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isabella2296
Admin
Number of posts : 45
Location : You'd really like to know, wouldn't you?
Registration date : 2008-08-16
| | Subject: 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
Developer
Number of posts : 36
Age : 29
Location : An insignificant little blue-green planet 93 million miles from the sun Sol.
Registration date : 2008-08-16
| | Subject: 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
Admin
Number of posts : 45
Location : You'd really like to know, wouldn't you?
Registration date : 2008-08-16
| | Subject: 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
Developer
Number of posts : 36
Age : 29
Location : An insignificant little blue-green planet 93 million miles from the sun Sol.
Registration date : 2008-08-16
| | Subject: 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
Admin
Number of posts : 45
Location : You'd really like to know, wouldn't you?
Registration date : 2008-08-16
| | Subject: Re: Manipulations Sat Aug 16, 2008 11:33 am | |
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skittles
School Math Nerd
Number of posts : 40
Age : 29
Location : Lost
Registration date : 2008-08-13
| | Subject: 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
Developer
Number of posts : 36
Age : 29
Location : An insignificant little blue-green planet 93 million miles from the sun Sol.
Registration date : 2008-08-16
| | Subject: Re: Manipulations Sat Aug 16, 2008 7:30 pm | |
| I completely failed. The answer is 43/8. I divided wrong. | |
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skittles
School Math Nerd
Number of posts : 40
Age : 29
Location : Lost
Registration date : 2008-08-13
| | Subject: Re: Manipulations Sat Aug 16, 2008 9:48 pm | |
| ?? I got a way different answer. | |
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isabella2296
Admin
Number of posts : 45
Location : You'd really like to know, wouldn't you?
Registration date : 2008-08-16
| | Subject: 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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skittles
School Math Nerd
Number of posts : 40
Age : 29
Location : Lost
Registration date : 2008-08-13
| | Subject: 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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isabella2296
Admin
Number of posts : 45
Location : You'd really like to know, wouldn't you?
Registration date : 2008-08-16
| | Subject: Re: Manipulations Sun Aug 17, 2008 9:30 am | |
| Correct! More on solving these in my blog.  | |
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AIME15
Hardcore TMF user
Number of posts : 163
Age : 27
Location : Pleasanton, CA
Registration date : 2008-08-13
| | Subject: Re: Manipulations Sun Aug 17, 2008 1:44 pm | |
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