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Ishaan94
 2 years ago
Find all real triples \((x,y, z)\) that satisfy \(x^4+y^4+z^4−4xyz = −1\).
Ishaan94
 2 years ago
Find all real triples \((x,y, z)\) that satisfy \(x^4+y^4+z^4−4xyz = −1\).

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mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.0It is just a surface in a 3 dimensional space.

KingGeorge
 2 years ago
Best ResponseYou've already chosen the best response.0I've seen a method to solve problems like this easily, but I can't remember it right now. I'll try and look it up later to try and find more information.

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1341442265927:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1341442357328:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.0dw:1341442492121:dw

mahmit2012
 2 years ago
Best ResponseYou've already chosen the best response.0So I just found out min(f)=1 so as I told in above that is a surface in a space.

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.1hi @Ishaan94 Using Completing the Square \[x^4+y^4+z^44xyz+1=0\\x^42x^2y^2+y^4+2x^2y^2+z^4+2z^22z^24xyz+1=0\\x^42x^2y^2+y^4+2(x^2y^22xyz+z^2)+z^42z^2+1=0\\(x^2y^2)^2+2(xyz)^2+(z^21)^2=0\] so \[x^2y^2=xyz=z^21=0\]

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.1only triples that satisfies the equation \[(x,y,z)=(1,1,1),(−1,−1,1),(1,−1,−1),(−1,1,−1)\]

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.0Thanks @mukushla if it's not too much to ask did you take training or something for the olympiads?

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.1welcome my friend no im a chemical engineering student and just love math

mukushla
 2 years ago
Best ResponseYou've already chosen the best response.1do u wanna get ready for Math Olympiad or something?

Ishaan94
 2 years ago
Best ResponseYou've already chosen the best response.0no i am already past my high school but i love solving math problems.
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