2003 USAMO Problems/Problem 5
Problem
Let ,
,
be positive real numbers. Prove that
![$\dfrac{(2a + b + c)^2}{2a^2 + (b + c)^2} + \dfrac{(2b + c + a)^2}{2b^2 + (c + a)^2} + \dfrac{(2c + a + b)^2}{2c^2 + (a + b)^2} \le 8.$](http://latex.artofproblemsolving.com/d/2/b/d2be8552ac3b2dcfb8d235a80ddc4d812b2f2155.png)
Solution
solution by paladin8:
WLOG, assume .
Then the LHS becomes .
Notice , so
.
So as desired.
2nd solution:
Because this inequality is symmetric, let's examine the first term on the left side of the inquality.
let and
. So
.
Note that . So Let
,
. QM-AM gives us $\sqrt{\frac{m^2+z^2}{2}$ (Error compiling LaTeX. Unknown error_msg)
.
Squaring both sides and rearranging the inequality gives us so
so
thus
.
Performing the same operation on the two other terms on the left and adding the results together completes the proof.