USAMO 2003 Problem #5
by KingSmasher3, Mar 29, 2013, 5:21 AM
Let 
, 
, 
be positive real numbers. Prove that
![\[ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.\]](//latex.artofproblemsolving.com/f/a/4/fa4784e725984994a726b6139706f1fba3e83689.png)
_______
The inequalities is homogeneous, so let
We want to prove
![\[\sum\frac{(a+1)^2}{2a^2+(1-a)^2} \le 8.
\]](//latex.artofproblemsolving.com/e/4/1/e41e37274bdb1d6eb1aa18fb0ae88050fac5c7ff.png)
That is,
Now note that
![\[\sum\frac{8a+2}{3\left(a-\frac{1}{3}\right)^2 +\frac{2}{3}} \le \sum \frac{3}{2}(8a+2) = 12(a+b+c)+9=21.
\]](//latex.artofproblemsolving.com/7/1/b/71b86291ff82a6604596ce5cd4e4f527959d1e53.png)
Equality holds with 
We are done.
, 
, 
be positive real numbers. Prove that![\[ \dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.\]](http://latex.artofproblemsolving.com/f/a/4/fa4784e725984994a726b6139706f1fba3e83689.png)
_______
The inequalities is homogeneous, so let
We want to prove![\[\sum\frac{(a+1)^2}{2a^2+(1-a)^2} \le 8.
\]](http://latex.artofproblemsolving.com/e/4/1/e41e37274bdb1d6eb1aa18fb0ae88050fac5c7ff.png)
That is,\[\begin{align*}\sum\frac{a^2+2a+1}{3a^2-2a+1} &\le 8 \\ \iff \sum\frac{3a^2-6a+3}{3a^2-2a+1} &\le 24 \\ \iff \sum\frac{3a^2-6a+3}{3a^2-2a+1} -3 &\le 21 \\ \iff \sum\frac{8a+2}{3\left(a-\frac{1}{3}\right)^2 +\frac{2}{3}} &\le 21 \end{align}\]Now note that
![\[\sum\frac{8a+2}{3\left(a-\frac{1}{3}\right)^2 +\frac{2}{3}} \le \sum \frac{3}{2}(8a+2) = 12(a+b+c)+9=21.
\]](http://latex.artofproblemsolving.com/7/1/b/71b86291ff82a6604596ce5cd4e4f527959d1e53.png)
Equality holds with 
We are done.This post has been edited 2 times. Last edited by KingSmasher3, Mar 29, 2013, 5:31 AM
March 2014
October 2013