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 Post subject: Challenge Problem #3Posted: Tue, 12 Aug 2003 20:22:03 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7961
Location: Lexington, MA
Challenge Problem #3

Solve:

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 Post subject: Posted: Tue, 12 Aug 2003 21:54:31 UTC
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Joined: Sun, 4 May 2003 16:04:19 UTC
Posts: 2906
WooHoo!

let the equation be:
a-b=2
(a-b)^2=a^2-2ab+b^2=4

a^3-b^3=56=(a-b)(a^2+ab+b^2)=(2)(4+3ab)
so 56=2(4+3ab) and
ab=8

(a-b)^2+4ab=(a+b)^2=4+32=36

if a+b=6, and a=4,b=2
6x+28=4^3, x=6
if a+b=-6
a=-2,b=-4
6x+28=-8
x=-6

_________________
Has anyone noticed that the below is WRONG? Otherwise this statement would be true:

where

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 Post subject: Posted: Tue, 12 Aug 2003 23:09:56 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Mon, 19 May 2003 19:55:19 UTC
Posts: 7961
Location: Lexington, MA
Nice going Bugz!
I was sure that this was your kind of problem.

The book has a more devious solution.
It's quite a learning experience.
It follows immediately after the wavy line.

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Theorem: If then
(A proof is given at the end.)

We have:

Let

From the theorem, we have:

********************

Proof

Cube both sides:

Since , we have:

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 Post subject: Posted: Tue, 12 Aug 2003 23:42:48 UTC
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Joined: Sun, 4 May 2003 16:04:19 UTC
Posts: 2906
yes, i do like the book's solution

_________________
Has anyone noticed that the below is WRONG? Otherwise this statement would be true:

where

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 Post subject: Posted: Wed, 13 Aug 2003 03:14:44 UTC
 S.O.S. Oldtimer

Joined: Mon, 19 May 2003 18:41:04 UTC
Posts: 210
Wow...I think both solutions are really clever!

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 Post subject: Posted: Wed, 13 Aug 2003 07:35:04 UTC
 Member of the 'S.O.S. Math' Hall of Fame

Joined: Fri, 2 May 2003 16:33:24 UTC
Posts: 781
Location: Hong Kong
There's a easier proof for a+b+c=0=>a^3+b^3+c^3=3abc by observing that
a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)

_________________
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\___ \ / _ \ / _` | '__ / _ \ '__ |_ /
___) | (_) | (_| | | | __/ | / /
|____/ \___ / \__,_|_| \__|_ | /___|

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 Post subject: Posted: Fri, 15 Aug 2003 06:10:43 UTC
 S.O.S. Oldtimer

Joined: Sun, 29 Jun 2003 18:38:23 UTC
Posts: 166
Location: Earth
Yes... cuz everyone can observe that.

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