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[r]
(1)X]`XGhGpG Problem Solving
(2)3
Panagiote Ligouras
X]`XGG
hsnliyhpjG
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Problem Solving
Old and New Problems for Mathematical Olympiads
(3)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 to (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
(4)7
Preface……….………… pg. 5
Contents……… 7
Part I Problems with variable ……… ……… 11
Problems with variables ……… ……… 11
Problems with variables ……… ……… 11
Problems with variables ……… ……… 11
Problems with variables ……… ……… 11
Problems with variables ……… ……… 11
Problems with n > 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
(5)9
(6)11 - AI1 - 1 -Let a be real number ( a R) Prove that:
1
) 2, ) 2,
a a a b a a
a a
! " # ! $ % &
- AI1 - 2 -Let a be non-negative real number (a "0) Prove that:
5
3
a %a ! "a !
Titu Andreescu, 2004 - AI1 - 3 -Let a be non-negative real number (a "0) Prove that:
' (2 ' (
2
5 a % !a "2 1!a
Panagiote Ligouras, 2009 - AI1 - 3 -Let a be non-negative real number (a " ) Prove that:0
' 2 ( '3 3 (' (3
2 a !1 " a !1 a!1
- AI1 - 4 -Let a be non-negative real number (a "0) Prove that:
' ( '3 (' (' (3
3
4 a !1 " a !1 a !1 a!1 , a"0
Panagiote Ligouras, 2009
- AI1 - 5 -Let a be non-negative real number (a "0) Prove that:
' ( '4 (' (4
3
(7)13
1
1 ,
4
8 a
a a
) *) *
! ! " # #
+ ,+ ,
%
- -
Gazeta Matematica, 2002 - AI1 - 12 -Let a be real number ( a R) such that % & & Prove that:1 a
41%a2 !41% !a 41! $a 3
The Mathscope, n.216.3 - AI1 - 13 -Let a be positive real number (a #0) Prove that:
3
2 4
2
1
1
a a
a a
a
! " % ! " ! !
CRUX, 35(4), p 254, 2009 - AI1 - 14 -Let a be non-negative real number (a "0) Prove that:
3 2
a! a! a $ !a
The Mathscope, n.332.5 - AI1 - 15 -Let a be non-negative real number (a "0) Prove that:
3 2 1
a! a ! a $ a!
Panagiote Ligouras, Mathlinks, 2009 - AI1 - 16 -Let a be real number such that a " % Prove that:2
3
2 2
a! ! a! ! a! $ !a
Panagiote Ligouras, Mathlinks, 2009
(8)-15
P r o b l e m s w i t h v a r i a b l e s
- AI2 - 21 -Let a, b be non-negative real numbers ( ,a b "0) Prove that:
2
a! "b ab
- AI2 - 22 -Let a, b be real numbers ( ,a b R) Prove that:
2 2
a !b " ab
- AI2 - 23 -Let ,a b "0 Prove that:
' (
3
a !b "ab a!b
- AI2 - 24 -Let ,a b "0 Prove that:
' (
4 2
a !b "ab a !b
- AI2 - 25 -Let ,a b "0 Prove that:
' ( ' (
5 3 2
a !b "ab a !b "a b a!b
- AI2 - 26 -Let ,a b R Prove that:
' (2
4
(9)17 - AI2 - 33 -Let ,a b #0 Prove that:
2
2
a b a b
a b
! " ! !
- AI2 - 34 -Let ,a b #0 Prove that:
2
1 1
a b
a b
a b
! $ ) ! *
+ ,
! -
- AI2 - 35 -Let ,a b #0 Prove that:
4
ab a b
a b
! $ !
- AI2 - 36 -Let ,a b #0 Prove that:
4 1
a b! $a!b
- AI2 - 37 -Let ,a b #0 such that a b! / Prove that:1
2
1 25
2
a b
a b
) ! * !) ! * "
+ , + ,
- -
- AI2 - 38 -Let ,a b "0 and n N* Prove that:
1
n n n n
(10)19 - AI2 - 45 -Let ,a b R* Prove that:
2 2
3 3
2 3
2 3
a b b a
a b
a b b a
! !
! $
!
! !
The Mathscope, n 285.2 - AI2 - 46 -Let ,a b #0 Prove that:
4 2
1 a b
ab"a !b !a !b
Russia, 1995 - AI2 - 47 -Let ,a b #0and k " Prove that:1
' (2 ' (2 ' (
a kb b ka a b
a k ab b
b ka a kb
! ! !
! "
! % !
! !
MathLinks, t=255035, 2009 - AI2 - 48 -Let ,a b#0, a b! / Prove that:1
1 2
, 1, , 1% a !1% b " 1% a b! / a b#
- AI2 - 49 -Let ,a b#0, a b! / Prove that:2
2 1 1
b a
a b
a ! ! b ! $
- AI2 - 50 -Let a b, #0, m N m, "2, r R rm, "1, ab/r2 Prove that:
' ( ' ( ' (
1
1 a m b m r m
! "
! ! !
(11)21 - AI2 - 57 -Let ,a b #0 Prove that:
3
2
a !b " ! !a b ab%
Norman Schaumberger, N.Y.S.M.T.J., n.153, 1984 - AI2 - 58 -Let ,a b #0 Prove that:
1
b a
a !b #
Kàlmàn Szabò, T.M.G., n 67.H, 1983 - AI2 - 59 -Let ,a b R such that a b &, Prove that:
1 ab! # !a b
PARABOLA, Q591, 1984 - AI2 - 60 -Let ,a b "0 Prove that:
' (2 ' (2
1 1
1
1 a b ab
! " !
! !
- AI2 - 61 -Let 0$x y, $ Prove that:1
' ( '2 (2 ' (' (
2
1!x ! 1!y ! 1%x ! %1 y " !1 1%xy
- AI2 - 62 -Let ,a b #0 Prove that:
' ( 3
32 a b 1 a b, ,a b
a b b a
) *
! + ! ," ! #
-
(12)23 - AI2 - 69 -Let a b! $1, ,a b# Prove that:0
3 2
1 1
20
a !b !a b!ab "
Cao Minh Quang - AI2 - 70 -Let ab/1, ,a b# Prove that:0
3
1
1
a b
b! a"
! !
Le Thanh Hai - AI2 - 71 -Let a b " Prove that:,
2 2
3
8
a b ab a b
a b
) % * !
! "
+ ,
+ , !
-
Yugoslavia, 1991 - AI2 - 72 -Let , 1,3
2
x y 30
4 Prove that:
2
3
x !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 xn!yn /1 Prove that
' (' (
2
4
1
1 1
1 1
n k n k
k k
k k
x y
x y / x / y
) ! * ) ! *
#++ , +, +6 ,,
% % -7 ! -7 ! .
(13)25
P r o b l e m s w i t h v a r i a b l e s
- AI3 - 79 -Let a, b, and c be positive real numbers such that a2!b2!c2 / Prove that:
2 2
3
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&a b c, , &1 and
1
ab bc! !ca/ Prove that:
2 2
3
1 1
a b c
a ! b ! c "
% % %
- AI3 - 81 -Let a, b, and c be positive real numbers such that 0&a b c, , &1 and
1
ab bc! !ca/ Prove that:
2 2
2 2
3 1
1 1
a b c a b c
a b c
a b c
) % % % * ! ! " ++ ! ! ,,
% % % - .
- AI3 - 82 -Let a, b, and c be real numbers such that ab bc! !ca/1 and
1 a b c, ,
% & & Prove that:
' (' (
2
2
9
1
cyc
a b
a b
!
"
% %
7
(14)27 For positive real numbers a, b, and c with a b c! ! / , show that1
1 1
2
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
1
abc / , then prove that
' ( ' ( ' (
2 2
2 2
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 , ,a b c # Prove that:1
' ( ' ( ' (
4 4
2 2 48
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 a b c #, , and
ab bc! !ca/abc Prove that:
' ( ' ( ' (
4 4
2 2
4 4 4 54
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 a b c! ! / Prove that:
1 1 27
3
1 ab bc ca
$ ! ! $
% % %
(15)29 - AI3 - 99 -Let a, b, c be non-negative real numbers, no two of which are zero Prove that
' (' ( ' (' ( ' (' (
3 3
2 2 2 2 2 2
1
2 2 2
a b c
a b c a 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
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
0
cyc
a bc p a b c q
a b c
0 % ! ! !
2
"
2 ! !
2
4
7
Panagiote Ligouras, 2009 - AI3 - 102 -If a, b, c are non-negative real numbers, then
2 2
2 2 2 2 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
1 1
(16)31 Let a, b, c be positive reals such that a! b! c/ Prove that3
2 2
2 2
2 2
a b c
b !a! c !b! a !c"
- AI3 - 110 -Let a, b, c be positive reals such that a b c! ! / Prove that
2 2
3 3 1,
2 2
a b c
a 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
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
2 2 2 3
2 2 2
a b c a b c
a ab b b bc c c ca a
! !
! ! "
% ! % ! % !
Nguyen Viet Anh - AI3 - 113 -Let a, b, c be positive real numbers Prove that
1
2 2
a b c
a b! ! b c! ! c!a$
Gazeta Matematic , 2000 - AI3 - 114 -Let a, b, c be positive real numbers Prove that
1 1
(17)33 - AI3 - 121 -Show that for all positive reals a, b, c,
3 3
2 2
3
2 2
a b b c c a
a c b a c b
! ! !
) * !) * !) * "
+ ! , + ! , + ! ,
- - -
MOP, 2004 - AI3 - 122 -Let a, b, c be positive real numbers Prove that
3
2 2
a b c
a! b c! !a b! ! c! a b c! ! "
- AI3 - 123 -Let a, b, c be positive real numbers Prove that
3
2 2
a b c
a b c! ! !a! b c! !a b! ! c$
M&Y-9, 2004 - AI3 - 124 -Let a, b, c be positive real numbers Prove that
2 2
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 3
9
a b c! ! / a b c and , ,a b c R!, prove that
2 2
ab bc ca
a b! ! c! a b c! ! !a! b c! "
Shaastra Online Math Contest, 2009 - AI3 - 126 -Let a, b, c be positive real numbers Prove that
3
2 2
ab c ca b bc a a b c
a b c a b c a b c
! ! ! ! ! !
! ! $
(18)35 Let a, b, c be positive real numbers Prove that
2 2 1 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
) *
! ! ! ! ! $ ! ! ! ! !
+ ,
+ ,
! ! ! ! ! ! - ! ! ! .
Panagiote Ligouras, MathLinks, t=259342, 2009 - AI3 - 134 -Let a, b, c be positive real numbers Prove that
' (2 ' (2 ' (2 ' (
2 2
b c a c a b a b c
abc a b c
a b c a b c a b c
! ! !
! ! " ! !
! ! ! ! ! !
MathLinks, t=259361, 2009 - AI3 - 135 -Let a, b, c be positive real numbers Prove that
' ('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 -Let a, b, c be positive real numbers Prove that
' ( ' (' (
' ( ' (' (
' ( ' (' (
3
2 2
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 -Let a, b, c be positive real numbers Prove that
' ( ' (
' (' ( ' (
2
2
cyc
ab a c b c
abc a b c
a b c a b c
! !
" ! !
! ! ! !
7
MathLinks, t=259603, 2009 - AI3 - 138 -Let a, b, c be positive real numbers Prove that
3
2 2
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 ! ! "
! ! ! ! ! ! ! ! !
(19)37 - AI3 - 145 -Let a, b, and c be positive real numbers such that abc / Prove that 1
' ( ' ( ' ( ' ( ' ( ' (
2 2
1 1
1
2 2
a a ab b b bc c c ca
a b b c c a
! ! ! ! ! !
! ! "
! ! !
MathLinks, t=258676, 2009 - AI3 - 146 -Let a, b, and c be positive real numbers such that abc / Prove that 1
' (' ( ' ( ' (' ( ' ( ' (' ( ' (
2 2
1 1 1
4
2 2
a a ab a b b b bc b c c c ca c a
a b b c c a
! ! ! ! ! ! ! ! ! ! ! !
! ! "
! ! !
Panagiote Ligouras, MathLinks, t=307913, 2009 - AI3 - 147 -Let a, b, and c be positive real numbers Prove that
' ( ' ( ' (
2 2
2 2
2 2
3
2 2
ac ab b ab bc c bc ca a
c b a c b a
! ! ! ! ! ! ! ! "
! ! !
Panagiote Ligouras, MathLinks, t=307910, 2009 - AI3 - 148 -Let a, b, and c be positive real numbers such that abc / Prove that 1
' ( ' ( ' ( ' ( ' ( ' (
2 2
1 2
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 -Let a, b, and c be positive real numbers such that abc / Prove that 1
' (
' (
2 2
2
1
3
cyc
a c a ab bc a b c
ab bc ca
a b
0 ! ! ! ! ! !
2 ! "
! !
2 !
4
7
(20)39 Let a, b, and c be positive real numbers Prove that
' ( ' (' (
2
2
4
2
2
cyc cyc
a b ab
a c c b
c 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(
1 1 11
7
2a b 2b c 2c a a b c ! ! "
! !
% % %
Pham Kim Hung - AI3 - 157 -Let a, b, and c be positive real numbers such that a2!b2!c2 / Then
2 2 3
1 2
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 a b c! ! / Prove that3
3
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
4 4 4
c a b
a ab b b bc c c ca a
! ! "
! ! ! ! ! !
Pham Kim Hung & Vo Quoc Ba Can - AI3 - 160 -Let a, b, and c be positive real numbers Prove that
2 2 2
4 4 4
a b c
a ab b b bc c c ca a
! ! $
! ! ! ! ! !
(21)41 - AI3 - 167 -Let a, b, and c be non-negative numbers such that ab bc! !ac8 Then0
' (
' (
2 2
6
4 4
a b c
a b b c a c
ab bc ac
c ab a bc b ca
! !
! ! !
! ! "
! !
! ! !
Michael Rozenberg, MathLinks, t=262789, 2009 - AI3 - 168 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that
2
1
3
4
cyc a bc a
"
! ! !
7
Panagiote Ligouras, MathLinks, t=310812, 2009 - AI3 - 169 -Let a, b, and c be positive real numbers Prove that
2 2
1 1 a b c
abc
a b c
! ! ! ! "
- AI3 - 170 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
2 2
2 2
1 1
a b c
a !b !c " ! !
Romania-TST, Iasi & Bucharest, 2006 - AI3 - 171 -Suppose that a, b, and c are positive reals satisfying a2!b2!c2 / Prove that
' 3 3(
2 2
2
1 1
3
a b c
abc
a b c
! !
! ! " !
Ho-joo Lee, CRUX, n 2532 - AI3 - 172 -Let a, b, c be positive real numbers such that a b c! ! / Prove that1
2 2
1 1 28
1
27
abc
a b c
) ! ! ! *"
+ ,
-
(22)43 Let a, b, and c be positive real numbers Prove that
' (
' (
2 2
2 2 2
9
4 4
ab bc ca
a b c a b c a b c
c b a
b c a a b c
! !
! ! !
) *
! ! " !+ ! ! ,%
- ! !
Panagiote Ligouras, ML, t=313586, 2009 - AI3 - 180 -Let a, b, and c be positive real numbers Prove that
2 2 2
2 2 2
10
3
3
a b c a b c ab bc ca
ab bc ca
b c a a b c
! ! ! !
! ! " % %
! ! ! !
Panagiote Ligouras, ML, t=320792, 2009 - AI3 - 181 -Let a, b, and c be positive real numbers Prove that
' (' (' (
2 2
2 2
120 18
a b c abc
a b b c c a b !c !a " % ! ! !
Tran Quoc Anh, ML t=293216, 2009 - AI3 - 182 -Let a, b, and c be positive real numbers Prove that
2 2
3
2 2
1
9
2
a b c a b c
b c a
b c a
) *
! ! ! " ! !
+ ,
+ ,
-
- AI3 - 183 -Let a, b, and c be positive real numbers Prove that
2 2
3
2 2
3
1
2
a b c a b c
b c a
b c a
) *
! ! ! " ! ! "
+ ,
+ ,
-
- AI3 - 184 -Let a, b, and c be positive real numbers Prove that
2 2
2 2 3
a b c a b c
(23)45 - AI3 - 191 -Let a, b, and c be positive real numbers such that abc / Prove that 1
' ( ' ( ' (
2 2
3
1 1
c a b
a !b !b !c !c !a "
- AI3 - 192 -Let x, y, and z be positive real numbers Prove that
' ( ' (
2
2 2
2
2
2 2 2
9
2 2
x z z y y x
xy yz zx
z x y xyz x y y z z x
! ! ) *
) ! * !) ! * ! ! " + , + , + , + ,
- - - ! !
Panagiote Ligouras, MathLinks, t=310024, 2009 - AI3 - 193 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that
9
2 2
1 1
1 1
a b c
) % *) % *) % *"
+ ,+ ,+ ,
- - -
- AI3 - 194 -Let a, b, and c be non-negative real numbers such that ab bc! !ca/ Prove that
2 2
1 1
2 1!a !1!b !1!c "
Vasile Cirtoaje, MathLinks, 2005 - AI3 - 195 -If , ,a b c $1and a b c! ! / Prove that1
2 2
1 1 27
10 1!a !1!b !1!c $
Titu Andreescu & Gabriel Dospinescu - AI3 - 196 -Let a, b, and c be positive real numbers such that a2!b2!c2 / Then1
' (2
2 2
3
1 1
a b c
a a b b c c b ! !c ! !a ! " ! !
(24)47 Let a, b, and c be positive reals such that ab bc! !ca/ Prove that
2 2
1
2
16
1 1
ab bc ca a b c b c a
b c a a b c
a b c
0 ) *
! ! $ 2 ! ! ! + ! ! ,! 3
! ! ! -
MathLinks, t=259357, 2009 - AI3 - 204 -Let a, b, and c be positive real numbers with ab bc! !ca/ Prove that
2 2
3
1 1
ab bc ca
c ! !a ! !b ! "
Titu Zvonaru, Mathematical Mayhem, 34(4) M299, 2008 - AI3 - 205 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that3
2 2
2 2
3
1 1
a b c
b ! !c ! !a ! "
Faruk Zejnulahi & Sefket Arslanagic, CRUX, n 2994 - AI3 - 206 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that3
2 2
1 1
3
1 1
a b c
b c a
! ! ! ! ! "
! ! !
- AI3 - 207 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that1
2 2
2 2
1 1
2
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 ab bc! !ca/ Prove that
2 2
2 2
1 1
2
1 1
a b c
a b c
% % %
! ! $
! ! !
(25)49 - AI3 - 215 -Let a, b, and c be positive real numbers Prove that
' (' ( ' (' 2 (
2 3
1 1
1
6 max ,
1 cyc cyc
cyc
a bc a ab c a b
a
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
1 1
2
1 1
a b c
! ! $
! ! !
EXCALIBUR, 6(2) & 6(3), 2001 - AI3 - 217 -Let r be a real number, 0& $ , and let a, b, and c be positive real r numbers such that abc/r3 Prove that
2 2
1 1
1 1
a b c r
! ! $
! ! ! !
Arkady Alt, CRUX, 34(3), n 3329, 2008 - AI3 - 218 -Let a, b, and c be positive reals such that ab bc! !ca/ Prove that
2 2
3
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
3
1 1
a b c
a b c
! ! $
(26)51 Let a, b, and c be non-negative reals such that ab bc! !ca/ Prove that
2 2
1 1
1
2 2
a ! !b ! !c ! $
- AI3 - 226 -Let a, b, and c be non-negative reals such that abc / Prove that 1
2 2
7 7
1
2 2
a b c
a b c
% ! % ! % "
! ! !
Vasile Cirtoaje, MathLinks, t=245046, 2008 - AI3 - 227 -Let a, b, and c be positive real numbers Prove that
2 2
1 1
2 2 ab bc ca
a ! bc!b ! ca!c ! ab" ! !
Vasile Cirtoaje, MathLinks, 2005 - AI3 - 228 -Let a, b, and c be positive real numbers Prove that
2 2
2 2 2 2
a b c
a ! bc!b ! ca!c ! ab"
Romania, 1997 - AI3 - 229 -Let a, b, and c be positive real numbers Prove that
2 2 2 2
bc ca ab
a ! bc!b ! ca!c ! ab$
Romania, 1997 - AI3 - 230 -Let a, b, and c be positive real numbers Prove that
2 2
2 2
2 2
1
2 2
a bc b ca c ab
a bc b ca c ab
% % %
! ! "
! ! !
(27)53 - AI3 - 237 -Let a, b, and c be positive real numbers Prove that
2 2 2 2
a b c a b c
ab bc ca
a 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 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
2 2
a b c a b c
ab bc ca
a bc b ca c ab
! !
! ! $
! !
! ! !
Tran Quoc Anh, MathLinks, t=224981, 2008 - AI3 - 240 -Let a, b, and c be positive real numbers Prove that
2 2
1 1
2 2
a b c a b c
ab bc ca
a bc b ca c ab
% ! % ! % $ ! ! %
! !
! ! !
Panagiote Ligouras, ML, t=322098, 2010 - AI3 - 241 -Let a, b, and c be positive real numbers Prove or disprove that
' ( ' (' (
2
2 1 1 2
2
cyc
a a bc a b c ab bc ca
ab bc ca
a bc
! ! % ! ! ! ! ! %
$
! !
!
7
Panagiote Ligouras, MathLinks, t=251776, 2009 - AI3 - 242 -Let a, b, and c be positive real numbers Prove that
2 2
1 1
3 3
a b c
a b c abc
! !
! ! $
! ! !
(28)55 Let a, b, and c be positive real numbers such that abc / Prove that 1
3 3
2 2 2
4 4
a b c
a ! ab b! !b ! bc!c !c ! ca!a "
José Luis Dìaz-Barrero, SSMJ, n 5054, 2009 - AI3 - 249 -Let a, b, and c be positive real numbers such that abc $ Prove that 1
2 2
3
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
1 1
a ab b b bc c c ca a a b c
! ! "
! ! ! ! ! ! ! !
Vasile Cirtoaje, Gazeta M!"#$!"%& -B, 9, 2000 - AI3 - 251 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that
2 2 2
1 1
abc a !ab b! !b !bc!c !c !ca!a $
- AI3 - 252 -Let a, b, and c be positive real numbers Prove that
' ( ' ( ' (
2 2
2 2 2
3
a b c
a b c
b a ab b c b bc c a c ca a
! ! "
! !
! ! ! ! ! !
- AI3 - 253 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that
' ( ' ( ' (
2 2
2 2 2 2
a b b c c a
b a ab b c b bc c a c ca a
! ! ! ! ! "
! ! ! ! ! !
(29)57 - AI3 - 260 -Let a, b, c be non-negative reals, no two of which are zero Prove that
' ( ' ( ' (
2 2 2 2
a b c b c a c a b
b bc c c ca a a ab b
! ! !
! ! "
! ! ! ! ! !
- AI3 - 261 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2 2 2
a bc b ca c ab
b bc c c ca a a ab b
! ! !
! ! "
! ! ! ! ! !
Vasile Cirtoaje, ML, 2005 - AI3 - 262 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2 2
2 3
5
a bc b ca c ab
b bc c c ca a a ab b
! ! ! ! ! "
! ! ! ! ! !
- AI3 - 263 -Let a, b, and c be positive real numbers Prove that
2 2
2 2 2
5 5
7
a bc b ca c ab
b bc c c ca a a ab b
! ! ! ! ! "
! ! ! ! ! !
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
pa qbc pb qca pc qab
p q
b bc c c ca a a ab b
! ! ! ! ! " !
! ! ! ! ! !
Panagiote Ligouras, 2009 - AI3 - 265 -Let a, b, and c be positive real numbers Prove that
' (
' (
2 2 2 2 2
2 2 2 2 2
4
3 3
3
ab bc ca
a b c b c a c a b
a ab b b bc c c ca a a b c
! !
! % ! ! % ! ! % ! $
! ! ! ! ! ! ! !
(30)59 Let a, b, and c be positive real numbers Prove that
' (
3 3 3
2 2 2
2
a b b c c a
a b c
a ab b b bc c c ca a
! ! !
! ! " ! !
! ! ! ! ! !
- AI3 - 272 -Let a, b, c be positive real numbers and k > Prove that
' (
3 3 3
2 2 2
1
ka b kb c kc a k
a b c
a ab b b bc c c ca a
! ! ! ! ! " ! ! !
! ! ! ! ! !
Panagiote Ligouras, 2009 - AI3 - 273 -Let a, b, and c be positive real numbers such that abc / Prove that 1
3 3 3
2 2 2 2
a b b c c a
a ab b b bc c c ca a
! ! ! ! ! "
! ! ! ! ! !
- AI3 - 274 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that
3 3
2 2 2
a b c
a !ab b! !b !bc!c !c !ca!a "
- AI3 - 275 -Let a, b, and c be positive real numbers such that abc / Prove that 1
3 3
2 2 2
a b c
a !ab b! !b !bc!c !c !ca!a "
José Luis Dìaz-Barrero, SSMJ 2009(2) n 5054 - AI3 - 276 -Let a, b, c be positive real numbers such that abc / and k > Prove that 1
3 3 3
2 2 2
ka b kb c kc a
k
a ab b b bc c c ca a
! ! !
! ! " !
! ! ! ! ! !
(31)61 - AI3 - 282 -Let a, b, and c be positive real numbers Prove that
3 3
2 2 2
3
a b c a b c
a ab b b bc c c ca a
! !
! ! "
! ! ! ! ! !
Le Trung Kien - AI3 - 283 -If a, b, c >0, prove that
2 2 2 3 3
ab bc ca a b c
a ab b b bc c c ca a
! ! ! !
$
! ! ! ! ! ! ! !
Murray S Klamkin, CRUX n 805 - AI3 - 284 -Let a, b, c be non-negative reals, no two of which are zero Prove that
' (
2 2 2 2 2
1 1 15
2
b c c a a b a b c ab ac ca
! ! "
! ! ! ! ! ! ! !
- AI3 - 285 -Let a, b, c be non-negative reals, no two of which are zero Prove that
' (
2 2 2 2
1 1 10
b c c a a b a b c
! ! "
! ! ! ! !
- AI3 - 286 -Let a, b, and c be positive real numbers such that a b c! ! / Prove that
2 2 2
1 1
2
b !c !c !a !a !b "
- AI3 - 287 -Let a, b, c be non-negative real numbers Prove that
2 2 2 2 2
1 1
ab bc ca
a !b !b !c !c !a " ! ! %a !b !c
(32)63 Let a, b, and c be positive real numbers such that abc / Then 1
3 3
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 -Let a, b, and c be positive real numbers Prove that
2 2 2
1 1
a b b c c a
a b c
a b b c c a
! ! !
! ! $ ! !
! ! !
- AI3 - 296 -Let a, b, and c be positive real numbers Prove that
' ( ' ( ' (
2 2 2
1 5 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 b
b c c a a b
) *
% % % % % % % % %
! ! " ++ ! ! ,,
! ! !
! ! ! - .
Panagiote Ligouras, MathLinks, t=326767, 2010 - AI3 - 297 -If c" " # , thenb a
2
2 2 2
2
ac ba cb a b c
b c a
a b b c c a
) *
! ! $ + ! ! ,
! ! ! -
Panagiote Ligouras, MathLinks, t=322519, 2010 - AI3 - 298 -Let a, b, c be non-negative reals, no two of which are zero Prove that
' ( ' ( ' (
2 2 2
2 2
6
a a b c b a b c c a b c
b c c a a b
! ! ! ! ! !
! ! "
! ! !
- AI3 - 299 -Let a, b, c be non-negative reals, no two of which are zero Prove that
' (2 ' (2 ' (2
2 2 2
3
a b c a b c a b c
b c c a a b
% ! ! % ! ! %
! ! "
! ! !
(33)65 - AI3 - 306 -Let a, b, and c be positive real numbers Prove that
3 3 3
2 2 2 2 2
1
a b c a b c
a b b c c a b c a
) *
! ! $ ++ ! ! ,,
! ! ! - .
Panagiote Ligouras, MathLinks, t=311797, 2009 - AI3 - 307 -Let a, b, and c be positive real numbers Prove that
2 2
2 2
cyc cyc cyc cyc
a a b a a b
b c b
b c b
) * ) *
) * ! ) * !
+ , + ,
+ , "+ ,
+ ! ,+ , + ! ,+ ,
-7 -@ -7 -@
Panagiote Ligouras, ML, t=251794, 2009 - AI3 - 308 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2 2
2 2
2
a bc b ac c ab
b c c a a b
! ! !
! ! "
! ! !
- AI3 - 309 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2 2
16 16 16
10
a bc b ac c ab
b c c a a b
! ! !
! ! "
! ! !
Vasile Cirtoaje, ML, 2005 - AI3 - 310 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2 2
3
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
4 4
4
ab bc ca bc ca ab ca ab bc
b c c a a b
! ! ! ! ! ! ! ! "
(34)67 Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2 2
2 2
3
a bc b ac c ab
b c c a a b
! ! !
! ! "
! ! !
Vo Quoc Ba Can & Vu Dinh Quy - AI3 - 319 -Let a, b, c be positive real numbers Prove that
2 2
2 2 2
4 4
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 -Let a, b, c be positive real numbers Prove that
2
2
2
8
cyc
a bc a bc
b c
) ! ! ! *
+ , " !
+ ! ,
-
7
Panagiote Ligouras, 2009 - AI3 - 321 -Let a, b, c be positive real numbers Prove that
' (' ( ' (' ( ' (' (
2 2
2 2 2 2 2 2 2
a b c a b c
abc
a b a c b a b c c a c b
! !
! ! $
! ! ! ! ! !
Romania, local MO, 1988 - AI3 - 322 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
3 3
2 2 2
9
a bc b ac c ab abc
a b c
b c c a a b
! ! ! ! ! "
! !
! ! !
(35)69 - AI3 - 329 -Let a, b, c be positive real numbers Prove that
' 2 2( ' (
2 2
2 2 2
3 1 1 1 2
4
a b c a b c
abc a b b c c a a b c
! ! ! !
# ! ! #
! !
! ! !
Zdravko F Starc, RSME n.128, 2009 - AI3 - 330 -Let a, b, c be positive real numbers Prove that
2 2
2 2
2 2 2
2 2
3 2
4
a b c a b c a b c
abc a b b c c a a b c
) ! ! ! *# ! ! ! ! ! # ) ! ! ! *
+ , + ,
+ , ! ! ! - ! ! .
-
Panagiote Ligouras, MathLinks, t=326771, 2010 - AI3 - 331 -Let a, b, c be positive real numbers such that a2!b2!c2/ Prove that
2 1 2 1 2 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 abc / Prove that 1
2 2 2
2 2 2 2 2 2
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 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
2 2
3
a b c
b !c!c !a!a !b "
Pham Kim Hung - AI3 - 334 -Let a, b, c be positive reals such that a b c! ! / and abc / Prove that 1
' (' 2( ' (' 2( ' (' 2(
3
ab bc ca
a b a b b c b c c a c a
! ! $
(36)71 Let a, b, c be positive real numbers Prove that
2 2
1
a b c a b c
bc ca ab
a bc b ca c ab
) *
! ! $ + ! ! ,
! ! ! -
Baltic way, 2003 - AI3 - 342 -Let a, b, c be positive real numbers Prove that
2 2
bc ca ab a b c
b c c a a b
a !bc!b !ca!c !ab$ ! ! ! ! !
Pham Huu Duc, Mathematical Reflections n J60, 2007 - AI3 - 343 -Let a, b, c be positive real numbers Prove that
2 2
2 2
a b c
a !bc!b !ca!c !ab$
Canada - AI3 - 344 -Let a, b, and c be positive real numbers Prove that
2 2
2 2 23
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
1 1
b c c a a b
a b c
a bc b ca c ab
! ! ! ! ! $ ! !
! ! !
Ho-Joo Lee, CRUX n 2580 - AI3 - 346 -Let a, b, and c be positive real numbers Prove that
' ( ' (2 ' (
2 2
2 2
2 2 2
9
3 3 a b c a b c a b c
a a b b c c
a bc b ca c ab a b c ab bc ca
! ! ! ! ! ! ! !
! ! ! ! ! $
! ! ! ! ! ! ! !
(37)73 - AI3 - 353 -Let a, b, and c be positive real numbers Prove that
2 2 2 3
2 2
21
4
b bc c c ca a a ab b a b c
abc
a bc b ca c ab
! ! ! ! ! ! ! !
! ! " %
! ! !
Panagiote Ligouras, ML, t=249913, 2009 - AI3 - 354 -Let a, b, and c be positive real numbers Prove that
2 3
2
3 33 2
cyc
b bc c a b c
abc a bc
! ! ! !
" % !
7
Panagiote Ligouras, ML, t=329755, 2010 - AI3 - 355 -Let a, b, and c be positive real numbers Prove that
' (
' (
3 3
2 2
9
a b c
ab bc ca
c ab a bc b ca a b c
! !
! ! "
! ! ! ! !
MIC Shortlist, ML, t=259607, 2009 - AI3 - 356 -Let a, b, and c be non-negative real numbers Prove that
2 2
1 1
a b c
a bc b ca c ab
! ! "
! !
! ! !
Pham Kim Hung - AI3 - 357 -Let a, b, and c be non-negative real numbers Prove that
2 2
1 1 2
ab bc ca
a bc b ca c ab
! ! "
! !
! ! !
Pham Kim Hung - AI3 - 358 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2
a bc b ca c ab
a bc b ca c ab
% % %
! ! "
! ! !
(38)75 Let a, b, and c be positive real numbers such that , ,a b c " % Prove that 1
2 2
2 2
1 1
2
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 a b c, , $1 and a b c! ! / Prove that
2 2
1 1
a ! ! !c b ! ! !a c ! ! "b
Pham Thanh Nam - AI3 - 367 -Prove that if the real numbers a, b and c satisfy a2!b2!c2/ then3
' (2
2 2
2 2 12
2 2
a b c
a 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 a b c! ! / Prove that
'a2%ab b! 2('b2%bc!c2('c2%ca!a2($12
Pham Kim Hung, ML, 2006 - AI3 - 369 -Let a, b, c be non-negative real numbers such that a2!b2!c2 / Then
' 2(' 2(' 2(
8 a %ab b! b %bc!c c %ca!a $1
Pham Kim Hung - AI3 - 370 -Let a, b, c be non-negative reals, no two of which are zero Prove that
' (
2 2 2 2
1 1 12
b bc c c ca a a ab b a b c
! ! "
% ! % ! % ! ! !
(39)77 - AI3 - 377 -Let a, b, c be non-negative reals, no two of which are zero Prove that
2 2
2 2 2
3 3
5
a bc b ca c ab
b bc c c ca a a ab b
% ! % ! % "
% ! % ! % !
Panagiote Ligouras, ML, t=319775, 2009 - AI3 - 378 -Let a, b, and c be positive real numbers such that a4!b4!c4 / Then
3 3
2 2 2
3
a b c
abc b %bc!c !c %ca!a !a %ab b! $
- AI3 - 379 -Let a, b, and c be positive real numbers Prove that
' (
3 3
2 2 2
3 ab bc ca
a b c
a b c
b bc c c ca a a ab b
! !
! ! "
! !
% ! % ! % !
Sefket Arslanagic, CRUX n 2927 - AI3 - 380 -Let a, b, c be non-negative reals, no two of which are zero Prove that
3
2 2
cyc cyc cyc
a ab
a
a b c
b %bc c! " " ! !
7 7 7
- AI3 - 381 -Let a, b, and c be positive real numbers such that a4!b4!c4 / Then
3 3
2 2 2
3
a b b c c a ab bc ca
abc
b 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
2
a b c! ! / Prove that
3 3
2 2 2
8 8
26
a b c
b bc c c ca a a ab b
! ! !
! ! "
% ! % ! % !
(40)79 Let a, b, and c be positive real numbers such that abc / Then 1
2 2
3 3 3
1 ab bc ca 18
c a b a b c
! ! !
! ! "
! !
Hong Kong, 2000 - AI3 - 390 -Let a, b, and c be positive real numbers Prove that
' (
3 3
27
a b c
b c a a b c
! ! "
! !
- AI3 - 391 -Let a, b, and c be positive real numbers such that a2!b2!c2 / Then
2 2
3 3
a b c
b !c !a "
MathLinks, t=259844, 2009 - AI3 - 392 -Let a, b, c be positive real numbers such that ak!bk!ck / , k > Then 3
2 2
3 3
a b c
b !c !a "
ML, t=259844, 2009 - AI3 - 393 -Let a, b, c be positive real numbers such that a b c! ! / Prove that3
3 3
3
1 1
a b c
b ! !c ! !a ! $
Bin Zhao, ML, 2006 - AI3 - 394 -Let a, b, c be positive real numbers such that a2!b2!c2/ Prove that3
2 2
3 3
3
1 1
a b c
b ! !c ! !a ! "
(41)81 - AI3 - 401 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
' (
3 3
1
2 2
a b c
abc b ! !c ! !a ! " !
MathLinks, t=264216, 2009 - AI3 - 402 -For positive real numbers x, y, z holds
2 2
1 1
2
1 1
x ! ! y ! ! z ! / Prove
the inequality:
3 3
1 1
3
2 2
x ! ! y ! ! z ! &
Serbia, Junior Balkan Team Selection Test, 2009 - AI3 - 403 -Let x y z, , R! and x! ! / Prove that:y z
' (
3 3
3 3
1
9 27
8 8
x y z
xy yz zx y ! !z ! !x ! " ! ! !
Iran, National Math Olympiad, 2008 - AI3 - 404 -Let a, b, c be positive real numbers such that a b c! ! $ Prove that
3 3
27 31
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 a b c! ! /2 and k " 1 Prove that
3 3 3
ka kb kc
b !c !c !a !a !b "
(42)83 If a" " # , thenb c
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 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
5 5 5
3 3 3
5
a a b b b c c c a
a b b c c a
! ! ! ! ! ! ! ! "
! ! !
Panagiote Ligouras, ML, t=318005, 2009 - AI3 - 413 -Let a, b, c be positive real numbers such that a b c! ! / Prove that1
5 5 5
3 3 3
3 3 3
2
a a b b b c c c a
a b b c c a
! ! ! ! ! ! ! ! "
! ! !
Panagiote Ligouras, MathLinks, t=250356, 2009 - AI3 - 414 -Let a, b, c be positive real numbers Prove that
3 3
3 3 3
3 3
6
a abc b abc c abc
b c c a a b
! ! !
! ! "
! ! !
Tran Quoc Anh, ML, t=262924, 2009 - AI3 - 415 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
5 5 5
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
! ! ! ! ! ! ! ! !
! ! "
! ! !
Panagiote Ligouras, ML, t=297732, 2009 - AI3 - 416 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
5 5 5
3 3 3
3 3 3 3 3
7
b a c abc a b c abc b c a abc
b c c a a b
! ! ! ! ! ! ! ! ! ! ! "
! ! !
(43)85 - AI3 - 423 -If a b c #, , with a b c! ! / , prove that1
7 7 7
5 5 5
1
a b b c c a
a b b c c a
! ! ! ! ! "
! ! !
- AI3 - 424 -Let , ,a b c #0 and abc /1
5 5 5
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 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
1 1
9
a! ! "b c
- AI3 - 426 -Let a, b, c be positive real numbers Prove that
1 1 1
2
a b c a b b c c a
) *
! ! " + ! ! ,
! ! !
-
- AI3 - 427 -Let a, b, c be positive real numbers Prove that
1 1 1
4
a b c a b b c c a a b c
) *
! ! " + ! ! ,%
! ! ! ! !
-
- AI3 - 428 -If a b c #, , 0, prove that
8 8
3 3
1 1 a b c
a b c a b c
! !
! ! $
(44)87 Let a, b, c be positive real numbers Prove that
2 2 2 2 2
1 1
6
3 3
a b c
a b c a b c b c a c a b
) *
! ! " + ! ! ,
! ! ! ! ! !
-
- AI3 - 436 -Let a, b, c be positive real numbers Prove that
3 3 3
1 1 1
6
4 4
a b c a b b c c a
) *
! ! " + ! ! ,
! ! ! ! ! !
-
- AI3 - 437 -Let a, b, c be positive real numbers Prove that
1 36
a! ! "b c a b c! !
- AI3 - 438 -Let a, b, c be positive real numbers such that 12"21ab!2bc!8ca Then
1 15
2
a! ! "b c
Tran Nam Dung, Vietnam TST, 2001 - AI3 - 439 -Let a, b, c be positive real numbers Prove that
4
a a a
b!c " b c!
- AI3 - 440 -Let a, b, c be positive real numbers Prove that
a b a c b c
b a b c a c
! !
! " !
(45)89 - AI3 - 447 -Let a, b, c be positive real numbers Prove that
1
a 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" " Prove that b a
a b c b c a
b! !c a "a! !b c
- AI3 - 449 -Let a, b, c be positive real numbers Prove that
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 -Let a, b, c be positive real numbers Prove that
3
a b c a b c
b! !c a "b c! !a!c!a b! !
MathLinks, t=262433, 2009 - AI3 - 452 -Let a, b, c be positive real numbers Prove that
1 2
2
a b c b b a b c
b c a a c b c a b
! !
) *
! ! " + ! ! ! ,
! ! !
-
(46)91 Let a, b, c be positive real numbers Prove that
2 2
5
4
a b c a b c
b c a ab bc ca
! !
! ! " !
! !
Panagiote Ligouras, MathLinks, t=315855, 2009 - AI3 - 460 -Let a, b, c be positive real numbers such that abc $ Prove that 1
a b c
a b c c! ! " ! !a b
EXCALIBUR, 5(4), 2000 - AI3 - 461 -Let a, b, c be positive real numbers such that abc / Prove that 1
' 1(' 1(' 1(
a b c
a b c
b! !c a " ! ! ! %
Aaron Pixton - AI3 - 462 -Let a, b, c be positive real numbers such that abc / Prove that 1
' (
1 1
2
a b c
a b c
b c a a b c
0 ) *1
! ! " 2 ! ! !+ ! ! ,3
-
4
- AI3 - 463 -Let a, b, c be positive real numbers Prove that
' (
2
1 1
a b c
a b c
b c a a b c
) ! ! * " ! ! ) ! ! *
+ , + ,
- -
Great Britain Olympiad, 2005 - AI3 - 464 -Let a, b, c be positive real numbers Prove that
2
a b c a b b c c a
b c a c a b
! ! !
) ! ! * " ) ! ! *
+ , + ,
- -
(47)93 - AI3 - 471 -Let a, b, c be positive real numbers, show that
3 3
2
ab bc ca
a b c
c ! a ! b # ! !
China-TST, 2008 - AI3 - 472 -Let a, b, c be positive real numbers such that a2!b2!c2/ Prove that3
3 3
2
3
ab bc ca a b c
a b c
c a b
! !
! ! # ! ! ! !
Panagiote Ligouras, MathLinks, t=310241, 2009 - AI3 - 473 -Let a, b, c be positive real numbers such that a b c! ! / Prove that8
6 3
4
ab bc ca
a b c
c ! a ! b # ! !
Panagiote Ligouras, ML, t=311790, 2009 - AI3 - 474 -If a, b, c > and a b c! ! ! abc/ , then4
ab bc ca
a b c c ! a ! b # ! !
China-TST, 2007 - AI3 - 475 -Let a, b, c be positive real numbers Prove that
' (2
2 2 4 a b
a b c
a b c
b c a a b c
% ! ! " ! ! !
! !
BMO, 2005 - AI3 - 476 -Let a, b, c be positive real numbers Prove that
' (' 2 2(
2 2 a b c a b c
a b c
b c a ab bc ca
! ! ! !
! ! "
! !
(48)95 Let , ,a b c #0 such that a b c! ! / Prove:1
' (
2 2
2 2
3
a b c
a b c
b ! c ! a " ! !
Croatia TST 2007 - AI3 - 484 -Given positive real numbers a, b, c, prove that
' ( ' ( ' (
2 2
2 2 37 19
6
a b c ab bc ca
a b c
b c a a b c
! ! % ! !
! ! "
! !
Michael Rozenberg, ML, t=296853, 2009 - AI3 - 485 -Let a, b, c be positive real numbers Prove that
2 2
2 2
a b c a b c
b c a
b !c !a " ! !
1st Nordic Mathematical Olympiad - AI3 - 486 -Given positive real numbers a, b, c, prove that
' (
2 2
2 2 2
9
12 ab bc ca
a b c
b c a a b c
! !
! ! " %
! !
- AI3 - 487 -Let a, b, c be positive real numbers Prove that
2 2 2
2 2 2
1
3
2
a b c b c a a b c b c a
b c a a b c
b c a a b c
) * ) * ) * ) *
! ! ! ! ! " ! ! ! ! ! !
+ , + , + , + ,
+ , + , - . - .
- -
- AI3 - 488 -Let a, b, c be positive real numbers Prove that
' ( ' (
2 2
2 2
2 2 2
2 1 1 1
3
a b c
a b c a b c
a b c
b c a a b c
) * ) *
! ! " ! ! + ! ! ,% ! ! + ! ! ,
- -
(49)97 - AI3 - 495 -Let a, b, c be positive real numbers Prove that
3 3
3 3 , , ,
a b c a b c
a b c
b c a
b !c !a " ! ! #
- AI3 - 496 -Let a, b, c be positive real numbers Prove that
' ( '3 ( '3 (3
2 2
a b c a b c a b c
a b c
a b c
% ! ! % ! ! %
! ! " ! !
- AI3 - 497 -Suppose that a, b, c are positive real numbers and a5!b5!c5 / Then3
4 4
3 3
a b c
b !c !a "
- AI3 - 498 -Prove that for all positive real numbers a, b, c
2
5 5 6
3
3
a b c a b c
b c a
) ! ! *
! ! " ++ ,,
-
Nguyen Thuc Vu Hoang, MathLinks, t=298305, 2009 - AI3 - 499 -Let a, b, c be positive real numbers Prove that
3
2
a b c
b! c ! a "
- AI3 - 500 -Let a, b, c be positive real numbers Prove that
3
2
a b c
(50)99 Prove that
3
a b b c c a
c a b
! ! !
! ! "
where a, b, c are positive real numbers.
'#()#"*+,-.!/!0%12*3%#*WURZEL, n A50 - AI3 - 508 -Let a, b, c be positive real numbers Prove that
3
6
a b b c c a a b c
c a b abc
! ! ! ! ! " 6 ! !
Pham Huu Duc, Mathemtical Reflections n S41, 2007 - AI3 - 509 -Let a, b, c be positive real numbers Prove that
' (
' (' (' (
3
16
a b c a 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 -Let a, b, c be positive real numbers Prove that
' (
' (' (' (
4
2
a b b c c a
a b c
c a b abc a b b c c a
! ! !
! ! " ! !
! ! !
Panagiote Ligouras, MathLinks, t=252271, 2009 - AI3 - 511 -Let a, b, c be positive real numbers Prove that
' (
' (' (' (
3
4
2 a b c
a b b c c a
c a b a b b c c a
! !
! ! !
! ! "
! ! !
Panagiote Ligouras, MathLinks, t=308751, 2009 - AI3 - 512 -Let a, b, c be positive real numbers Prove that
' (
' (' (' (
4
3
9
2 a b c
a b b c c a
c a b abc a b b c c a
! !
! ! !
! ! "
! ! !
(51)101 - AI3 - 519 -Let a, b, c be positive real numbers, prove that
' (' (' (
8
4
a 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
3
b c c a a b
a b c
a b c
! ! !
! ! " ! ! !
EXCALIBUR, 5(3) & 5(4), 2000 - AI3 - 521 -Let a, b, c be positive real numbers, prove that
' ( '3 ( '3 (3 ' (
a b b c c a
ab bc ca
c a b
! ! !
! ! " ! !
- AI3 - 522 -Let a, b, c be positive real numbers, prove that
' ( '3 ( '3 (3 ' (
2 2
8
a b b c c a
a b c
c a b
! ! !
! ! " ! !
Romania, order 8, 2003 - AI3 - 523 -Let a, b, c be positive real numbers, prove that
' ( '3 ( '3 (3 ' ( ' (
2 2
2
a b b c c a
a b c a b c
c a b
! ! !
! ! " ! ! ! ! !
Panagiote Ligouras, MathLinks, t=297731, 2009 - AI3 - 524 -Let a, b, c be positive real numbers such that abc / , prove that1
2 2 2
6
a b b c c a
c a b
! ! !
! ! "
(52)103 If a, b, c are positive numbers, then
' 2 2(
2 2 6 a b c
a b c
a b c
b c a a b c
! !
! ! ! ! ! "
! !
Pham Huu Duc, MathLinks, 2006 - AI3 - 532 -Let a, b, c be positive real numbers Prove that
3 3 3
6
a b ab b c bc c a ca
abc
c a b
! ! ! ! ! "
- AI3 - 533 -Let a, b, c be positive real numbers Prove that
1 1 1
2
a b c a b b c c a
) *
! ! " + ! ! ,
! ! !
-
- AI3 - 534 -Let a, b, c be positive real numbers such that a" " Prove thatb c
2 2
2 2
a b b c c a
a b c
c ! a ! b " ! !
Vietnam, 29ts MO, 1991 - AI3 - 535 -Let a, b, c be positive real numbers such that a b c! ! / Prove that6
2 2
1 1 75
4
a b c
b c a
) ! * !) ! * !) ! * "
+ , + , + ,
- - -
ARML, 1987 - AI3 - 536 -Let a, b, c be positive real numbers such that ab bc! !ca/ Then1
2 2
1 1
16
a b c
b c a
) ! * !) ! * !) ! * "
+ , + , + ,
- - -
(53)105
' (
1 1
7 a b c
a! ! !b c ! ! "
- AI3 - 543 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
2 2
4
a b b c c a
c a b
! ! !
! ! "
- AI3 - 544 -Let a, b, c be positive real numbers Prove that
1 1 1
1 1 a b c
a b c b c a
) ! *) ! *) ! * )" ! *) ! *) ! *
+ ,+ ,+ , + ,+ ,+ ,
- - - - - -
- AI3 - 545 -Let a, b, c be positive real numbers such that a b c! ! / Prove that
1 1
1 1 64
a b c
) ! *) ! *) ! *"
+ ,+ ,+ ,
- - -
Yugoslavia, 30th MO, 1989 - AI3 - 546 -Let a, b, c be arbitrary positive real numbers Prove that
' (
3
1 a b c a b c
b c a abc
! ! ) ! *) ! *) ! *" !
+ ,+ ,+ ,
- - -
APMO, 1998 - AI3 - 547 -Let a, b, c be arbitrary positive real numbers Prove that
' (' (' (
2 2
1 a b c a b c
b c a
) *) *) *
! ! ! " ! ! ! + ,+ ,+ ,
+ ,+ ,+ , - - -
(54)107
a b c
a b c b c a
! ! ! ! ! $ ! ! !
Todor Mitev, CRUX, n 3236 - AI3 - 555 -Let a, b, c be positive real numbers such that abc / Prove that 1
2 2 1
6
a b c a b c
a b c
) *
! ! ! " + ! ! ! ! ! ,
-
Vasile Cirtoaje, ML, 2006 - AI3 - 556 -Let a, b, c be positive real numbers such that abc / Prove that 1
' ( 1
a b c
a b c
b c a a b c
) *
! ! " ! ! !+ ! ! ,%
-
EXCALIBUR, 12(4), 2007 - AI3 - 557 -Let a, b, c be positive real numbers such that abc / Prove that1
2 2 a b c 9 2 1
a b c a b c
b c a a b c
) *
! ! ! ! ! ! " + ! ! ! ! ! ,
-
Panagiote Ligouras, MathLinks, t=252656, 2009 - AI3 - 558 -Let a, b, c be positive real numbers such that abc / Prove that1
2 1
2
a b c
a b c
b c a a b c
) * ) *
! ! ! + ! ! ," !+ ! ! ,
- -
- AI3 - 559 -Let a, b, c be positive real numbers Prove that
'a b c( 1
a b c
) *
! ! + ! ! ,"
-
(55)109 If y"a b c, , " # , thenx
'2 (' ( ' ( 1
x y x y
a b c
xy a b c
! ! ) *
" ! ! + ! ! ,"