2003 JBMO Problems/Problem 4
Problem
Let . Prove that
Solution
Since and , we have that and are always positive.
Hence, and must also be positive.
From the inequality , we obtain that and, analogously, . Similarly, and .
Now,
Substituting and , we now need to prove .
We have
By Cauchy-Schwarz,
Since , we have .
Thus,
So,