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February 1st, 2018, 05:41 PM  #1 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  How did they get those two equations?
I am just wondering how they went from equation (3) to the pair of equations $3uv+p=0$ and $u^3+v^3=q$. I understand if those are true then the original equation will be true, but I was wondering why they chose $3uv+p=0$ instead of $u+v=0$. Why did they set $u^3+v^3$ equal to $q$? This step just seems mysterious to me. 
February 1st, 2018, 05:51 PM  #2 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. 
OMG nevermind since $x=u+v$, then we don't want it to be $0$. Sorry for asking a dumb question. However, I still don't really understand why they chose $q$ to be $u^3+v^3$ but maybe I'm just tired.

February 1st, 2018, 07:09 PM  #3 
Global Moderator Joined: Dec 2006 Posts: 20,919 Thanks: 2202 
That's a reasonable question. Somehow, they had reason to believe that would be a good strategy.

February 1st, 2018, 08:49 PM  #4  
Senior Member Joined: Oct 2009 Posts: 850 Thanks: 325  Quote:
 
February 2nd, 2018, 10:51 AM  #5 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems. 
This reminds me of a topic I was searching for on the internet; the difference between competition math and research math. Lately, I've been practicing a lot of problems from math competitions, and it sort of puts you in this habit of looking at math and asking to yourself, what am I "supposed" to do? You begin to think that every math problem has a path that you are supposed to take. But, I realize that research math is not like this. When I first read this proof, my thought was, how did he know that that was what he was "supposed to do"? But, in research math there is no "supposed to do". Anyways, thank you both for your responses 
February 2nd, 2018, 10:52 AM  #6 
Senior Member Joined: Oct 2016 From: Arizona Posts: 209 Thanks: 37 Math Focus: I'm still deciding, but my recent focus has been olympiad problems and math journal problems.  

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