If `x^y=y^z=z^x` and `xz=y^2` then which of the following is correct?

If x^(ln)[(y)/(z)].y^(ln[xz]^(2)),z^(ln[(x)/(y)])=y^(4ln y) for any x>1,y>1 and z>1, then which one of the following is correct?

If x/(y+z) = y/(z+x) = z/(x+y) then which one of the following is correct ?

If x,y,z, are in A.P. and x^(2),y^(2),z^(2) are in H.P., then which of the following is correct ?

If x,y,z, are in A.P. and x^(2),y^(2),z^(2) are in H.P., then which of the following is correct ?

If |{:(y,x,y+z),(z,y,x+y),(x,z,z+x):}|=0 , then which one of the following is correct ?

If |(y,x,y+z),(z,y,x+y),(x,z,z+x)|=0 , then which one of the following is correct ?

If, for positive real numbers x, y, z, the numbers x+y, 2y and y + z are in harmonic progression, then which one of the following is correct ?