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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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 Don Coppersmith Posts: 60 Registered: 2/2/06
Re: x + y + z >= xyz implies x^2 + y^2 + z^2 >= Axyz
Posted: Oct 25, 2013 4:20 PM
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My solution assumed the product xyz is positive.
If xyz is negative or 0, then the result is immediate
because T is nonnegative.

> A = sqrt(3).
> Assume x,y,z nonnegative.
>
> Let T = x^2+y^2+z^2,
> S = x+y+z,
> P = xyz.
>
> Express T^2-3SP as the sum of squares:
> T^2-3SP = (x^2-yz)^2 + (y^2-zx)^2 + (z^2-xy)^2
> + (1/2)[(xz-yz)^2 + (xy-yz)^2 + (xy-xz)^2]
>
> Then T^2-3SP >= 0,
> T^2 >= 3SP >= 3P^2
> T >= sqrt(3)P
> An extremal example is x=y=z=sqrt(3).
>
> Don Coppersmith
>

> > 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.
> >
> > 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

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