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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

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 Phil Carmody Posts: 2,219 Registered: 12/7/04
Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Posted: Nov 1, 2013 4:48 AM

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

>
> As a number of people have surmised, the maximum possible
> value of A is sqrt(3).
>
> Here's a proof ...

...
> contradiction, which completes the proof.

That is a lovely mathematical puzzle, thanks for posting it!

Phil
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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