Conditions to check when this system is well defined Given $s,i \in [0,1]$ I need to check if this system can be verified:
$$\begin{cases}
s = 1 - \frac{\min\{x,y,z\}}{i} \\
\frac{x+y+z}{3} = i
\end{cases}$$
where $x,y,z \in [0,1]$ are unknown.
For example if $s=1$ and $i=1$ the system is not verified since $i$ should be the average of three real numbers in $[0,1]$, hence $\min\{x,y,z\} = 1$ and $s$ should be $1-1 = 0 \neq 1$.
 A: The problem is symmetric in $\,x,y,z\,$ so it can be assumed WLOG that $\,x \le y \le z\,$, then $\,\min\{x,y,z\} = x\,$ and the system becomes:
$$
\begin{cases}
s = 1 - \frac{x}{i} \\
\frac{x+y+z}{3} = i
\end{cases}
\;\;\iff\;\;
\begin{cases}
si = i - x \\
x+y+z = 3i
\end{cases}
\;\;\iff\;\;
\begin{cases}
\begin{align}
x = (1-s)i \quad\tag{1} \\
y+z = (s+2)i \quad\tag{2}
\end{align}
\end{cases}
$$
The first equation $(1)$ is always satisfiable since the RHS $\,(1-s)i \in [0,1]\,$ is within the allowed range for $\,x\,$.
For $\,(2)\,$ to have eligible solutions $\,y,z \in [x, 1]\,$, the necessary and sufficient condition is that $\,y+z \in [2x, 2] \iff 2x \le y+z \le 2\,$, which by $\,(1)$-$(2)\,$ is:
$$\require{cancel}
2(1-s)\,i \;\le\; (s+2)\,i \;\le\; 2 \;\;\iff\;\;
\begin{cases}
\begin{align}
2(1-s)\le s+2 \;\;&\iff\;\; 0 \le 3s \quad\tag{3} \\[5px]
(s+2)i \le 2 \;\;& \tag{4}
\end{align}
\end{cases}
$$
Condition $(3)$ is always satisfied, so in the end the condition for solvability is $(4)\,$.
