Prove that `[(x^2+y^2+z^2)/(x+y+z)]^(x+y+z)> x^x y^y z^z >[(x+y+z)/3]^(x+y+z)(x ,y ,z >0)`

Prove that |[x, y, z],[x^2, y^2, z^2], [x^3, y^3, z^3]|=xyz(x-y)(y-z)(z-x)

Prove that 64(x+y+z)^3-(2x+y+z)^3-(x+2y+z)^3-(x+y+2z)^3=3(3x+3y+2z)(3x+2y+3z)(2x+3y+3z)

If a^(x)=(x+y+z)^(y), a^(y)=(x+y+z)^(z), a^(z)=(x+y+z)^(x) , then :

Prove that |(y+z, x,y),(z+x, z, x),(x+y, y, z)| = (x+y+z)(x-z)^(2) .

Prove that Det [[x + y + 2z, x, y], [z, y + z + 2x, y], [z, x, z + x + 2y]] = 2 (x + y + z) ^ 3

Prove that |[x+y, y+z, z+x] , [z+x, x+y, y+z] , [y+z, z+x, x+y]|=2|[x,y,z] , [z,x,y] , [y,z,x]|

Show that Delta=|[(y+z)^2, x y , z x],[ x y, (x+z)^2, y z],[ x z, y z, (x+y)^2]|=2 x y z(x+y+z)^3

Prove: |(z, x, y),( z^2,x^2,y^2),(z^4,x^4,y^4)|=|(x, y, z),( x^2,y^2,z^2),(x^4,y^4,z^4)|=|(x^2,y^2,z^2),(x^4,y^4,z^4),(x, y, z)|=x y z(x-y)(y-z)(z-x)(x+y+z) .

Multiply: x^2+y^2+z^2-x y+x z+y z\ by x+y-z