4. Find all real solutions to the system of equations

$$
\left\{\begin{array}{l}
\sqrt{x-997}+\sqrt{y-932}+\sqrt{z-796}=100 \\
\sqrt{x-1237}+\sqrt{y-1121}+\sqrt{3045-z}=90 \\
\sqrt{x-1621}+\sqrt{2805-y}+\sqrt{z-997}=80 \\
\sqrt{2102-x}+\sqrt{y-1237}+\sqrt{z-932}=70
\end{array}\right.
$$

(L. S. Korechkova, A. A. Tessler)
Answer: $x=y=z=2021$.

Solution. First, we prove that the solution is unique if it exists. Let $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ be two different solutions and, without loss of generality, $x_{1} \leqslant x_{2}$. Then there are four possible cases: $y_{1} \leqslant y_{2}$ and $z_{1} \leqslant z_{2}$ (with at least one of the three inequalities being strict); $y_{1} \leqslant y_{2}$ and $z_{1}>z_{2} ; y_{1}>y_{2}$ and $z_{1} \leqslant z_{2} ; y_{1}>y_{2}$ and $z_{1}>z_{2}$. Each case contradicts the corresponding equation due to monotonicity.

The solution itself can be found by assuming it is an integer and all roots are extracted as integers. For example, let $x=1621+a^{2}=1237+b^{2}$ for integers $a, b \geqslant 0$, then $(b+a)(b-a)=384$; if we try all possible values of $a$, it becomes clear that only $a=20$ allows the other roots to be extracted. Additionally, one can notice that some roots look similar (e.g., $\sqrt{x-997}$ and $\sqrt{z-997}$), so it is convenient to look for a solution where $x=y=z$. After this, the numbers 1121, 1621, 2102 can help guess the answer.

Criteria. 2 points for finding the solution (verifying the calculations that it fits or somehow motivating its discovery is not required); 5 points for proving uniqueness.