5. A vessel is filled with cold water. The following procedure is performed with this vessel: three quarters of the cold water are poured out and the vessel is refilled to its original volume with hot water. As a result, the temperature of the water in the vessel increases by $24^{\circ} \mathrm{C}$. After this, the procedure is repeated several times: three quarters of the volume of water are poured out and the vessel is refilled to its original volume with hot water (of the same temperature).

A) After how many procedures will the temperature of the water in the vessel differ from the temperature of the hot water by $3^{\circ} \mathrm{C}$?

B) Is it possible to achieve a situation where the temperature of the water in the vessel differs from the temperature of the hot water by exactly $1^{\circ} \mathrm{C}$?
Answer: A) $n=4$; B) impossible.

Instructions. From the heat balance equation:

$$
\begin{aligned}
& t_{1}-t_{c}=t_{h}-t_{1} \Rightarrow 2 t_{1}=t_{c}+t_{h} \\
& t_{2}-t_{1}=t_{h}-t_{2} \Rightarrow 2 t_{2}=t_{h}+t_{1} \Rightarrow t_{2}=t_{h}+\frac{t_{h}+t_{c}}{2} \Rightarrow t_{2}=\frac{t_{c}+3 t_{h}}{4}
\end{aligned}
$$

Similarly, $t_{3}=\frac{t_{c}+7 t_{h}}{8}$

$t_{n}=\frac{t_{c}+\left(2^{n}-1\right) t_{h}}{2^{n}} \Rightarrow 2^{n}=\frac{t_{h}-t_{c}}{t_{h}-t_{n}}$

From the problem statement and (1), it follows that $t_{h}-t_{c}=48$. From (4), we get that: A) $n=4$; B) impossible.

All problems were evaluated according to the following criteria:

20 points - correct (not necessarily the same as above) solution and correct answer;

15 points - mostly correct solution and correct answer, but there are defects in the justification

10 points - solution is mostly correct, but contains computational errors.