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 Manipulations

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isabella2296
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PostSubject: Manipulations   Sat Aug 16, 2008 8:07 am

Source: AMC 10

a + a^-1 = 4

Find a^4 + a^-4


We really need LaTeX.
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herefishyfishy1
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 8:13 am

You square both sides of x+x^-1=4. You get x^2+2+x^-2=16. Then that means x^2+x^-2=14. Square both sides again. x^4+2+x^-4=196. Therefore, x^4+x^-4=194.

The answer is 194!
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isabella2296
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 8:15 am

Correct!

How about this one?

x + y = 2

x^2 + y^2 = 3

Find x^3 + y^3.

(Yes, this is from Problems of the Day.)
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herefishyfishy1
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 8:24 am

86? Because if you divide the first equation into the second, you get x-y=3/2, and you can solve x=7/4 and y=1/4. And then you cube them, and add.
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isabella2296
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 11:33 am

Sorry, that's incorrect.
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 2:35 pm

edit: will post solution later.. just realized where this problem was from
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herefishyfishy1
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 7:30 pm

I completely failed. The answer is 43/8. I divided wrong.
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PostSubject: Re: Manipulations   Sat Aug 16, 2008 9:48 pm

?? I got a way different answer.
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isabella2296
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PostSubject: Re: Manipulations   Sun Aug 17, 2008 8:39 am

Sorry, not 48/3. It is 5. Anyone have a solution, or should I post it?
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PostSubject: Re: Manipulations   Sun Aug 17, 2008 9:05 am

x+y=2
x^2+y^2=3

cube x+y=2 to get x^3+y^3+3x^2y+3xy^2=8.
square x+y=2 to get x^2+2xy+y^2=4. Subracting x^2+y^2=3 from this, you find xy to be 1/2.
Now look at 3x^2y+3xy^2. This can be factored into 3xy(x+y), Since x+y=2, and xy=1/2, 3xy(x+y)=3. Subract this from x^3+y^3+3x^2y+3xy^2=8 to get x^3+y^3=5.
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PostSubject: Re: Manipulations   Sun Aug 17, 2008 9:30 am

Correct! More on solving these in my blog. Smile
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AIME15
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PostSubject: Re: Manipulations   Sun Aug 17, 2008 1:44 pm

For those who don't know about isabella's blog, the URL is:

http://www.artofproblemsolving.com/Forum/weblog.php?w=1172
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