The number of real solutions of the system of equations x=2z21+z2,y=2x21+x2,z=2y21+y2  is:

The given system of equations are :  x=2z21+z2          ……( i ) y=2x21+x2          ……( ii ) z=2y21+y2          ……( iii )Now,  x=2z1+z2.z            =2z1+z2.2y21+y2             [from Eq.(iii)]          =2z(1+z2).2y(1+y2).y           =2z1+z2.2y1+y2.2x21+x2      [from Eq. (ii)]∴  1=(2z1+z2)(2y1+y2)(2x1+x2) Let x=tan  α,y=tanβ  and z=tanγ       1=(2tanγ1+tan2γ)(2tanβ1+tan2β)(2tanα1+tan2α)       ∴  1=sin2αsin2βsin2γ          (∵sin2α=2tanα1+tan2α) It is possible only when             sin2α=sin2β=sin2γ=1 ∴    α=β=γ=450 ∴    x=y=z=1 ∴   Solution of the given equation is (1,1,1) .