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Topic: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Replies: 20   Last Post: Nov 1, 2013 4:48 AM

 Messages: [ Previous | Next ]
 David C. Ullrich Posts: 3,555 Registered: 12/13/04
Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Posted: Oct 25, 2013 10:36 AM

On Fri, 25 Oct 2013 07:20:29 -0500, quasi <quasi@null.set> wrote:

>Here's a nice challenge problem which I adapted from a past
>competition problem ...
>
>Problem:
>
>Find, with proof, the largest real number A such that
>
> x + y + z >= xyz
>
>implies
>
> x^2 + y^2 + z^2 >= Axyz

I'm assuming you meant to restrict to x, y, z >= 0.
The largest such A is 0.

If x = y = 1 then

>My solution is elementary (avoids Calculus), but if a method
>based on Calculus yields an easy resolution, that would be worth
>seeing as well.
>
>quasi

Date Subject Author
10/25/13 quasi
10/25/13 David C. Ullrich
10/25/13 quasi
10/26/13 Karl-Olav Nyberg
10/26/13 quasi
10/26/13 Peter Percival
10/25/13 David C. Ullrich
10/25/13 Don Coppersmith
10/25/13 Don Coppersmith
10/25/13 gnasher729
10/25/13 Peter Percival
10/27/13 gnasher729
10/27/13 dan.ms.chaos@gmail.com
10/27/13 Peter Percival
10/27/13 fom
10/30/13 quasi
10/30/13 Peter Percival
10/30/13 quasi
10/31/13 quasi
10/31/13 quasi
11/1/13 Phil Carmody