## Ishaan94 3 years ago Find all real triples \((x,y, z)\) that satisfy \(x^4+y^4+z^4−4xyz = −1\).

1. Ishaan94

@KingGeorge

2. mahmit2012

It is just a surface in a 3 dimensional space.

3. KingGeorge

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.

4. mahmit2012

|dw:1341442265927:dw|

5. mahmit2012

|dw:1341442357328:dw|

6. mahmit2012

|dw:1341442492121:dw|

7. mahmit2012

So I just found out min(f)=-1 so as I told in above that is a surface in a space.

8. mukushla

hi @Ishaan94 Using Completing the Square \[x^4+y^4+z^4-4xyz+1=0\\x^4-2x^2y^2+y^4+2x^2y^2+z^4+2z^2-2z^2-4xyz+1=0\\x^4-2x^2y^2+y^4+2(x^2y^2-2xyz+z^2)+z^4-2z^2+1=0\\(x^2-y^2)^2+2(xy-z)^2+(z^2-1)^2=0\] so \[x^2-y^2=xy-z=z^2-1=0\]

9. mukushla

only triples that satisfies the equation \[(x,y,z)=(1,1,1),(−1,−1,1),(1,−1,−1),(−1,1,−1)\]

10. Ishaan94

Thanks @mukushla if it's not too much to ask did you take training or something for the olympiads?

11. mukushla

welcome my friend no im a chemical engineering student and just love math

12. mukushla

13. Ishaan94

no i am already past my high school but i love solving math problems.

14. mukushla

just like me :D