- Prove that if and , then

.- Similarly to the previous problem:

Prove that if and , then

.- Let's generalize the problems above:

Prove that if , , , and then

.Find the maximum possible value of .

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- Jan 1st 2008, 07:49 AMjames_bondInequality
- Prove that if and , then

. - Similarly to the previous problem:

Prove that if and , then

. - Let's generalize the problems above:

Prove that if , , , and then

.Find the maximum possible value of .

- Prove that if and , then
- Jan 1st 2008, 08:48 AMIsomorphism
- Jan 1st 2008, 09:07 AMIsomorphism
I post the remaining separately because a small trick there, wont work any longer ;)

I will directly attack the generalization,

*Claim:*

__Proof:__

Denote LHS of your inequality by I,

AM-HM says:

-------------(*)

Applying CS:

But

So .

But by (*),

Thus

Equality for all s equal.

K cant get any better than that because we showed equality achieving ;)