Inequality

[kinu]

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]

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]

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]

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]

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]

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]

I tried taking cases, Jeff, but gave up after 40 minutes or so. :|

[Subhotosh Khan]

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: