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

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mahmit2012Best ResponseYou've already chosen the best response.0
It is just a surface in a 3 dimensional space.
 one year ago

KingGeorgeBest ResponseYou've already chosen the best response.0
I'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.
 one year ago

mahmit2012Best ResponseYou've already chosen the best response.0
dw:1341442265927:dw
 one year ago

mahmit2012Best ResponseYou've already chosen the best response.0
dw:1341442357328:dw
 one year ago

mahmit2012Best ResponseYou've already chosen the best response.0
dw:1341442492121:dw
 one year ago

mahmit2012Best ResponseYou've already chosen the best response.0
So I just found out min(f)=1 so as I told in above that is a surface in a space.
 one year ago

mukushlaBest ResponseYou've already chosen the best response.1
hi @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 \[x^2y^2=xyz=z^21=0\]
 one year ago

mukushlaBest ResponseYou've already chosen the best response.1
only triples that satisfies the equation \[(x,y,z)=(1,1,1),(−1,−1,1),(1,−1,−1),(−1,1,−1)\]
 one year ago

Ishaan94Best ResponseYou've already chosen the best response.0
Thanks @mukushla if it's not too much to ask did you take training or something for the olympiads?
 one year ago

mukushlaBest ResponseYou've already chosen the best response.1
welcome my friend no im a chemical engineering student and just love math
 one year ago

mukushlaBest ResponseYou've already chosen the best response.1
do u wanna get ready for Math Olympiad or something?
 one year ago

Ishaan94Best ResponseYou've already chosen the best response.0
no i am already past my high school but i love solving math problems.
 one year ago
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