2003 USAMO Problems/Problem 5
Problem
Let , , be positive real numbers. Prove that
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.