Inequality

[kinu]

\(\displaystyle a,b,c>0\) and \(\displaystyle a,b,c\) are distinct real number
then prove that \(\displaystyle \displaystyle \frac{a^2+1}{b+c}+\frac{b^2+1}{c+a}+\frac{c^2+1}{a+b}\geq 3\)

 

[kinu]

\(\displaystyle a,b,c>0\) and \(\displaystyle a,b,c\) are distinct real number
then prove that \(\displaystyle \displaystyle \frac{a^2+1}{b+c}+\frac{b^2+1}{c+a}+\frac{c^2+1}{a+b}\geq 3\)

goi it.

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

 

[Deleted member 4993]

Re:

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]

Re: Re:

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. :|

 

[Deleted member 4993]

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: