Prove the second inequality assuming the first

[ehrenfest]

Homework Statement

http://www.math.cornell.edu/~putnam/ineqs.pdf

Can someone give me a hint on problem 1?

Homework Equations

The Attempt at a Solution

 

[Hurkyl]

You were already given a hint: prove the second inequality assuming the first. Second hint below:




















Isn't the expression

$$\left( \frac{ \frac{1}{a_1} + \cdots + \frac{1}{a_n} }{n} \right)^{-1}$$

already in a form to which the first inequality is applicable?

 

[ehrenfest]

Yes. I read the problem. I spent 30 minutes manipulating both the first and the second inequality. Maybe I am missing something obvious, but I just don't see how to manipulate correctly. I tried logarithms, finding common denominators...

 

[Hurkyl]

Bah, you saw it before I edited my post.

 

[ehrenfest]

Wow. I feel stupid now. :)