Problem 8.2. (15 points) Real numbers $x_{1}, x_{2}, x_{3}, x_{4}$ are such that

$$
\left\{\begin{array}{l}
x_{1}+x_{2} \geqslant 12 \\
x_{1}+x_{3} \geqslant 13 \\
x_{1}+x_{4} \geqslant 14 \\
x_{3}+x_{4} \geqslant 22 \\
x_{2}+x_{3} \geqslant 23 \\
x_{2}+x_{4} \geq 24
\end{array}\right.
$$

What is the smallest value that the sum $x_{1}+x_{2}+x_{3}+x_{4}$ can take?
Answer: 37.

Solution. By adding the second equality to the last one, we get $x_{1}+x_{2}+x_{3}+x_{4} \geqslant 37$.

It is also worth noting that the value of the expression $x_{1}+x_{2}+x_{3}+x_{4}$ can be equal to 37, for example, when $x_{1}=1, x_{2}=11, x_{3}=12, x_{4}=13$. It is easy to verify that such numbers satisfy all the conditions of the problem.

## Criteria

The points for the estimate and the example are summed up:

+7 p. Estimate - it is proven that the considered sum is not less than 37.

+8 p. Example - numbers are provided for which the sum is 37 (or it is otherwise proven that such numbers exist).