Multivariate-Multi-objective Optimization Problem: $x+y+z = 1$ and $x^2+y^2+z^2 = 1$ $x, y, z $ are three distinct positive real numbers such that 
$$\begin{cases}&x + y + z = 1\\
&{x^2} + {y^2} + {z^2} = 1\\
&x \ne y \ne z\\
&0 < x,y,z < 1
 \end{cases}$$
Is there any solution for $x, y, z $ ? If yes, how we can find the solution. Thank you.

Since $0<x,y,z<1$, you have that $x^2<x, y^2<y$ and $z^2<z$. Plugging in the first two equations you get $$1=x^2+y^2+z^2<x+y+z=1$$ which is a contradiction.
