Problem 9.8. Integers $n$ and $m$ satisfy the inequalities $3 n - m \leq 26$, $3 m - 2 n < 46$. What can $2 n + m$ be? List all possible options.
Answer: 36.

Solution. Since the numbers $m$ and $n$ are integers, the values of the expressions in the condition $3 n-m$, $n+m$, $3 m-2 n$ are also integers. Then

$$
\left\{\begin{array}{l}
3 n-m \leqslant 4 \\
n+m \geqslant 27 \\
3 m-2 n \leqslant 45
\end{array}\right.
$$

By adding three times the first inequality to the third, we get $7 n \leqslant 57$, from which $n \leqslant 8$. By adding twice the first inequality to three times the third, we get $7 m \leqslant 143$, from which $m \leqslant 20$.

Thus, $n \leqslant 8, m \leqslant 20$, and $n+m \geqslant 27$. This is possible in only three cases.

- Suppose $n=7, m=20$. Then the condition $3 m-2 n \leqslant 45$ is not satisfied.
- Suppose $n=8, m=19$. Then the condition $3 n-m \leqslant 4$ is not satisfied.
- Suppose $n=8, m=20$. It is easy to see that all conditions are satisfied in this case. Therefore, $2 n+m=2 \cdot 8+20=36$.

Another solution. Let's represent the given inequalities as regions on the coordinate plane Omn, as shown in Fig. 8. Each inequality corresponds to a certain half-plane. For example, the inequality $3 n-m<5$ is the half-plane located below the line $3 n-m=5$.

The intersection of these half-planes is a triangle, inside which lies one integer point $m=20, n=8$ (points on the boundary of the triangle do not work, as the inequalities are strict), from which we immediately get the unique answer $2 n+m=36$.

![](https://cdn.mathpix.com/cropped/2024_05_06_2fdcf97aa9799d0d4cd6g-35.jpg?height=543&width=1007&top_left_y=88&top_left_x=219)

Fig. 8: to the solution of problem 9.8

## 10th grade