Anatolia mathematics olympiads

[devrimci_kürt]

x^4 +y^4+z^4 =4xyz-1

denkleminin bütün gerçel çözümlerini bulunuz

 

 

x^4 +y^4+z^4 =4xyz-1

Find all real solutions of the equation

[devrimci_kürt]

We first arrange this equation

(x^4 +2.x^2+1) + (y^4 -2y^2.z^2+z^4)+(2y^2.z^2-4xyz+2x^2)=0

 

(x^2-1)^2 +(y^2-z^2)^2+ 2(yz-x)^2=0

condition for satisfying

(x^2-1)^2=0

(y^2-z^2)^2=0

2(yz-x)^2=0

hence

(x, y, z) = (1, 1, 1) , (1, -1, -1) , (-1, -1, 1) , (-1, 1, -1)