! ! "

! ! !


------------------------------- AI3 - 147 -------------------------------

Let a, b, and c be positive real numbers. Prove that

' ( ' ( ' (

2 2 2

2 2 2

2 2 2 4

32 2 2

ac ab b ab bc c bc ca a

c b a c b a

! ! ! ! ! !! ! "

! ! !


------------------------------- AI3 - 148 -------------------------------


' (' (

' (' (

' (' (2 2 2

1 2 1 2 1 22

2 2 2

a c a ab bc b a b bc ca c b c ca ab

a b b c c a

! ! ! ! ! ! ! ! ! ! ! !! ! "

! ! !

Panagiote Ligouras, ML, t=313583, 2009

------------------------------- AI3 - 149 -------------------------------


' (' (

2 2 2

2

1 23

2cyc

a c a ab bc a b c

ab bc caa b

0 1! ! ! ! ! !2 3 ! "! !2 3!4 5

7Panagiote Ligouras, MathLinks, t=258695, 2009


39

------------------------------- AI3 - 155 -------------------------------


' ( ' (' (

2 2

2

42

2 2 32cyc cyc

a b ab

a c c bc b

!! "

! !!7 7

Panagiote Ligouras, MathLinks, t= 320116, 2009

------------------------------- AI3 - 156 -------------------------------

Let a, b, and c be three arbitrary real numbers. Prove that

' ( ' ( ' ( ' (2 2 2 2 2 2

1 1 1 11

72 2 2 a b ca b b c c a! ! "

! !% % %

Pham Kim Hung

------------------------------- AI3 - 157 -------------------------------

Let a, b, and c be positive real numbers such that 2 2 2 1a b c! ! / . Then

2 2 2 3

1 2 1 2 1 2 5

a b c

bc ca ab! ! "

! ! !Raja Oktovin, MathLinks, t=248249, 2008

------------------------------- AI3 - 158 -------------------------------

Let a, b, c be non-negative real numbers such that 3a b c! ! / . Prove that

31 2 1 2 1 2

a b c

bc ca ab! ! "

! ! !

Pham Kim Hung

------------------------------- AI3 - 159 -------------------------------

Let a, b, c be non-negative real numbers. Prove that

2 2 2 2 2 21

4 4 4 4 4 4

c a b

a ab b b bc c c ca a! ! "

! ! ! ! ! !

Pham Kim Hung & Vo Quoc Ba Can

------------------------------- AI3 - 160 -------------------------------


2 2 2 2 2 21

4 4 4 4 4 4

a b c

a ab b b bc c c ca a! ! $

! ! ! ! ! !Bin Zhao, Mathematical Reflections, n. O32, 2006


41

------------------------------- AI3 - 167 -------------------------------

Let a, b, and c be non-negative numbers such that 0ab bc ac! ! 8 . Then

' (' (2 2 2

6

54 4 4

a b ca b b c a c

ab bc acc ab a bc b ca

! !! ! !! ! "

! !! ! !

Michael Rozenberg, MathLinks, t=262789, 2009

------------------------------- AI3 - 168 -------------------------------

Let a, b, and c be positive real numbers such that 3a b c! ! / . Prove that

2

1 2

34 7 8cyc a bc a"

! ! !7


------------------------------- AI3 - 169 -------------------------------


2 2 2

1 1 1 a b c

abca b c

! !! ! "

------------------------------- AI3 - 170 -------------------------------

Let a, b, c be positive real numbers such that 3a b c! ! / . Prove that

2 2 2

2 2 2

1 1 1a b c

a b c! ! " ! !

Romania-TST, Iasi & Bucharest, 2006

------------------------------- AI3 - 171 -------------------------------

Suppose that a, b, and c are positive reals satisfying 2 2 2 1a b c! ! / . Prove

that

' (3 3 3

2 2 2

21 1 13

a b c

abca b c

! !! ! " !

Ho-joo Lee, CRUX, n. 2532

------------------------------- AI3 - 172 -------------------------------


2 2 2

1 1 1 281

27abc

a b c

) *! ! ! "+ ,

- .Walther Janous, CRUX, Klamkin-02


43

------------------------------- AI3 - 179 -------------------------------


' (' (

2 2 2

2 2 2 2 2 2

99

4 4

ab bc caa b c a b c a b c

c b ab c a a b c

! !! ! !) *! ! " ! ! ! %+ ,

- . ! !


------------------------------- AI3 - 180 -------------------------------


2 2 2 2 2 2

2 2 2 2 2 2

103 1

3

a b c a b c ab bc ca

ab bc cab c a a b c

! ! ! !! ! " 6 % 6 %

! ! ! !Panagiote Ligouras, ML, t=320792, 2009

------------------------------- AI3 - 181 -------------------------------


' (' (' (

2 2 2

2 2 2

12018

a b c abc

a b b c c ab c a! ! " %

! ! !

Tran Quoc Anh, ML t=293216, 2009

------------------------------- AI3 - 182 -------------------------------


2 2 2

32 2 2

19 2 3

2

a b c a b c

b c ab c a

) *! ! ! " ! 6 ! 6+ ,+ ,

- .

------------------------------- AI3 - 183 -------------------------------


2 2 2

32 2 2

31 2 3 6

2

a b c a b c

b c ab c a

) *! ! ! " ! 6 ! 6 "+ ,+ ,

- .

------------------------------- AI3 - 184 -------------------------------


2 2 2

2 2 2 3 3 3

a b c a b c

b c a b c a! ! $ ! !


45

------------------------------- AI3 - 191 -------------------------------


' ( ' ( ' (2 2 2

3

21 1 1

c a b

a b b c c a! ! "

! ! !

------------------------------- AI3 - 192 -------------------------------

Let x, y, and z be positive real numbers. Prove that

' (' (

22 2 222 2

2 2 2 2 2 2

92 2 2

x z z y y xxy yz zx

z x y xyz x y y z z x

! !) *) * ) *! ! ! ! ! "+ ,+ , + , + ,- . - . ! !- .


------------------------------- AI3 - 193 -------------------------------


9

2 2 2

1 1 11 1 1 2

a b c

) *) *) *% % % "+ ,+ ,+ ,

- .- .- .

------------------------------- AI3 - 194 -------------------------------

Let a, b, and c be non-negative real numbers such that 3ab bc ca! ! / .

Prove that

2 2 2

1 1 1 3

21 1 1a b c! ! "

! ! !Vasile Cirtoaje, MathLinks, 2005

------------------------------- AI3 - 195 -------------------------------

If , , 1a b c $ and 1a b c! ! / . Prove that

2 2 2

1 1 1 27

101 1 1a b c! ! $

! ! !Titu Andreescu & Gabriel Dospinescu

------------------------------- AI3 - 196 -------------------------------


' (22 2 2

3

41 1 1

a b ca a b b c c

b c a! ! " ! !

! ! !Hellenic-IMO-TST, 2002


47

------------------------------- AI3 - 203 -------------------------------

Let a, b, and c be positive reals such that 1ab bc ca! ! / . Prove that

2 2 2

12 3

161 1 1

ab bc ca a b c b c a

b c a a b ca b c

0 1) *! ! $ ! ! ! ! ! !2 3+ ,

! ! ! - .4 5MathLinks, t=259357, 2009

------------------------------- AI3 - 204 -------------------------------

Let a, b, and c be positive real numbers with 3ab bc ca! ! / . Prove that

2 2 2

3

21 1 1

ab bc ca

c a b! ! "

! ! !Titu Zvonaru, Mathematical Mayhem, 34(4) M299, 2008

------------------------------- AI3 - 205 -------------------------------


2 2 2

2 2 2

3

21 1 1

a b c

b c a! ! "

! ! !Faruk Zejnulahi & Sefket Arslanagic, CRUX, n. 2994

------------------------------- AI3 - 206 -------------------------------


2 2 2

1 1 13

1 1 1

a b c

b c a

! ! !! ! "

! ! !

------------------------------- AI3 - 207 -------------------------------


2 2 2

2 2 2

1 1 1 7

21 1 1

a b c

b c a

! ! !! ! $

! ! !MathLinks, t=260118, 2009

------------------------------- AI3 - 208 -------------------------------

Let a, b, and c be positive real numbers such that 1ab bc ca! ! / . Prove

that2 2 2

2 2 2

1 1 1 3

21 1 1

a b c

a b c

% % %! ! $

! ! !Gazeta Matematic , 2003


49

------------------------------- AI3 - 215 -------------------------------


' (' ( ' (' (2 2 23

2 3 3 3

1 1 1 116 max ,

1 1 1cyc cyccyc

a bc a ab c a ba

a a a b

9 :! ! ! !) *! ; ;"+ , < =+ ,! ! !; ;- . > ?

7 7@

Mihàly Bencze, CRUX, 34(4) n. 3349, 2008

------------------------------- AI3 - 216 -------------------------------

Let a, b, and c be positive real numbers such that a b c abc! ! / . Then

2 2 2

1 1 1 3

21 1 1a b c! ! $

! ! !

EXCALIBUR, 6(2) & 6(3), 2001

------------------------------- AI3 - 217 -------------------------------

Let r be a real number, 0 1r& $ , and let a, b, and c be positive real

numbers such that 3abc r/ . Prove that

2 2 2 2

1 1 1 3

1 1 1 1a b c r! ! $

! ! ! !

Arkady Alt, CRUX, 34(3), n. 3329, 2008

------------------------------- AI3 - 218 -------------------------------

Let a, b, and c be positive reals such that 1ab bc ca! ! / . Prove that

2 2 2

3

21 1 1

a b c

a b c! ! $

! ! !

------------------------------- AI3 - 219 -------------------------------

Let a, b, and c be positive real numbers such that a b c abc! ! / . Then

2 2 2

3 3

21 1 1

a b c

a b c! ! $

! ! !


51

------------------------------- AI3 - 225 -------------------------------

Let a, b, and c be non-negative reals such that 3ab bc ca! ! / . Prove that

2 2 2

1 1 11

2 2 2a b c! ! $

! ! !

------------------------------- AI3 - 226 -------------------------------

Let a, b, and c be non-negative reals such that 1abc / . Prove that

2 2 2

7 6 7 6 7 61

2 2 2

a b c

a b c

% % %! ! "

! ! !Vasile Cirtoaje, MathLinks, t=245046, 2008

------------------------------- AI3 - 227 -------------------------------


2 2 2

1 1 1 2

2 2 2 ab bc caa bc b ca c ab! ! "

! !! ! !Vasile Cirtoaje, MathLinks, 2005

------------------------------- AI3 - 228 -------------------------------


2 2 2

2 2 21

2 2 2

a b c

a bc b ca c ab! ! "

! ! !Romania, 1997

------------------------------- AI3 - 229 -------------------------------


2 2 21

2 2 2

bc ca ab

a bc b ca c ab! ! $

! ! !Romania, 1997

------------------------------- AI3 - 230 -------------------------------


2 2 2

2 2 2

2 2 21

2 2 2

a bc b ca c ab

a bc b ca c ab

% % %! ! "

! ! !Panagiote Ligouras, MathLinks, t=310029, 2009


53

------------------------------- AI3 - 237 -------------------------------


2 2 22 2 2

a b c a b c

ab bc caa bc b ca c ab

! !! ! $

! !! ! !

Ho Phu Thai & Da Nang, Mathematical Reflections, n. O39, 2007

------------------------------- AI3 - 238 -------------------------------

Prove that for all non-negative real numbers a, b, c, we have

2 2 2

2 2 2

2 2 22 2

2 2 2

a bc b ca c ab

a bc b ca c ab

! ! !! ! "

! ! !Pham Kim Hung

------------------------------- AI3 - 239 -------------------------------

Let a, b, and c be non-negative real numbers. Prove that

2 2 22 2 2

a b c a b c


! !! ! $

! !! ! !Tran Quoc Anh, MathLinks, t=224981, 2008

------------------------------- AI3 - 240 -------------------------------


2 2 2

1 1 1 2

2 2 2

a b c a b c


% % % ! ! %! ! $

! !! ! !Panagiote Ligouras, ML, t=322098, 2010

------------------------------- AI3 - 241 -------------------------------

Let a, b, and c be positive real numbers. Prove or disprove that

' ( ' (' (2

2

2 1 1 1 2

2cyc

a a bc a b c ab bc ca

ab bc caa bc

! ! % ! ! ! ! ! %$

! !!7Panagiote Ligouras, MathLinks, t=251776, 2009

------------------------------- AI3 - 242 -------------------------------


2 2 2

1 1 1

3 3 3 4

a b c

a b c abc

! !! ! $

! ! !Romania, 2003


55

------------------------------- AI3 - 248 -------------------------------


3 3 3

2 2 2 2 2 21

4 4 4

a b c


! ! ! ! ! !José Luis Dìaz-Barrero, SSMJ, n. 5054, 2009

------------------------------- AI3 - 249 -------------------------------

Let a, b, and c be positive real numbers such that 1abc $ . Prove that

2 2 2

3

2

a b c

b b c c a a! ! "

! ! !Cao Minh Quang, CRUX, 35(2), n. 3421, 2009

------------------------------- AI3 - 250 -------------------------------

Let a, b, c be non-negative reals, no two of which are zero. Prove that

' (2 2 2 2 2 2 2

1 1 1 9

a ab b b bc c c ca a a b c! ! "

! ! ! ! ! ! ! !

Vasile Cirtoaje, Gazeta M!"#$!"%& -B, 9, 2000

------------------------------- AI3 - 251 -------------------------------


2 2 2 2 2 2

1 1 1 1

abca ab b b bc c c ca a! ! $

! ! ! ! ! !

------------------------------- AI3 - 252 -------------------------------


' ( ' ( ' (2 2 2

2 2 2 2 2 2

3a b c

a b cb a ab b c b bc c a c ca a! ! "

! !! ! ! ! ! !

------------------------------- AI3 - 253 -------------------------------


' ( ' ( ' (2 2 2

2 2 2 2 2 22

a b b c c a

b a ab b c b bc c a c ca a

! ! !! ! "

! ! ! ! ! !



57

------------------------------- AI3 - 260 -------------------------------


' ( ' ( ' (2 2 2 2 2 2

2a b c b c a c a b

b bc c c ca a a ab b

! ! !! ! "

! ! ! ! ! !

------------------------------- AI3 - 261 -------------------------------


2 2 2

2 2 2 2 2 22

a bc b ca c ab


! ! !! ! "

! ! ! ! ! !Vasile Cirtoaje, ML, 2005

------------------------------- AI3 - 262 -------------------------------


2 2 2

2 2 2 2 2 2

2 3 2 3 2 35

a bc b ca c ab


! ! !! ! "

! ! ! ! ! !

------------------------------- AI3 - 263 -------------------------------


2 2 2

2 2 2 2 2 2

5 2 5 2 5 27

a bc b ca c ab


! ! !! ! "

! ! ! ! ! !Panagiote Ligouras, ML, t=322104, 2010

------------------------------- AI3 - 264 -------------------------------

Let a, b, and c be positive real numbers and p, q >0. Prove that

2 2 2

2 2 2 2 2 2

pa qbc pb qca pc qabp q


! ! !! ! " !

! ! ! ! ! !Panagiote Ligouras, 2009

------------------------------- AI3 - 265 -------------------------------


' (' (

2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

43 3 3 1

33

ab bc caa b c b c a c a b

a ab b b bc c c ca a a b c

! !! % ! % ! %! ! ! $

! ! ! ! ! ! ! !



59

------------------------------- AI3 - 271 -------------------------------


' (3 3 3 3 3 3

2 2 2 2 2 2

2

3

a b b c c aa b c


! ! !! ! " ! !

! ! ! ! ! !

------------------------------- AI3 - 272 -------------------------------

Let a, b, c be positive real numbers and k > 0. Prove that

' (3 3 3 3 3 3

2 2 2 2 2 2

1

3

ka b kb c kc a ka b c


! ! ! !! ! " ! !


------------------------------- AI3 - 273 -------------------------------


3 3 3 3 3 3

2 2 2 2 2 22

a b b c c a


! ! !! ! "

! ! ! ! ! !

------------------------------- AI3 - 274 -------------------------------


3 3 3

2 2 2 2 2 21

a b c


! ! ! ! ! !

------------------------------- AI3 - 275 -------------------------------


3 3 3

2 2 2 2 2 21

a b c


! ! ! ! ! !José Luis Dìaz-Barrero, SSMJ 2009(2) n. 5054

------------------------------- AI3 - 276 -------------------------------

Let a, b, c be positive real numbers such that 1abc / and k > 0. Prove that

3 3 3 3 3 3

2 2 2 2 2 21

ka b kb c kc ak


! ! !! ! " !



61

------------------------------- AI3 - 282 -------------------------------


3 3 3

2 2 2 2 2 2 3

a b c a b c


! !! ! "

! ! ! ! ! !Le Trung Kien

------------------------------- AI3 - 283 -------------------------------

If a, b, c >0, prove that

2 2 2 2 2 2 3 3

ab bc ca a b c


! ! ! !$

! ! ! ! ! ! ! !

Murray S. Klamkin, CRUX n. 805

------------------------------- AI3 - 284 -------------------------------


' (2 2 2 2 2 2 2 2 2

1 1 1 15

2b c c a a b a b c ab ac ca! ! "

! ! ! ! ! ! ! !

------------------------------- AI3 - 285 -------------------------------


' (2 2 2 2 2 2 2

1 1 1 10

b c c a a b a b c! ! "

! ! ! ! !

------------------------------- AI3 - 286 -------------------------------


2 2 2 2 2 2

1 1 1 5

2b c c a a b! ! "

! ! !

------------------------------- AI3 - 287 -------------------------------

Let a, b, c be non-negative real numbers. Prove that

2 2 2 2 2 2 2 2 2

1 1 1 6 8

ab bc caa b b c c a a b c! ! " %

! !! ! ! ! !Pham Kim Hung


63

------------------------------- AI3 - 294 -------------------------------

Let a, b, and c be positive real numbers such that 1abc / . Then

3 3 3

2 2 2 2 2 2

15

12

a b c a b c

a b b c c a

! ! !! ! $

! ! !Panagiote Ligouras, ML, t=322517, 2010

------------------------------- AI3 - 295 -------------------------------


2 2 2 2 2 2

1 1 1a b b c c a

a b ca b b c c a

! ! !! ! $ ! !

! ! !

------------------------------- AI3 - 296 -------------------------------


' ( ' ( ' (2 2 2 2 2 2

1 8 5 5 8 5 5 8 5 5

10

a b b c c a a b c b c a c a b

a b c b c a c a bb c c a a b

) *% % % % % % % % %! ! " ! !+ ,+ ,! ! !! ! ! - .


------------------------------- AI3 - 297 -------------------------------

If 0c b a" " # , then

2

2 2 2 2 2 2

2

3

ac ba cb a b c

b c aa b b c c a

) *! ! $ ! !+ ,

! ! ! - .Panagiote Ligouras, MathLinks, t=322519, 2010

------------------------------- AI3 - 298 -------------------------------


' ( ' ( ' (2 2 2 2 2 2

2 2 26

a a b c b a b c c a b c

b c c a a b

! ! ! ! ! !! ! "

! ! !

------------------------------- AI3 - 299 -------------------------------


' ( ' ( ' (2 2 2

2 2 2 2 2 2

3

2

a b c a b c a b c

b c c a a b

% ! ! % ! ! %! ! "

! ! !Vasile Cirtoaje, ML, t=259748, 2009


65

------------------------------- AI3 - 306 -------------------------------


3 3 3 3 3 3

2 2 2 2 2 2 2 2 2

1

2

a b c a b c

a b b c c a b c a

) *! ! $ ! !+ ,+ ,! ! ! - .


------------------------------- AI3 - 307 -------------------------------


2 2 2

2 2 2cyc cyccyc cyc

a a b a a b

b c bb c b

) * ) *) * ) *! !+ , + ,+ , + ,"+ , + ,+ , + ,!!- . - .- . - .7 7@ @


------------------------------- AI3 - 308 -------------------------------


2 2 2

2 2 2 2 2 2

2 2 2 9

2

a bc b ac c ab

b c c a a b

! ! !! ! "

! ! !

------------------------------- AI3 - 309 -------------------------------


2 2 2

2 2 2 2 2 2

16 16 1610

a bc b ac c ab

b c c a a b

! ! !! ! "

! ! !Vasile Cirtoaje, ML, 2005

------------------------------- AI3 - 310 -------------------------------


2 2 2 2 2 2

3

2

ab bc ca bc ca ab ca ab bc

b c c a a b

% ! % ! % !! ! "

! ! !

------------------------------- AI3 - 311 -------------------------------

Let a, b, c be non-negative reals, no two of which are zero. Then

2 2 2 2 2 2

4 4 44

ab bc ca bc ca ab ca ab bc

b c c a a b

! ! ! ! ! !! ! "

! ! !


67

------------------------------- AI3 - 318 -------------------------------


2 2 2

2 2 2 2 2 2

2 2 23

a bc b ac c ab

b c c a a b

! ! !! ! "

! ! !Vo Quoc Ba Can & Vu Dinh Quy

------------------------------- AI3 - 319 -------------------------------


2 2 2

2 2 2 2 2 2

4 4 42 2

a bc b ac c ab

b c c a a b

! ! !! ! " !

! ! !Vo Quoc Ba Can, CRUX 35(4) n. 3419, 2009

------------------------------- AI3 - 320 -------------------------------


2 2

2 2

2 2 48 2

cyc

a bc a bc

b c

) *! ! !+ , " !+ ,!- .

7Panagiote Ligouras, 2009

------------------------------- AI3 - 321 -------------------------------


' (' ( ' (' ( ' (' (2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2

a b c a b c

abca b a c b a b c c a c b

! !! ! $

! ! ! ! ! !

Romania, local MO, 1988

------------------------------- AI3 - 322 -------------------------------


2 2 2 3

3 3 32 2 2 2 2 2

9a bc b ac c ab abc

a b cb c c a a b

! ! !! ! "

! !! ! !

Pham Huu Duc, Mathematical Reflections, n S27, 2006


69

------------------------------- AI3 - 329 -------------------------------


' ( ' (2 2 2

2 2 22 2 2 2 2 2

3 21 1 1

4

a b c a b c

abc a b ca b b c c a

! ! ! !# ! ! #

! !! ! !Zdravko F. Starc, RSME n.128, 2009

------------------------------- AI3 - 330 -------------------------------


2 2 2

2 2 22 2 2 2 2 2

2 1 2 1 2 13 2 2 1

4

a b c a b c a b c

abc a b ca b b c c a

) *! ! ! ! ! ! !) *! # ! ! # !+ , + ,+ , ! !- .! ! !- .


------------------------------- AI3 - 331 -------------------------------

Let a, b, c be positive real numbers such that 2 2 2 3a b c! ! / . Prove that

2 2 2 2 2 23

1 1 1

a b c

a b b c c a! ! $

! ! ! ! ! !Pham Kim Hung

------------------------------- AI3 - 332 -------------------------------

Let a, b, c be positive real numbers such that 1abc / . Prove that

2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 21 1 1 1 1 1

a b b c c a a b b c c a

a b b c c a a b b c c a

! ! ! ! ! !! ! " ! !

! ! ! ! ! ! ! ! ! ! ! !Jingjun Han, Mathematical Reflections n. J104, 2008

------------------------------- AI3 - 333 -------------------------------


2 2 2

3

2

a b c

b c c a a b! ! "

! ! !Pham Kim Hung

------------------------------- AI3 - 334 -------------------------------

Let a, b, c be positive reals such that 3a b c! ! / and 1abc / . Prove that

' (' ( ' (' ( ' (' (2 2 2 2 2 2

3

2

ab bc ca

a b a b b c b c c a c a! ! $

! ! ! ! ! !


71

------------------------------- AI3 - 341 -------------------------------


2 2 2

1

2

a b c a b c

bc ca aba bc b ca c ab

) *! ! $ ! !+ ,

! ! ! - .

Baltic way, 2003

------------------------------- AI3 - 342 -------------------------------


2 2 2

bc ca ab a b c

b c c a a ba bc b ca c ab! ! $ ! !

! ! !! ! !Pham Huu Duc, Mathematical Reflections n. J60, 2007

------------------------------- AI3 - 343 -------------------------------


2 2 2

2 2 22

a b c

a bc b ca c ab! ! $

! ! !Canada

------------------------------- AI3 - 344 -------------------------------


2 2 2

2 2 2 32

a b c a b c

a bc b ca c ab abc

! !! ! $

! ! !

Pham Huu Duc, CRUX 34(6) n. 3374, 2008

------------------------------- AI3 - 345 -------------------------------

Suppose that a, b, and c are positive real numbers. Prove that

2 2 2

1 1 1b c c a a b

a b ca bc b ca c ab

! ! !! ! $ ! !

! ! !Ho-Joo Lee, CRUX n. 2580

------------------------------- AI3 - 346 -------------------------------


' ( ' ( ' (2

2 2 22 2 2

2 2 2 2 2 2

93 3 3 a b c a b c a b ca a b b c c

a bc b ca c ab a b c ab bc ca

! ! ! ! ! ! ! !! ! !! ! $

! ! ! ! ! ! ! !Panagiote Ligouras, MathLinks, t=329754, 2010


73

------------------------------- AI3 - 353 -------------------------------


2 2 2 2 2 2 3 3 3

2 2 2

21

4 4

b bc c c ca a a ab b a b c

abca bc b ca c ab

! ! ! ! ! ! ! !! ! " %


------------------------------- AI3 - 354 -------------------------------


2 2 3 3 3

2

3 4 3 33

2 2cyc

b bc c a b c

abca bc

! ! ! !" %

!7Panagiote Ligouras, ML, t=329755, 2010

------------------------------- AI3 - 355 -------------------------------


' (' (

3 3 3

2 2 2 4

9

2

a b cab bc ca

c ab a bc b ca a b c

! !! ! "

! ! ! ! !

MIC Shortlist, ML, t=259607, 2009

------------------------------- AI3 - 356 -------------------------------


2 2 2

1 1 1 6

a b ca bc b ca c ab! ! "

! !! ! !Pham Kim Hung

------------------------------- AI3 - 357 -------------------------------


2 2 2

1 1 1 2 2

ab bc caa bc b ca c ab! ! "

! !! ! !Pham Kim Hung

------------------------------- AI3 - 358 -------------------------------


2 2 2

2 2 20

a bc b ca c ab

a bc b ca c ab

% % %! ! "

! ! !Vasile Cirtoaje, ML, 2005


75

------------------------------- AI3 - 365 -------------------------------

Let a, b, and c be positive real numbers such that , , 1a b c " % . Prove that

2 2 2

2 2 2

1 1 12

1 1 1

a b c

c b a c b a

! ! !! ! "

! ! ! ! ! !Laurentiu Panaitopol, JBMO, 2003

------------------------------- AI3 - 366 -------------------------------

Let a, b, c be real numbers such that 1 , , 1a b c% $ $ and 0a b c! ! / .

Prove that

2 2 21 1 1 3a c b a c b! ! ! ! ! ! ! ! "Pham Thanh Nam

------------------------------- AI3 - 367 -------------------------------

Prove that if the real numbers a, b and c satisfy 2 2 2 3a b c! ! / then

' (22 2 2

2 2 2 122 2 2

a b ca b c

c b a c b a

! !! ! "

! ! ! ! ! !.

When does the equality hold? Baltic Way, 2008

------------------------------- AI3 - 368 -------------------------------

Let a, b, c be non-negative real numbers such that 3a b c! ! / . Prove that

' (' (' (2 2 2 2 2 2 12a ab b b bc c c ca a% ! % ! % ! $

Pham Kim Hung, ML, 2006

------------------------------- AI3 - 369 -------------------------------

Let a, b, c be non-negative real numbers such that 2 2 2 2a b c! ! / . Then

' (' (' (2 2 2 2 2 28 1a ab b b bc c c ca a% ! % ! % ! $

Pham Kim Hung

------------------------------- AI3 - 370 -------------------------------


' (2 2 2 2 2 2 2

1 1 1 12

b bc c c ca a a ab b a b c! ! "

% ! % ! % ! ! !

Vasile Cirtoaje, 2006


77

------------------------------- AI3 - 377 -------------------------------


2 2 2

2 2 2 2 2 2

3 3 35

a bc b ca c ab


% % %! ! "

% ! % ! % !Panagiote Ligouras, ML, t=319775, 2009

------------------------------- AI3 - 378 -------------------------------


3 3 3

2 2 2 2 2 2

3a b c

abcb bc c c ca a a ab b! ! $

% ! % ! % !

------------------------------- AI3 - 379 -------------------------------


' (3 3 3

2 2 2 2 2 2

3 ab bc caa b c

a b cb bc c c ca a a ab b

! !! ! "

! !% ! % ! % !Sefket Arslanagic, CRUX n. 2927

------------------------------- AI3 - 380 -------------------------------


3

2 24

2cyc cyc cyc

a aba

a b cb bc c" "

! !% !7 7 7

------------------------------- AI3 - 381 -------------------------------


3 3 3

2 2 2 2 2 2

3a b b c c a ab bc ca

abcb bc c c ca a a ab b

! ! ! ! ! !! ! $

% ! % ! % !Panagiote Ligouras, MathLinks, t=329119, 2010

------------------------------- AI3 - 382 -------------------------------

Let a, b, c be non-negative reals, no two of which are zero, such that

2a b c! ! / . Prove that

3 3 3

2 2 2 2 2 2

8 8 826

a b c


! ! !! ! "

% ! % ! % !Panagiote Ligouras, ML, t=316360, 2009


79

------------------------------- AI3 - 389 -------------------------------

Let a, b, and c be positive real numbers such that 1abc / . Then

2 2 2

3 3 3 3 3 3

1 1 1 18ab bc ca

c a b a b c

! ! !! ! "

! !Hong Kong, 2000

------------------------------- AI3 - 390 -------------------------------


' (3 3 3 2

27a b c

b c a a b c! ! "

! !

------------------------------- AI3 - 391 -------------------------------


2 2 2

3 3 33

a b c

b c a! ! "


------------------------------- AI3 - 392 -------------------------------

Let a, b, c be positive real numbers such that 3k k ka b c! ! / , k > 0. Then

2 2 2

3 3 33

a b c

b c a! ! "

ML, t=259844, 2009

------------------------------- AI3 - 393 -------------------------------


3 3 3

3

21 1 1

a b c

b c a! ! $

! ! !Bin Zhao, ML, 2006

------------------------------- AI3 - 394 -------------------------------


2 2 2

3 3 3

3

21 1 1

a b c

b c a! ! "

! ! !ML, t=259844, 2009


81

------------------------------- AI3 - 401 -------------------------------


' (3 3 3

15

62 2 2

a b cabc

b c a! ! " !

! ! !MathLinks, t=264216, 2009

------------------------------- AI3 - 402 -------------------------------

For positive real numbers x, y, z holds2 2 2

1 1 1 1

21 1 1x y z! ! /

! ! !. Prove

the inequality:

3 3 3

1 1 1 1

32 2 2x y z! ! &

! ! !

Serbia, Junior Balkan Team Selection Test, 2009

------------------------------- AI3 - 403 -------------------------------

Let , ,x y z R! and 3x y z! ! / . Prove that:

' (3 3 3

3 3 3

1 2

9 278 8 8

x y zxy yz zx

y z x! ! " ! ! !

! ! !

Iran, National Math Olympiad, 2008

------------------------------- AI3 - 404 -------------------------------

Let a, b, c be positive real numbers such that 1a b c! ! $ . Prove that

3 2 3 2 3 2

27

311 1 1

a b c

a a b b c c! ! $

! ! ! ! ! !Cao Minh Quang, CRUX 35(2) n. 3421, 2009

------------------------------- AI3 - 405 -------------------------------

Let a, b, c be non-negative real numbers such that 2a b c! ! / and 1k " .

Prove that

3 3 3 3 3 32

k k ka b c

b c c a a b! ! "

! ! !Tran Quoc Anh, ML, t=292594, 2009


83

------------------------------- AI3 - 411 -------------------------------

If 0a b c" " # , then

3 3 3 3 3 3

3 3 3 3 3 3 3 3 3 3 3 3

a b b c c a ab bc ca

a b b c c a a b b c c a! ! " ! !

! ! ! ! ! !

------------------------------- AI3 - 412 -------------------------------


5 4 5 5 4 5 5 4 5

3 3 3 3 3 3

5

6

a a b b b c c c a

a b b c c a

! ! ! ! ! !! ! "


------------------------------- AI3 - 413 -------------------------------


5 4 5 5 4 5 5 4 5

3 3 3 3 3 3

3 2 3 3 2 3 3 2 32

a a b b b c c c a

a b b c c a

! ! ! ! ! !! ! "

! ! !Panagiote Ligouras, MathLinks, t=250356, 2009

------------------------------- AI3 - 414 -------------------------------


3 3 3

3 3 3 3 3 3

3 3 36

a abc b abc c abc

b c c a a b

! ! !! ! "

! ! !Tran Quoc Anh, ML, t=262924, 2009

------------------------------- AI3 - 415 -------------------------------


5 3 5 5 3 5 5 3 5

3 3 3 3 3 3

3 3 3 19

3

b a c abc a b c abc b c a abc

b c c a a b

! ! ! ! ! ! ! ! !! ! "


------------------------------- AI3 - 416 -------------------------------


5 3 5 5 3 5 5 3 5

3 3 3 3 3 3

3 3 3 3 3 3 3 3 37

b a c abc a b c abc b c a abc

b c c a a b

! ! ! ! ! ! ! ! !! ! "

! ! !

Panagiote Ligouras, MathLinks, t= 315857, 2009


85

------------------------------- AI3 - 423 -------------------------------

If , , 0a b c # with 1a b c! ! / , prove that

7 7 7 7 7 7

5 5 5 5 5 5

1

3

a b b c c a

a b b c c a

! ! !! ! "

! ! !

------------------------------- AI3 - 424 -------------------------------

Let , , 0a b c # and 1abc /

5 5 5 5 5 51

ab bc ca

a b ab b c bc c a ca! ! $

! ! ! ! ! !Unused problem in IMO 1996, EXC 2(4)+2(5), 1996

------------------------------- AI3 - 425 -------------------------------


1 1 19

a b c! ! "

------------------------------- AI3 - 426 -------------------------------


1 1 1 1 1 12

a b c a b b c c a

) *! ! " ! !+ ,! ! !- .

------------------------------- AI3 - 427 -------------------------------


1 1 1 1 1 1 94

a b c a b b c c a a b c

) *! ! " ! ! %+ ,! ! ! ! !- .

------------------------------- AI3 - 428 -------------------------------

If , , 0a b c # , prove that

8 8 8

3 3 3

1 1 1 a b c

a b c a b c

! !! ! $

G. C. Giri, CRUX n. 413


87

------------------------------- AI3 - 435 -------------------------------


2 2 2 2 2 2 2 2 2

1 1 16

3 2 3 2 3 2

a b c

a b c a b c b c a c a b

) *! ! " ! !+ ,

! ! ! ! ! !- .

------------------------------- AI3 - 436 -------------------------------


3 3 3 3 3 3

1 1 1 1 1 16

4 4 4a b c a b b c c a

) *! ! " ! !+ ,

! ! ! ! ! !- .

------------------------------- AI3 - 437 -------------------------------


1 4 9 36

a b c a b c! ! "

! !

------------------------------- AI3 - 438 -------------------------------

Let a, b, c be positive real numbers such that 12 21 2 8ab bc ca" ! ! . Then

1 2 3 15

2a b c! ! "

Tran Nam Dung, Vietnam TST, 2001

------------------------------- AI3 - 439 -------------------------------


4a a a

b c b c! "

!

------------------------------- AI3 - 440 -------------------------------


a b a c b c

b a b c a c

! !! " !

! !


89

------------------------------- AI3 - 447 -------------------------------


1a b c a b b c

b c a b c a b

! !! ! " ! !

! !

------------------------------- AI3 - 448 -------------------------------

Let a, b, c be positive real numbers such that c b a" " . Prove that

a b c b c a

b c a a b c! ! " ! !

------------------------------- AI3 - 449 -------------------------------


a b c a b b c c a

b c a a c b a c b

! ! !! ! " ! !

! ! !Indian-TST to the IMO 2002

------------------------------- AI3 - 450 -------------------------------

Let a, b, c, k be positive real numbers. Prove that

a b c a kb b kc c ka

b c a a kc b ka c kb

! ! !! ! " ! !

! ! !Nguyen Viet Anh

------------------------------- AI3 - 451 -------------------------------


3

2

a b c a b c

b c a b c a c a b! ! " ! ! !

! ! !


------------------------------- AI3 - 452 -------------------------------


1 2 2 5

2 2

a b c b b a b c

b c a a c b c a b

! !) *! ! " ! ! !+ ,! ! !- .



91

------------------------------- AI3 - 459 -------------------------------


2 2 2 52

4

a b c a b c

b c a ab bc ca

! !! ! " 6 !

! !


------------------------------- AI3 - 460 -------------------------------

Let a, b, c be positive real numbers such that 1abc $ . Prove that

a b ca b c

c a b! ! " ! !

EXCALIBUR, 5(4), 2000

------------------------------- AI3 - 461 -------------------------------


' (' (' (1 1 1 5a b c

a b cb c a! ! " ! ! ! %

Aaron Pixton

------------------------------- AI3 - 462 -------------------------------


' (1 1 1 1

2

a b ca b c

b c a a b c

0 1) *! ! " ! ! ! ! !2 3+ ,

- .4 5

------------------------------- AI3 - 463 -------------------------------


' (2

1 1 1a b ca b c

b c a a b c

) * ) *! ! " ! ! ! !+ , + ,

- . - .Great Britain Olympiad, 2005

------------------------------- AI3 - 464 -------------------------------


23

2

a b c a b b c c a

b c a c a b

! ! !) * ) *! ! " ! !+ , + ,

- . - .Romany-JTST, 2006


93

------------------------------- AI3 - 471 -------------------------------

Let a, b, c be positive real numbers, show that

3 3 3 32ab bc ca

a b cc a b! ! # 6 ! !

China-TST, 2008

------------------------------- AI3 - 472 -------------------------------


3 3 3 321

3 3

ab bc ca a b ca b c

c a b

! !! ! # ! ! 6 ! !


------------------------------- AI3 - 473 -------------------------------


6 3 3 34ab bc ca

a b cc a b! ! # 6 ! !


------------------------------- AI3 - 474 -------------------------------

If a, b, c > 0 and 4a b c abc! ! ! / , then

ab bc caa b c

c a b! ! # ! !

China-TST, 2007

------------------------------- AI3 - 475 -------------------------------


' (22 2 2 4 a ba b ca b c

b c a a b c

%! ! " ! ! !

! !BMO, 2005

------------------------------- AI3 - 476 -------------------------------


' (' (2 2 22 2 2 a b c a b ca b c

b c a ab bc ca

! ! ! !! ! "

! !MathLinks, t=249523, 2009


95

------------------------------- AI3 - 483 -------------------------------

Let , , 0a b c # such that 1a b c! ! / . Prove:

' (2 2 2

2 2 23a b c

a b cb c a! ! " ! !

Croatia TST 2007

------------------------------- AI3 - 484 -------------------------------

Given positive real numbers a, b, c, prove that

' ( ' (' (

2 2 22 2 2 37 19

6

a b c ab bc caa b c

b c a a b c

! ! % ! !! ! "

! !Michael Rozenberg, ML, t=296853, 2009

------------------------------- AI3 - 485 -------------------------------


2 2 2

2 2 2

a b c a b c

b c ab c a! ! " ! !

1st Nordic Mathematical Olympiad

------------------------------- AI3 - 486 -------------------------------

Given positive real numbers a, b, c, prove that

' (2 2 2

2 2 2 2 2 2

912

ab bc caa b c

b c a a b c

! !! ! " %

! !

------------------------------- AI3 - 487 -------------------------------


2 2 2 2 2 2

2 2 2 2 2 2

1 13

2 2

a b c b c a a b c b c a

b c a a b cb c a a b c

) * ) * ) * ) *! ! ! ! ! " ! ! ! ! ! !+ , + , + , + ,+ , + , - . - .- . - .

------------------------------- AI3 - 488 -------------------------------


' ( ' (2 2 2

2 2 2

2 2 2 2 2 2

2 1 1 1 1 1 1

3

a b ca b c a b c

a b cb c a a b c

) * ) *! ! " ! ! ! ! % ! ! ! !+ , + ,

- . - .



97

------------------------------- AI3 - 495 -------------------------------


3 3 3

3 3 3, , , 0

a b c a b ca b c

b c ab c a! ! " ! ! #

------------------------------- AI3 - 496 -------------------------------


' ( ' ( ' (3 3 3

2 2 2a b c a b c a b ca b c

a b c

% ! ! % ! ! %! ! " ! !

------------------------------- AI3 - 497 -------------------------------

Suppose that a, b, c are positive real numbers and 5 5 5 3a b c! ! / . Then

4 4 4

3 3 33

a b c

b c a! ! "

------------------------------- AI3 - 498 -------------------------------

Prove that for all positive real numbers a, b, c

25 5 5 6 6 6

333

a b c a b c

b c a

) *! !! ! " 6 + ,+ ,

- .Nguyen Thuc Vu Hoang, MathLinks, t=298305, 2009

------------------------------- AI3 - 499 -------------------------------


33

2

a b c

b c a! ! "

------------------------------- AI3 - 500 -------------------------------


32 3 6a b c

b c a! 6 ! 6 "


99

------------------------------- AI3 - 507 -------------------------------

Prove that

3 2a b b c c a

c a b

! ! !! ! "

where a, b, c are positive real numbers.'#()#"*+,-.!/!0%12*3%#*WURZEL, n. A50

------------------------------- AI3 - 508 -------------------------------


36

a b b c c a a b c

c a b abc

! ! ! ! !! ! " 6

Pham Huu Duc, Mathemtical Reflections n. S41, 2007

------------------------------- AI3 - 509 -------------------------------


' (' (' (' (

316

3

a b ca b b c c a

c a b a b b c c a

! !! ! !! ! "

! ! !

Vo Quoc Ba Can, Mathemtical Reflections n. O43, 2007

------------------------------- AI3 - 510 -------------------------------


' (' (' (' (

43

22

a b b c c aa b c

c a b abc a b b c c a

! ! !! ! " ! ! 6

! ! !


------------------------------- AI3 - 511 -------------------------------


' (' (' (' (

3

46

2a b ca b b c c a

c a b a b b c c a

! !! ! !! ! " 6

! ! !


------------------------------- AI3 - 512 -------------------------------


' (' (' (' (

4

63

92

a b ca b b c c a

c a b abc a b b c c a

! !! ! !! ! " 6

! ! !



101

------------------------------- AI3 - 519 -------------------------------

Let a, b, c be positive real numbers, prove that

' (' (' (8

4 1a b b c c a a b c abc

c a b b c c a a b a b b c c a

! ! ! ) *! ! % ! ! " %+ ,! ! ! ! ! !- .

Cezar Lupu & Cosmin Pohoata, RSME, 11(1), n. 97, 2008

------------------------------- AI3 - 520 -------------------------------

For a, b, c > 0, if abc = 1, then show that

3b c c a a b

a b ca b c

! ! !! ! " ! ! !

EXCALIBUR, 5(3) & 5(4), 2000

------------------------------- AI3 - 521 -------------------------------


' ( ' ( ' ( ' (3 3 3

8a b b c c a

ab bc cac a b

! ! !! ! " ! !

------------------------------- AI3 - 522 -------------------------------


' ( ' ( ' ( ' (3 3 3

2 2 28a b b c c a

a b cc a b

! ! !! ! " ! !

Romania, order 8, 2003

------------------------------- AI3 - 523 -------------------------------


' ( ' ( ' ( ' ( ' (3 3 3

22 2 22 2a b b c c a

a b c a b cc a b

! ! !! ! " ! ! ! ! !


------------------------------- AI3 - 524 -------------------------------

Let a, b, c be positive real numbers such that 1abc / , prove that

2 2 2 2 2 2

6a b b c c a

c a b

! ! !! ! "

Gazeta Matematic , 2002


103

------------------------------- AI3 - 531 -------------------------------

If a, b, c are positive numbers, then

' (2 2 22 2 2 6 a b ca b c

a b cb c a a b c

! !! ! ! ! ! "

! !Pham Huu Duc, MathLinks, 2006

------------------------------- AI3 - 532 -------------------------------


3 3 3 3 3 3

6a b ab b c bc c a ca

abcc a b

! ! !! ! "

------------------------------- AI3 - 533 -------------------------------


1 1 1 1 1 12

a b c a b b c c a

) *! ! " ! !+ ,! ! !- .

------------------------------- AI3 - 534 -------------------------------

Let a, b, c be positive real numbers such that a b c" " . Prove that

2 2 22 2 2a b b c c aa b c

c a b! ! " ! !

Vietnam, 29ts MO, 1991

------------------------------- AI3 - 535 -------------------------------


2 2 21 1 1 75

4a b cb c a

) * ) * ) *! ! ! ! ! "+ , + , + ,

- . - . - .ARML, 1987

------------------------------- AI3 - 536 -------------------------------

Let a, b, c be positive real numbers such that 1ab bc ca! ! / . Then

2 2 21 1 1

16a b cb c a

) * ) * ) *! ! ! ! ! "+ , + , + ,

- . - . - .Mircea Becheanu, Mathematical Reflections, n. J57, 2007


105

' (1 1 1 47

3a b c

a b c! ! ! ! ! "

------------------------------- AI3 - 543 -------------------------------


2 2 2

4a b b c c a

c a b

! ! !! ! "

------------------------------- AI3 - 544 -------------------------------


1 1 1 1 1 11 1 1 a b ca b c b c a

) *) *) * ) *) *) *! ! ! " ! ! !+ ,+ ,+ , + ,+ ,+ ,

- .- .- . - .- .- .

------------------------------- AI3 - 545 -------------------------------


1 1 11 1 1 64a b c

) *) *) *! ! ! "+ ,+ ,+ ,

- .- .- .Yugoslavia, 30th MO, 1989

------------------------------- AI3 - 546 -------------------------------

Let a, b, c be arbitrary positive real numbers. Prove that

' (3

21 1 1 2

a b ca b c

b c a abc

! !) *) *) *! ! ! " !+ ,+ ,+ ,

- .- .- .APMO, 1998

------------------------------- AI3 - 547 -------------------------------


' (' (' (2 2 2

1 1 1 1 1 1a b c

a b cb c a

) *) *) *! ! ! " ! ! !+ ,+ ,+ ,+ ,+ ,+ ,

- .- .- .APMO

------------------------------- AI3 - 548 -------------------------------



107

1 1 13a b c

a b ca b c b c a

! ! ! ! ! $ ! ! !

Todor Mitev, CRUX, n. 3236

------------------------------- AI3 - 555 -------------------------------


2 2 2 3 1 1 16

2a b c a b c

a b c

) *! ! ! " ! ! ! ! !+ ,

- .Vasile Cirtoaje, ML, 2006

------------------------------- AI3 - 556 -------------------------------


' ( 1 1 13

a b ca b c

b c a a b c

) *! ! " ! ! ! ! ! %+ ,

- .EXCALIBUR, 12(4), 2007

------------------------------- AI3 - 557 -------------------------------


2 2 2 1 1 19 2

a b ca b c a b c

b c a a b c

) *! ! ! ! ! ! " ! ! ! ! !+ ,

- .Panagiote Ligouras, MathLinks, t=252656, 2009

------------------------------- AI3 - 558 -------------------------------


2 1 1 12

3

a b ca b c

b c a a b c

) * ) *! ! ! ! ! " ! ! !+ , + ,

- . - .

------------------------------- AI3 - 559 -------------------------------


' ( 1 1 19a b c

a b c

) *! ! ! ! "+ ,

- .

------------------------------- AI3 - 560 -------------------------------



109

If , , 0y a b c x" " # , then

' (' ( ' (2 2 1 1 1

9x y x y

a b cxy a b c

! ! ) *" ! ! ! ! "+ ,

- .

1691 Disequazioni Algebriche 2009-65_a_finale-Dispari (4).....

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