An interesting and useful inequalities

Completed prove.The inequality holds when abc 1... Email:crycry.tara1995@yahoo.com Thank you and best regards.

An interesting and useful inequality for math olympiad

CQT(crycry-tara1995)

10/7/2012 Problem(Vasile):Let a b c , , 0.prove that:

4 a b b c c a a

a b c     bc ($1)

I.Solution:

Because the inequality is homogeneous so we can assume a  b c 3 and

we must prove:

4

b b c c a abc

Letx y z, , be a permutation of a b c, , sastisfy x yz  xyxz yz

Also we have xyz 3, ,x y z,  0

So use arrangement inequality we have:

b b c c a abc xy x xz y yz z xyz y z

Use AM-GM inequality we have:

3

(

7 )

y xz  y xz  y   x z x z  (2) From (1)(2)(*) is true.so completed prove

The inequality holds when abc or c 0,a 2b or b 0,c 2a or

0, 2

a b c

Note that similar we also have :

3 27 2 2 2

4

abc  b ac ba cab c ($2)

The inequality holds when abc or c 0,b 2a or b 0,a 2c or

0, 2

a c b

*Use ($1) we have a solution for problem 5 in Canada math Olympiad 1999

http://www.artofproblemsolving.com/Forum/viewtopic.php?p=768&sid=da4 481f13b80aeb6f70cbabd32df19ae#p768

II.Application:

1.Problem 1(unknown in a book of can_hang2007):

Let a b, ,c 0,a  b c 3.Prove that:

2(b acb ac  a b c  a bc 19

*Solution(me):

)

2(ab bc ca  (a b c )(a b c)  4abc 19

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

3 2 2 2

19 2(

(a b c)   abca b b c c a)

Use a  b c 3 we need prove:

4

abca  cc 

It is inequality ($1) with a  b c 3

Completed prove.The inequality holds when abc 1 or c 0,b 1,a 2 or

0, 1, 2

b a c or a 0,c 1,b 2

2.Problem 2(nguoivn):

Let a b, ,c 0,a  b c 3.Prove that:

1

Posted in

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=470936

*Solution(unknown):

a

Use AM-GM inequality we have:

b  b    b 

c

a

So we have:

3

a

It is inequality ($2) with a  b c 3

Completed prove.The inequality holds when c 0,a 1,b 2 or

0, 1, 2

b c a or a 0,b 1,c 2

3.Problem 3(hungkhtn):

Let a b, ,c 0,a  b c 3.Prove that:

*Solution(unknown):

Use AM-GM inequality we have:

2 2

b c b c  c c       

c a c a  a a       

So we need prove:

2

ab bc ca

It is inequality ($2) with a  b c 3

Completed prove.the inequality holds when c 0,a 1,b 2 or

0, 1, 2

b c a or a 0,b 1,c 2

4.Problem 4(lilteevn):

Leta b c, ,  0,a  b c 3.Prove that:

3 3

a

b  c       abc

Posted in

*Solution(songvuive):

Use Cauchy inequality we have:

2

(a b c)

ab a c bc b a ca c b

 



ab a c bc b aca c b

So we need prove:

3 3(3  abc) ab a cbc b a ca c b

3

3

9  abcab a c bc b aca c b

 

Use AM-GM inequality we have: 3

1 1

3 abc abc 

Use Cauchy inequality we have:

3

3

a b c

c b a c b a b c ab ac c

So we need prove:

4 ab bc c a a c b

it is inequality ($2) with a  b c 3

Completed prove.The inequality holds when abc 1

5.Problem 5(nguoivn):

Let a b, ,c 0,a  b c 3.Prove that:

(ab  bc ca abacbc  16

*Solution(unknown):

(ab bc ca ab ac bc 5

Use AM-GM inequality we have:

2 (a b bc ca abacbc  abacbc) ab bc c a

abacbc  ab bc ca

27 abacbc  ab bc ca

So we need to prove:

(abacbc)  ab bc ca  2

We have:

(a ba cbc)  ab bc ca  a b ab  b c bc  a c c a  2abc a b c)

(a b) bc b( c) ca c( a) 6abc ab (3 ) (3 ) (3 ) 6



3(a b bc ca abc a b c  6abc 3(ab bc ca a bc)

So we need prove:

4

bc ca a

a b    b c

It is inequality ($2) with a  b c 3

Completed prove.The inequality holds when a 0,b 1,c 2or

0, 1, 2

b c a or c 0,a 1,b 2

6.Problem 6(sieubebuvietnam):

Let a b c, ,  0,a  b c 1.Prove that:

1

bcca ab abacbc

Posted in

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=469347

*Solution(me):

We have:

( ab bc ca )(ab ac bc) a b b c c a abc( b )

a

= 2 2 2

a

Use Cauchy inequality we have:

cbacba  c bca acb ab

2

2 2 2

(a b c)

a b c ab ac bc

 



So:

( ab bc ca )(ab ac bc)

b b c c

    abc(3 

2

2 2 2

(a b c)

a b c ab ac bc

 

2 2 2

2 2 2

abc a b c

b b c c a abc

a b c ab ac bc

81

*Solution(unknown)

Posted

36

Because the inequality is homogeneous so we can assume a  b c 3 and

we must prove:

2 2 2

) 3

abc a  c 

Assume a is max( , , )a b c

let

2

tbc

we have:

So we need prove:

2 2 2

a t a  t 

witha 2t 1,t 1

2 3 2

(4 6

(t t  t  t 1 0

   (it is true with t 1)

Now return our problem,use lemma with a  b c 1 we have :

2 2 2

8

1

a c b a  b c 

But:

2

a b c

b c ab ac b

So:

2 2 2

2 2 2

1

2 54 3

abc a b c

a b c ab ac bc

So if we want to prove (*) we must prove:

2 2 2 4

27

It is inequality ($1) with a  b c 1

Completed prove.The inequality holds when 1

3

abc

7.Problem 7(a problem in Inequalities with Beautiful Solutions):

Let a b c , , 0,a  b c 3.Prove that:

*Solution(can_hang or Vasile):

After expanding,this simplifies to:

3

(16 5 ) 5 ( )

64 r   r A r AB

rab c Aab    bb cc a

Use AM-GM inequality we have:

3

3 abc  abc  r

Use Schur inequality we have:

3

4

r

Use inequality ($2) with a  b c 3,we have:

4

So we need prove:

(16 5 )(4 )

4

It is true because r 1

Completed prove.The inequality holds when abc 1 or a 0,b 1,c 2 or

0, 1, 2

b c a or c 0,a 1,b 2

8.Problem 8(nguoivn):

Let a b, ,c 0,a  b c 3.Prove that:

*Solution(in a book of can_hang2007 and nguoivn):

Use AM-GM inequality we have:

Similar we have:

4 2

bc bc bc c a

c a



4 2

ca ca ca a b

a b





So we need prove:

2

cyc

ab  abc ab  ab ac bc

We have 2(ab a)( c b)( c)  2[(a b c ab)( acbc) abc]  6ab 2abc

So we need prove:

4 ab bc ca abc

It is ($2) inequality with a  b c 3

Completed prove.the inequality holds when abc 1 or a 0,b 1,c 2 or

0, 1, 2

b c a or c 0,a 1,b 2

III.Proposed problem:

1.Problem 1(nguoivn):

Let a b, ,c 0,a  b c 3.Prove that:

) (2 ) (2 ) 345 ( 2 a b bc c a  6

2.Problem 2(hungkhtn):

Let a b, ,c 0,a  b c 0.Prove that:

1

Hint:assume a  b c 3 and expand

3.Problem 3(a problem in Inequalities with Beautiful Solutions):

Let 0 a b c, ,  1.Prove that:

4

4.Problem 4(sieubebuvietnam):

Let a b c, ,  0,a  b c 1.Prove that:

2 2 2

1

bcca ab abc a b c

Hint(me):Posted in

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=469347

5.Problem 5(unknown):

Let a b, ,c 0,a  b c 3.Prove that:

1 1

cyc

a b

a b 

 Hint(me):posted in

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=52&t=468841

6.Problem 6(nguoivn+quykhtn-qa1):

Let a b, ,c 0,a  b c 3.Prove that:

(ab c bca cab 

Hint(quykhtn-qa1):posted in

http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=213849

Please contact me if there are some mistakes

Email:crycry.tara1995@yahoo.com

Thank you and best regards