1. There are 4 kg of a copper-tin alloy, in which $40\%$ is copper, and 6 kg of another copper-tin alloy, in which $30\%$ is copper. What masses of these alloys need to be taken so that after melting, 8 kg of an alloy containing $p\%$ copper is obtained? Find all $p$ for which the problem has a solution.
Answer: $0.8 p-24$ kg; $32-0.8 p$ kg; $32.5 \leqslant p \leqslant 35$.

Solution. If the first alloy is taken $x$ kg, then the second one is $(8-x)$ kg. The conditions of the problem impose restrictions on the possible values of $x$:

$$
\left\{\begin{array}{l}
0 \leqslant x \leqslant 4 ; \\
0 \leqslant 8-x \leqslant 6
\end{array} \Longleftrightarrow 2 \leqslant x \leqslant 4\right.
$$

Let's calculate the amount of copper in the new alloy:

$$
0.4 x+0.3(8-x)=\frac{p}{100} \cdot 8
$$

From this, $x=0.8 p-24$. Solving the double inequality $2 \leqslant 0.8 p-24 \leqslant 4$, we get the answer.

Evaluation. 12 points for a correct solution.