2001 IMO Shortlist Problems/A6
Prove that for all positive real numbers ,
The leader of the Bulgarian team had come up with the following generalization to the inequality:
We will use the Jenson's inequality.
Now, normalize the inequality by assuming
Consider the function . Note that this function is convex and monotonically decreasing which implies that if , then .
Thus, we have
Thus, we only need to show that i.e.
Which is true since
The last part follows by the AM-GM inequality.
Equality holds if
By Carlson's Inequality, we can know that
On the other hand, and
-- Haozhe Yang