Inequalities

For discussing Olympiad Level Number Theory problems
User avatar
SANZEED
Posts:550
Joined:Wed Dec 28, 2011 6:45 pm
Location:Mymensingh, Bangladesh
Inequalities

Unread post by SANZEED » Tue Jan 31, 2012 11:38 pm

Let $x,y,z$ be the non-negative numbers,no two of which are zero.Prove that

$(x^{2}-yz)/(x+y)+(y^{2}-zx)/(y+z)+(z^{2}-xy)/(z+x)>=0$
$\color{blue}{\textit{To}} \color{red}{\textit{ problems }} \color{blue}{\textit{I am encountering with-}} \color{green}{\textit{AVADA KEDAVRA!}}$

User avatar
Phlembac Adib Hasan
Posts:1016
Joined:Tue Nov 22, 2011 7:49 pm
Location:127.0.0.1
Contact:

Re: Inequalities

Unread post by Phlembac Adib Hasan » Wed Feb 01, 2012 7:40 pm

Well,it's a direct consequence of Muirhead's inequality.You can also use Holder and Re-arrangement together for easy-looking proofs.Here is a short proof using Cauchy.
\[\sum_{cyc}\frac{x^2-yz}{x+y}\geq 0\]
\[\Rightarrow \sum_{cyc}\frac{x^2}{x+y}\geq \sum_{cyc}\frac{yz}{x+y}\]
\[\Rightarrow \sum_{cyc}x^3 y+x^3 z+x^2z^2+x^2yz\geq \sum_{cyc}yz^3+xyz^2+y^2z^2+xy^2z\]
\[\Rightarrow \sum_{cyc}x^3 y+\sum_{cyc}xy^3+\sum_{cyc}x^2z^2+\sum_{sym}x^2yz\geq \sum_{cyc}xy^3+\sum_{cyc}x^2y^2+2\sum_{sym}x^2yz\]
\[\Rightarrow \sum_{cyc}x^3 y\geq \sum_{sym}x^2yz\]
Which is true by cauchy.\[\left ( \sum_{cyc}x^3 y \right )\left ( \sum_{sym} xyz^2\right )\geq \left ( \sum_{sym}x^2yz \right )^2\]
\[\Rightarrow \sum_{cyc}x^3 y\geq \sum_{sym}x^2yz\]
Welcome to BdMO Online Forum. Check out Forum Guides & Rules

Post Reply