4. In all other cases, $-\mathbf{0}$ points.

## Task 4

Maxim Andreevich, a former university lecturer, gives math lessons to groups of schoolchildren. The cost of one hour (60 minutes) of Maxim Andreevich's lesson with a group of schoolchildren is 3 thousand rubles (after all taxes). In addition to income from tutoring, he has a steady income from investment activities, which amounts to 14 thousand rubles per month. Maxim Andreevich dedicates his weekends entirely to his family and does not work. Maxim Andreevich is very disciplined and strictly monitors the distribution of his time and income. On working days, of which there are exactly 21 in each month, he sleeps for eight hours a day, works for some time, devotes some time to his hobby - creating metal engravings, and spends the rest of the time on rest and household chores. He considers his hobby an important part of his life, so if he works L hours in a day, he will definitely spend 2L hours on creating engravings. He does not sell the engravings but gives them to friends and acquaintances. In addition, Maxim Andreevich is an active philanthropist and donates money to a fund for helping sick children. Every working day, he sets aside exactly $\mathrm{k} / 3$ thousand rubles for charity if he spends a total of $\mathrm{k}$ hours on rest and household chores that day. He spends 70 thousand rubles per month on all household expenses, hobbies, and household management (excluding charitable donations), and he has no other expenses. Of course, Maxim Andreevich loves to rest and do household chores, but he has to work, and every working day he decides for himself how many lessons with students he will conduct that day.

(a) What is the maximum amount of money Maxim Andreevich can donate to charity each month?

(b) Since Maxim Andreevich is a very good math teacher, the parents of the students he teaches are willing to offer him a higher rate for additional lessons (beyond those he already conducts daily). Maxim Andreevich has long taken care of the days when he will no longer be able to work, so he confidently spends all the money he receives each month. Will Maxim Andreevich agree to their offer? If yes, what should the pay for the extra lessons be for him to agree to conduct additional lessons? If no, explain why he would refuse such an offer.
# Solution and Grading Scheme:

(a) Maxim Andreevich works $\left(24-8-2 L_{i}-k_{i}\right)=16-2 L_{i}-k_{i}$ hours a day, where $L_{i}$ is the number of hours he works as a tutor on the $i$-th working day, and $k_{i}$ is the number of hours he spends on rest and household chores on the $i$-th working day. The number of working hours in one working day can differ from the number of working hours in another working day.

Then $16-2 L_{i}-k_{i}=L_{i}$, or $L_{i}=\left(16-k_{i}\right) / 3$

His daily earnings from tutoring are $3 L_{i}=3\left(16-k_{i}\right) / 3=16-k_{i}$.

In a month, Maxim Andreevich earns from tutoring

$$
\begin{gathered}
3 L_{1}+3 L_{2}+\cdots+3 L_{20}=\left(16-k_{1}\right)+\left(16-k_{2}\right)+\cdots+\left(16-k_{20}\right) \\
3\left(L_{1}+L_{2}+\cdots+L_{20}\right)=21 * 16-\left(k_{1}+k_{2}+\cdots+k_{20}\right)
\end{gathered}
$$

Or $3 L^{*}=336-K^{*}$,

where $L^{*}$ is the total number of working hours in a month, and $K^{*}$ is the total number of hours in a month that Maxim Andreevich spends on rest and household chores.

His monthly expenses amount to $70+\left(k_{1}+k_{2}+\cdots+k_{20}\right) / 3=70+K^{*} / 3$. Since he spends all his earnings, including income from investment activities, then

$14+336-K^{*}=70+K^{*} / 3$, from which we find the maximum number of hours in a month that Maxim Andreevich spends on rest and household chores: $K^{*}=210$.

Then in a month, Maxim Andreevich will spend a maximum of $\frac{K^{*}}{3}=\frac{210}{3}=$ 70 thousand rubles on charity.

Full solution - 10 points.

For a partial solution assuming that the number of working hours, leisure, etc., are the same every day of the month - 6 points.

If only the equations for the number of hours are written - 2 points.

(b) If Maxim Andreevich declines the offer to work more, then in a month he will work $L^{*}=(336-70) / 3=42$ hours.

Suppose Maria Ivanovna would agree to work extra for a higher hourly wage A>3. Then her monthly earnings from tutoring would be

$3 * 42+A(\tilde{L}-42)=126-42 \mathrm{~A}+\mathrm{A} \tilde{L}$

Since $L_{i}=\left(16-k_{i}\right) / 3$, then with the new distribution of working time, as before

$\tilde{L}=L_{1}+L_{2}+\cdots+L_{20}=\left(16-k_{1}\right) / 3+\left(16-k_{2}\right) / 3+\cdots+\left(16-k_{20}\right) / 3=16 * 21 / 3-$ $\widetilde{K} / 3$

Then in a month, including investments, Maxim Andreevich will earn

$126-42 A+A(112-\widetilde{K} / 3)+14$ thousand rubles.

Since the maximum expenses will equal the income with the proportions in the distribution of time between work and rest, then

$$
\begin{gathered}
126-42 A+A(112-\widetilde{K} / 3)+14=70+\widetilde{K} / 3 \\
70(A+1)=\frac{\widetilde{K}}{3}(A+1)
\end{gathered}
$$

And, as in part (a), we get: $K^{*}=210$. That is, Maria Ivanovna will not agree to work more than the hours she works at a rate of 2 thousand rubles/hour, regardless of the increased wage rate.

Full justification - 10 points,

Partial justification - 5 points

## Task 5

Company "Zolotnik" is engaged in gold and precious stone mining. "Zolotnik" has signed a contract to process a square plot of land with an area of 10 km². The workers of "Zolotnik" suffer from chronic fatigue due to the very labor-intensive work, which leads to a 2-fold decrease in their productivity each day. The amount of land that any worker can process on the first day of work is $1 / 2$ km². The salary of one worker for the entire period of work is 3 monetary units (mon. units), and the daily rental cost of equipment for processing the entire plot of land is 7 mon. units. The contract with a team of workers, which can consist of any number of workers, can only be signed on the first day the contract for processing the plot of land comes into effect.

(a) Find the minimum number of workers required to process the specified plot of land.

(b) Find the number of workers who can process the plot of land in one day.

(c) If "Zolotnik" minimizes its costs for processing the plot of land, how many workers will the company hire and how long will the processing take? What will be the total costs of the company for renting equipment and paying workers?

(a) Answer: 11.

We will prove by contradiction that no smaller number of workers can complete the task. Suppose $n$ workers processed the entire plot of land in $k$ days, where $n \leq 10$. Then each of them processed no more than

$\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{k}}=1-\frac{1}{2^{k}}$

square kilometers of land, which is strictly less than 1.

Thus, in total, all workers processed no more than

$$
n\left(1-\frac{1}{2^{k}}\right)=n-\frac{n}{2^{k}} \leq 10-\frac{10}{2^{k}}10
$$

Thus, 11 is the minimum number of workers required to process the given plot of land.

## Full justification - 5 points

(b) Answer: 20.

Indeed, on the first day, each worker processes $\frac{1}{2}$ square kilometers of land, so to process the entire plot, $10: \frac{1}{2}=20$ workers are required.

## Full justification - 5 points

(c) Answer: the company will hire 14 workers for 2 days;