2004 JBMO Problems/Problem 1
Prove that the inequality holds for all real numbers and , not both equal to 0.
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 simplification we get: or, which is always true since .
Again, since the inequality is homogenous, we can assume WLOG that .
By AM-GM we gave and by QM-AM we have that .
Substituting we have
By Trivial Inequality,
Then by multiplying by on both sides, we use the Trivial Inequality again to obtain which means which after simplifying, proves the problem.