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
Since all terms are homogeneous, we may assume WLOG that .
Then the LHS becomes .
Notice , so
.
So , as desired.
See also
2003 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |