
Author  Message 

isabella2296 Admin
Number of posts : 45 Location : You'd really like to know, wouldn't you? Registration date : 20080816
 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. 

 
herefishyfishy1 Developer
Number of posts : 36 Age : 22 Location : An insignificant little bluegreen planet 93 million miles from the sun Sol. Registration date : 20080816
 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! 

 
isabella2296 Admin
Number of posts : 45 Location : You'd really like to know, wouldn't you? Registration date : 20080816
 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.) 

 
herefishyfishy1 Developer
Number of posts : 36 Age : 22 Location : An insignificant little bluegreen planet 93 million miles from the sun Sol. Registration date : 20080816
 Subject: Re: Manipulations Sat Aug 16, 2008 8:24 am  
 86? Because if you divide the first equation into the second, you get xy=3/2, and you can solve x=7/4 and y=1/4. And then you cube them, and add. 

 
isabella2296 Admin
Number of posts : 45 Location : You'd really like to know, wouldn't you? Registration date : 20080816
 Subject: Re: Manipulations Sat Aug 16, 2008 11:33 am  
 

 
skittles School Math Nerd
Number of posts : 40 Age : 22 Location : Lost Registration date : 20080813
 Subject: Re: Manipulations Sat Aug 16, 2008 2:35 pm  
 edit: will post solution later.. just realized where this problem was from 

 
herefishyfishy1 Developer
Number of posts : 36 Age : 22 Location : An insignificant little bluegreen planet 93 million miles from the sun Sol. Registration date : 20080816
 Subject: Re: Manipulations Sat Aug 16, 2008 7:30 pm  
 I completely failed. The answer is 43/8. I divided wrong. 

 
skittles School Math Nerd
Number of posts : 40 Age : 22 Location : Lost Registration date : 20080813
 Subject: Re: Manipulations Sat Aug 16, 2008 9:48 pm  
 ?? I got a way different answer. 

 
isabella2296 Admin
Number of posts : 45 Location : You'd really like to know, wouldn't you? Registration date : 20080816
 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? 

 
skittles School Math Nerd
Number of posts : 40 Age : 22 Location : Lost Registration date : 20080813
 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. 

 
isabella2296 Admin
Number of posts : 45 Location : You'd really like to know, wouldn't you? Registration date : 20080816
 Subject: Re: Manipulations Sun Aug 17, 2008 9:30 am  
 Correct! More on solving these in my blog. 

 
AIME15 Hardcore TMF user
Number of posts : 163 Age : 21 Location : Pleasanton, CA Registration date : 20080813
 Subject: 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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 Subject: Re: Manipulations  
 

 
