Reference: R. Todev, Nordic Mathematical Contests, 1987-2009.
Question:
Let a, b, and c be positive real numbers. Prove that .
Solution:
The arithmetic-geometric inequality yields
,
or …call this relation I.
On the other hand, the Cauchy-Schwarz inequality implies
….call this relation II.
We arrive at the inequality we desire by combining relations I and II. Hence, the proof. QED.
Cheers,
Nalin Pithwa.