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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

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.

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:

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