3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive, which means the largest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend

$$
\text { 4800+1500-4800*0.2=5340 rubles. }
$$

This is the most cost-effective way to make the purchases.

## Maximum 30 points

## Problem 5

Student Alexei informed his parents that he is already an adult and can manage his finances independently. His mother suggested he use a duplicate of the bank card linked to her account. To participate in a charitable New Year's program, Alexei wants to buy 40 "Joy" chocolate bars and donate them to a children's home. However, the bank, whose clients are Alexei's parents, has implemented a new system to protect against unauthorized card payments. The protection system analyzes the root mean square (RMS) value of the last 3 purchases (S) using the formula $S=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{3}}$, where $\mathrm{x}_{1}, \mathrm{x}_{2}$, and $\mathrm{x}_{3}$ are the amounts of the last 3 purchases, and compares the value of $S$ with the amount of the current purchase. If the current payment exceeds the value of $S$ by three times, the bank blocks the transaction and requires additional verification (e.g., a call from his mother to the call center). Over the past month, only payments for mobile phone service have been made, each for 300 rubles. How many minimum receipts should Alexei split the purchase into to buy all 40 "Joy" chocolate bars at a cost of 50 rubles each?
# Solution:

The root mean square value of the last purchases is $\sqrt{(300 * 300+300 * 300+300 * 300) / 3}=300$ rubles. Therefore, the permissible first purchase is no more than $300 * 3=900$ rubles, with which 18 chocolates can be bought. It remains to buy 22 chocolates for a total of $22 * 50=1100$ rubles.

For the second purchase, the average value considering the first purchase will be $\sqrt{(300 * 300+300 * 300+900 * 900) / 3}=\sqrt{330000}$, which is more than 500 rubles. Therefore, the second purchase can be for more than $500 * 3=1500$ rubles, which is sufficient to buy the remaining chocolates. The minimum number of receipts is 2.

## Maximum 15 points

15 points - a fully detailed correct solution and the correct answer.

10 points - correct calculation of the permissible amount of the first purchase, the correct number of chocolates in the first purchase, and the correct calculation of the amount for the second purchase.

5 points - correct calculation of the permissible amount of the first purchase.

## Appendix to Problem No. 1

Insurance is a system of relationships between insurers (insurance company) and insured individuals (for example, any citizen), which allows reducing property risks by insuring, for example, property against possible adverse events...

Property insurance is one of the types of insurance according to the Civil Code of the Russian Federation. Real estate is one of the types of property. Houses, apartments, and land plots are considered real estate.

## To insure a risk: