Question

x≠y≠z,x+1/y=y+1/z=z+1/x.求x立方 · y立方 ·z立方

Answer 1

令x+1/y=y+1/z=z+1/x=r,有

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

r=x+1/y=1/(r-z)+1/(r-1/z),化简,得

(r^2-1)z^2-(r^3-r)z+r^2-1=0

同理,有

(r^2-1)x^2-(r^3-r)x+r^2-1=0

(r^2-1)y^2-(r^3-r)y+r^2-1=0

所以有x,y,z都是方程(r^2-1)t^2-(r^3-r)t+r^2-1=0的根

而x≠y≠z,所以有方程恒为0,即r^2-1=r^3-r=0,r=1

即x+1/y=y+1/z=z+1/x=1

xy+1=y;yz+1=z;zx+1=x;

xy*yz*zx=(x-1)(y-1)(z-1)=xyz-xy-yz-zx+x+y+z-1=xyz,

而xyz≠0,故xyz=1

x^3 * y^3 * z^3 = 1