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kinu
07-10-2011, 12:24 AM
a,b,c>0 and a,b,c are distinct real number
then prove that \displaystyle \frac{a^2+1}{b+c}+\frac{b^2+1}{c+a}+\frac{c^2+1}{a +b}\geq 3

kinu
07-11-2011, 01:37 AM
a,b,c>0 and a,b,c are distinct real number
then prove that \displaystyle \frac{a^2+1}{b+c}+\frac{b^2+1}{c+a}+\frac{c^2+1}{a +b}\geq 3

goi it.

[spoiler:1vps9sjz]\displaystyle \frac{a^2}{b+c}+\frac{b^2}{c+a}+\frac{c^2}{a+b}+\l eft(\frac{1}{b+c}+\frac{1}{c+a}+\frac{1}{a+b}\righ t)\geq \frac{(a+b+c)^2}{2.(a+b+c)}+\frac{(1+1+1)^2}{2.(a+ b+c)}(Using C.S Inequality)
\geq \frac{(a+b+c)}{2}+\frac{9}{2.(a+b+c)}\geq 2.\sqrt{\frac{(a+b+c)}{2}.\frac{9}{2.(a+b+c)}}\geq 3
Using A.M\geq G.M[/spoiler:1vps9sjz]

mmm4444bot
07-11-2011, 03:24 PM
Can you please explain the meaning of the following notations (or just tell me the words that those five letters stand for)? Thank you. :)

C.S

A.M

G.M

Subhotosh Khan
07-11-2011, 04:17 PM
Can you please explain the meaning of the following notations (or just tell me the words that those five letters stand for)? Thank you. :)

C.S

A.M

G.M

I think I know the bottom two Arithmatic Mean and Geometric Mean.

pka
07-11-2011, 07:27 PM
Can you please explain the meaning of the following notations (or just tell me the words that those five letters stand for)? Thank you. :)

C.S

A.M

G.M

I think I know the bottom two Arithmatic Mean and Geometric Mean.

C.S is for Cauchy–Schwarz

JeffM
07-11-2011, 08:36 PM
I am really glad that someone solved this problem. I spent hours trying to find a proof that involved only algebra. Obviously I failed.

mmm4444bot
07-12-2011, 12:34 AM
I tried taking cases, Jeff, but gave up after 40 minutes or so. :|

Subhotosh Khan
07-12-2011, 09:48 AM
That must be one of those International Math Olympiad problem.

Application of CS inequality did not cross my mind - as evident by the fact that I did not know what CS did stand for. :mrgreen: