Multivariate-Multi-objective Optimization Problem: $x+y+z = 1$ and $x^2+y^2+z^2 = 1$

Question

$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.

Answer 1

$$\begin{cases}x + y + z = 1\\0\leq x,y,z \leq 1\end {cases}$$ is the equation for a tetrahedron whose vertices are located in the three axis. The vertices are at $x = 1, y = 1, z = 1$.

$$x^2 + y^2 + z^2 = 1$$ is the equation for a sphere with radius 1.

The sphere and that tetrahedron only intersect in the tetrahedron's vertices. i.e. on $(1,0,0), (0,1,0)$ and $(0,0,1) $.

Since you have restricted that the coordinates are all positive $> 0$, and strictly less than 1, there is no solution for your system.

Answer 2

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.