physics chemistry maths science forums
become expert I help I sign up I login
refer a friend - earn nickels!!   
 advanced
 
Home
Ask & Discuss Questions
Study Material
Experts Zone
Hang Out!

Ask & Discuss Questions with Community & Experts

 Moderation Team
 90 chars left    advanced
Ask iit jee aieee pet cbse icse state board community Community Discussion Question: An inequality
Forum Index -> Algebra like the article? email it to a friend.  
Author Message
[Post New]13 Mar 2008 21:45:26 IST
    Subject: An inequality
[Up] View Post
konichiwa2x (2224)

Blazing goIITian

Olaaa!! Perrrfect answer. 418  [485 rates]
konichiwa2x's Avatar
total posts: 648    
offline Offline
If and

prove  
 
PS. This is not a challenge. I am just curious to know how some of you would approach the problem.
Guide to latex:
http://www.goiit.com/posts/list/community-shelf-a-guide-to-latex-48056.htm

JEE and OLYMPIA INFINATUM
http://iit-redefined.theforum.name/index.php
    
[Post New]13 Mar 2008 22:36:23 IST
    Subject: Re:An inequality
Accepted Answer [?] [Up] View Post
hsbhatt (3694)

Forum Expert Blazing goIITian

Olaaa!! Perrrfect answer. 700  [797 rates]
hsbhatt's Avatar
total posts: 1091    
offline Offline
\text{WLOG, assume} \ a\geq b\geq c \\ \\ \text {Hence} \ \frac{a^2}{b^2+c^2} \geq \frac {b^2} {c^2+a^2} \geq \frac {c^2} {a^2+b^2} \\ \\ \text {So, we can use Chebyshev
 this reply: 5 points  (with 1 Olaaa!! Perrrfect answer.   in 1 votes )   [?]
 
You have to be logged on to rate
  
[Post New]14 Mar 2008 12:49:31 IST
    Subject: Re:An inequality
[Up] View Post
konichiwa2x (2224)

Blazing goIITian

Olaaa!! Perrrfect answer. 418  [485 rates]
konichiwa2x's Avatar
total posts: 648    
offline Offline
nice sir!
 
 
And similarly for the other two. We also know a+b+c=1
 
Hence it is sufficient to prove
 
To finish the proof, since we'll have:

 
 
Guide to latex:
http://www.goiit.com/posts/list/community-shelf-a-guide-to-latex-48056.htm

JEE and OLYMPIA INFINATUM
http://iit-redefined.theforum.name/index.php
 this reply: 0 points  (with 0 Olaaa!! Perrrfect answer.   in 0 votes )   [?]
 
You have to be logged on to rate
  
[Post New]14 Mar 2008 13:17:43 IST
    Subject: Re:An inequality
[Up] View Post
hsbhatt (3694)

Forum Expert Blazing goIITian

Olaaa!! Perrrfect answer. 700  [797 rates]
hsbhatt's Avatar
total posts: 1091    
offline Offline
\text{I must confess I am in love with Rearrangement Inequality. Hence this proof} \\ \\ \text {We have seen in the previous post of mine that} \\ \\ \text {If we assume} \ a \geq b \geq c \ \text{then} \\ \\ \frac {a^2} {b^2+c^2} \geq \frac {b^2} {c^2+a^2} \geq \frac {c^2} {a^2+b^2} \\ \\ \text{If we call the given expression as I, then from RI, we have} \\ \\ I \geq \sum \frac {a b^2} {b^2+c^2} \ \text {and also} \\ \\ I \geq \sum \frac {a c^2} {b^2+c^2} \\ \\ \text {adding the above two} \\ \\ 2I \geq (a+b+c) \\ \\ \text {Hence} I \geq \frac{1} {2}
 this reply: 2 points  (with 0 Olaaa!! Perrrfect answer.   in 1 votes )   [?]
 
You have to be logged on to rate
  
[Post New]14 Mar 2008 13:29:35 IST
    Subject: Re:An inequality
[Up] View Post
vineetnegi (107)

Cool goIITian

Olaaa!! Perrrfect answer. 19  [25 rates]
vineetnegi's Avatar
total posts: 95    
offline Offline
My method is similiar to sir bhatt
but still as you need diff method i think i might post it
by cauchhy swartz inequality
(a+b+c)^2<=(sum)((aa+bb)/a+(bb+cc)/b+(cc+aa)/c)
1<=s(1+bb/a+cc/b+aa/c)
let a<b<c
a^2<b^2<c^2
1/c<1/b<1/a
min value by rearrange.inequality
a+b+c=1
1/(1+1)<=s
s>=1/2
 
sorry i didnt see that you dont want sol. through re. inequality.
those who dont believe in god closes the gates of miracles in their life
 this reply: 2 points  (with 0 Olaaa!! Perrrfect answer.   in 1 votes )   [?]
 
You have to be logged on to rate
  
 
Forum Index -> Algebra
Go to:   

Top Offers for goIITians
Correspondence Courses
Brilliant Tutorials
Narayana Institute
Aakash Institute
Classroom/Crash Courses
Narayana - Kota , Delhi , Others
Brilliant Tutorials - Class , Crash
Aakash Institute - Medical , Engg
Online Test Series
Brilliant Tutorials
Narayana Institute
Aakash Institute
Mahesh Tutorials
AMITY      Sri Chaitanya