2004 JBMO Problems/Problem 1
Problem
Prove that the inequality holds for all real numbers and , not both equal to 0.
Solution
Since the inequality is homogeneous, we can assume WLOG that xy = 1.
Now, substituting , we have:
, thus we have
Now squaring both sides of the inequality, we get:
after cross multiplication and some simplification we get:
or,
The above is always true since .