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2. When insuring property, the insurance amount cannot exceed its actual value (insurance value) at the time of concluding the insurance contract.
Insurance tariff - the rate of the insurance premium or the insurance premium (insurance premium) expressed in rubles, payable per unit of the insurance amount, which is us... | # Solution:
In accordance with the instruction, the base rate is $0.2\%$ of the insurance amount, apply a reducing factor for the absence of a change in ownership over the past 3 years $(0.8)$ and an increasing factor for the absence of certificates from the PND and ND $(1.3)$.
In total: $0.2 * 0.8 * 1.3=0.208\%$
Th... | 31200 | Other | math-word-problem | Yes | Yes | olympiads | false |
3. Maria Ivanovna decided to use the services of an online clothing store and purchase summer clothing: trousers, a skirt, a jacket, and a blouse. Being a regular customer of this store, Maria Ivanovna received information about two ongoing promotions. The first promotion allows the customer to use an electronic coupon... | Solution:
(a) Maria Ivanovna can make one purchase, using only one of the promotions, or she can "split" the selected items into two purchases, using both promotions in this case.
Let's consider all possible options:
1) One purchase. In this case, Maria Ivanovna can save either 900 rubles by using the "third item fr... | 6265 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
Task 11. (16 points)
The Dorokhov family plans to purchase a vacation package to Crimea. The family plans to travel with the mother, father, and their eldest daughter Polina, who is 5 years old. They carefully studied all the offers and chose the "Bristol" hotel. The head of the family approached two travel agencies, ... | # Solution:
Cost of the tour with the company "Globus"
$(3 * 25400) *(1-0.02)=74676$ rubles.
Cost of the tour with the company "Around the World"
$(11400+2 * 23500) * 1.01=58984$ rubles.
Answer: 58984 | 58984 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 12. (16 points)
The Vasilievs' family budget consists of the following income items:
- parents' salary after income tax deduction - 71000 rubles;
- income from renting out property - 11000 rubles;
- daughter's scholarship - 2600 rubles
The average monthly expenses of the family include:
- utility payments - 84... | # Solution:
family income
$71000+11000+2600=84600$ rubles
average monthly expenses
$8400+18000+3200+2200+18000=49800$ rubles
expenses for forming a financial safety cushion
$(84600-49800) * 0.1=3480$ rubles
the amount the Petrovs can save monthly for the upcoming vacation
$84600-49800-3480=31320$ rubles
## Ans... | 31320 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 13. (8 points)
Natalia Petrovna has returned from her vacation, which she spent traveling through countries in North America. She has a certain amount of money left in foreign currency.
Natalia Petrovna familiarized herself with the exchange rates at the nearest banks: "Rebirth" and "Garnet." She decided to take... | # Solution:
1) cost of currency at Bank "Vozrozhdenie":
$$
120 * 74.9 + 80 * 59.3 + 10 * 3.7 = 13769 \text{ RUB}
$$
2) cost of currency at Bank "Garant":
$$
120 * 74.5 + 80 * 60.1 + 10 * 3.6 = 13784 \text{ RUB}
$$
Answer: 13784 | 13784 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
Task 14. (8 points)
To attend the section, Mikhail needs to purchase a tennis racket and a set of tennis balls. Official store websites have product catalogs. Mikhail studied the offers and compiled a list of stores where the items of interest are available:
| Item | Store | |
| :--- | :---: | :---: |
| | Higher Le... | # Solution:
1) cost of purchase in the store "Higher League":
$$
\text { 5600+254=5854 rub. }
$$
1) cost of purchase in the store "Sport-guru": $(2700+200)^{*} 0.95+400=6005$ rub.
## Answer: 5854 | 5854 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 15. (8 points)
The fast-food network "Wings and Legs" offers summer jobs to schoolchildren. The salary is 25000 rubles per month. Those who work well receive a monthly bonus of 5000 rubles.
How much will a schoolchild who works well at "Wings and Legs" earn per month (receive after tax) after the income tax is d... | Solution:
The total earnings will be 25000 rubles + 5000 rubles $=30000$ rubles
Income tax $13 \%-3900$ rubles
The net payment will be 30000 rubles - 3900 rubles $=26100$ rubles
## Correct answer: 26100
## 2nd Option | 26100 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 11. (16 points)
One way to save on utility bills is to use the night tariff (from 23:00 to 07:00). To apply this tariff, a multi-tariff meter needs to be installed.
The Romanov family is considering purchasing a multi-tariff meter to reduce their utility bills. The cost of the meter is 3500 rubles. The installat... | # Solution:
2) use of a multi-tariff meter:
$$
3500+1100+(230 * 3.4+(300-230) * 5.2) * 12 * 3=45856 \text { rub. }
$$
3) use of a typical meter
$$
300 * 4.6 * 12 * 3=49680 \text { rub. } \quad \text {. } \quad \text {. }
$$
the savings will be
$49680-45856=3824$ rub.
## Answer: 3824 | 3824 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 12. (16 points)
The budget of the Petrovs consists of the following income items:
- parents' salary after income tax deduction - 56000 rubles;
- grandmother's pension - 14300 rubles;
- son's scholarship - 2500 rubles
Average monthly expenses of the family include:
- utility payments - 9800 rubles;
- food - 210... | # Solution:
family income
$56000+14300+2500=72800$ rubles.
average monthly expenses
$9800+21000+3200+5200+15000=54200$ rubles.
expenses for forming a financial safety cushion
$(72800-54200) * 0.1=1860$ rubles.
the amount the Petrovs can save monthly for the upcoming vacation
$72800-54200-1860=16740$ rubles.
An... | 16740 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 13. (8 points)
Maxim Viktorovich returned from a trip to Asian countries. He has a certain amount of money in foreign currency left.
Maxim Viktorovich familiarized himself with the exchange rates at the nearest banks: "Voskhod" and "Elfa". He decided to take advantage of the most favorable offer. What amount wil... | # Solution:
1) cost of currency at "Voskhod" bank:
$110 * 11.7 + 80 * 72.1 + 50 * 9.7 = 7540$ rubles.
2) cost of currency at "Alpha" bank: $110 * 11.6 + 80 * 71.9 + 50 * 10.1 = 7533$ rubles.
Answer: 7540. | 7540 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 14. (8 points)
Elena decided to get a pet - a budgerigar. She faced the question of where to buy a cage and a bath more cost-effectively.
On the official websites of the stores, product catalogs are posted. Elena studied the offers and compiled a list of stores where the items she is interested in are available:... | # Solution:
2) cost of purchase in the "Zoimir" store: $4500+510=5010$ rubles
3) cost of purchase in the "Zooidea" store: $(3700+680) * 0.95+400=4561$ rubles
## Answer: 4561 | 4561 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
Task 15. (8 points)
Announcement: "Have free time and want to earn money? Write now and earn up to 2500 rubles a day working as a courier with the service 'Food.There-Here!'. Delivery of food from stores, cafes, and restaurants.
How much will a school student working as a courier with the service 'Food.There-Here!' e... | # Solution:
The total earnings will be (1250 rubles * 4 days) * 4 weeks = 20000 rubles
Income tax 13% - 2600 rubles
The amount of earnings (net pay) will be 20000 rubles - 2600 rubles = 17400 rubles
Correct answer: 17400 | 17400 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. There are several technologies for paying with bank cards: chip, magnetic stripe, paypass, cvc. Arrange the actions performed with a bank card in the order corresponding to the payment technologies.
1 - tap
2 - pay online
3 - swipe
4 - insert into terminal | Answer in the form of an integer, for example 1234.
Answer: 4312 | 4312 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
# Task 1. (4 points)
The price of a new 3D printer is 625000 rubles. Under normal operating conditions, its resale value decreases by $20 \%$ in the first year, and then by $8 \%$ each subsequent year. After how many years will the resale value of the printer be less than 400000 rubles? | Solution:
Let's calculate the cost of the printer year by year:
1 year $=625000 * 0.8=500000$ rubles
2 year $=500000 * 0.92=460000$ rubles (1 point)
3 year $=460000 * 0.92=423200$ rubles (1 point)
4 year $=423200 * 0.92=397694$ rubles. (1 point)
Answer: in 4 years. ( $\mathbf{1}$ point) | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 4. (8 points)
Kolya's parents give him pocket money once a month, calculating it as follows: 100 rubles for each A in math, 50 rubles for a B, 50 rubles are deducted for a C, and 200 rubles are deducted for a D. If the amount turns out to be negative, Kolya simply gets nothing. The math teacher gives a grade... | # Solution:
If Kolya received a final grade of 2 for the quarter, then for each 5 he received more than 5 2s, for each 4 - more than 3 2s, and for each 3 - more than 1 2. This means that the number of 2s was greater than the total number of all other grades combined, so Kolya could receive money in at most one of the ... | 250 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 6. (10 points)
Vasily is planning to graduate from college in a year. Only 270 out of 300 third-year students successfully pass their exams and complete their bachelor's degree. If Vasily ends up among the 30 expelled students, he will have to work with a monthly salary of 25,000 rubles. It is also known tha... | # Solution:
In 4 years after graduating from school, Fedor will earn $25000 + 3000 * 4 = 37000$ rubles (2 points).
The expected salary of Vasily is the expected value of the salary Vasily can earn under all possible scenarios (2 points). It will be 270/300 * $(1 / 5 * 60000 + 1 / 10 * 80$ $000 + 1 / 20 * 25000 + (1 -... | 45025 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 1. (4 points)
In the run-up to the New Year, a fair is being held at the school where students exchange festive toys. As a result, the following exchange norms have been established:
1 Christmas tree ornament can be exchanged for 2 crackers, 5 sparklers can be exchanged for 2 garlands, and 4 Christmas tree ... | # Solution:
a) 10 sparklers $=4$ garlands = 16 ornaments = $\mathbf{32}$ crackers. (1 point)
b) Convert everything to crackers. In the first case, we have $\mathbf{11}$ crackers. In the second case, 2 sparklers $=4 / 5$ garlands $=16 / 5$ ornaments $=32 / 5$ crackers $=\mathbf{6.4}$ crackers. Answer: 5 ornaments and ... | 32 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
# Task 3. (8 points)
In the Sidorov family, there are 3 people: the father works as a programmer with an hourly rate of 1500 rubles. The mother works as a hairdresser at home and charges 1200 rubles per haircut, which takes her 1.5 hours. The son tutors in mathematics and earns 450 rubles per academic hour (45 minutes... | # Solution
In this problem, there are 2 possible interpretations, both of which were counted as correct.
In one case, it is assumed that 8 hours are spent on work on average over the month, in the other that no more than 8 hours are spent on work each day.
## First Case:
1) Determine the hourly wage for each family... | 19600 | Other | math-word-problem | Yes | Yes | olympiads | false |
# Problem 4. (10 points)
On December 31 at 16:35, Misha realized he had no New Year's gifts for his entire family. He wants to give different gifts to his mother, father, brother, and sister. Each of the gifts is available in 4 stores: Romashka, Odynachik, Nezabudka, and Lysichka, which close at 20:00. The journey fro... | # Solution:
Notice that in each of the stores, there is a "unique" gift with the lowest price.
If Misha managed to visit all 4 stores, he would spend the minimum amount of $980+750+900+800=3430$ rubles. However, visiting any three stores would take Misha at least $30 * 3+25+30+35=180$ minutes. Considering the additio... | 3435 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive than a, which means the highest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend $4800 + 1500 - 4800 * 0.2 = 5340$ rubles. This is the most profitable way to make t... | # Solution:
The average value of the last purchases is $(785+2033+88+3742+1058) / 5 = 1541.2$ rubles. Therefore, an acceptable purchase would be no more than $1541.2 * 3 = 4623.6$ rubles. With this amount, one can buy $4623.6 / 55 \approx 84$ chocolates.
## Maximum 20 points
20 points - fully detailed solution and c... | 84 | Other | math-word-problem | Yes | Yes | olympiads | false |
1. What was NOT used as money?
1) gold
2) stones
3) horses
4) dried fish
5) mollusk scales
6) all of the above were used | Answer: 6. All of the above were used as money. | 6 | Logic and Puzzles | MCQ | Yes | Yes | olympiads | false |
2. A stationery store is running a promotion: there is a sticker on each notebook, and for every 5 stickers, a customer can get another notebook (also with a sticker). Fifth-grader Katya thinks she needs to buy as many notebooks as possible before the new semester. Each notebook costs 4 rubles, and Katya has 150 rubles... | Answer: 46.
1) Katya buys 37 notebooks for 148 rubles.
2) For 35 stickers, Katya receives 7 more notebooks, after which she has notebooks and 9 stickers.
3) For 5 stickers, Katya receives a notebook, after which she has 45 notebooks and 5 stickers.
4) For 5 stickers, Katya receives the last 46th notebook. | 46 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. How much did the US dollar exchange rate change over the 2014 year (from January 1, 2014 to December 31, 2014)? Give the answer in rubles, rounding to the nearest whole number (the answer is a whole number). | Answer: 24. On January 1, 2014, the dollar was worth 32.6587, and on December 31, it was 56.2584.
$56.2584-32.6587=23.5997$. Since rounding was required, the answer is 24.
Note: This problem could have been solved using the internet. For example, the website https://news.yandex.ru/quotes/region/1.html | 24 | Other | math-word-problem | Yes | Yes | olympiads | false |
5. Vanya decided to give Masha a bouquet of an odd number of flowers for her birthday, consisting of yellow and red tulips, so that the number of flowers of one color differs from the number of flowers of the other by exactly one. Yellow tulips cost 50 rubles each, and red ones cost 31 rubles. What is the largest numbe... | Answer: 15.
A bouquet with one more red tulip than yellow ones is cheaper than a bouquet with the same total number of flowers but one more yellow tulip. Therefore, Vanya should buy a bouquet with one more red tulip. The remaining flowers can be paired into red and yellow tulips, with each pair costing 81 rubles. Let'... | 15 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. What is a sign of a financial pyramid?
1) an offer of income significantly above average
2) incomplete information about the company
3) aggressive advertising
4) all of the above | Answer: 4. All of the above are signs of a financial pyramid. | 4 | Other | MCQ | Yes | Yes | olympiads | false |
Problem 12. (6 points)
Victor received a large sum of money as a birthday gift in the amount of 45 thousand rubles. The young man decided to save this part of his savings in dollars on a currency deposit. The term of the deposit agreement was 2 years, with an interest rate of 4.7% per annum, compounded quarterly. On t... | Answer: 873 USD.
## Comment:
45000 RUB / 56.60 RUB $\times(1+4.7\% / 4 \text { quarters })^{2 \text { years } \times 4 \text { quarters }}=873$ USD. | 873 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Task 17-19. (2 points per Task)
Alena opened a multi-currency deposit at "Severny" Bank for 3 years. The deposit involves the deposit of funds in three currencies: euros, dollars, and rubles. At the beginning of the deposit agreement, Alena's account contained 3000 euros, 4000 dollars, and 240000 rubles. The interes... | Answer 17: $3280;
Answer 18: 4040 euros,
Answer 19: 301492 rubles.
## Comment:
1 year
Euros: 3000 euros $\times(1+2.1 \%)=3063$ euros.
Dollars: 4000 dollars $\times(1+2.1 \%)=4084$ dollars.
Rubles: 240000 rubles $\times(1+7.9 \%)=258960$ rubles.
2 year
Euros: (3063 euros - 1000 euros $) \times(1+2.1 \%)=2106$ ... | 3280 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 6. (4 points)
Ivan bought a used car from 2010 for 90,000 rubles with an engine power of 150 hp and registered it on January 29. On August 21 of the same year, the citizen sold his car and a month later bought a horse and a cart for 7,500 rubles. The transport tax rate is set at 20 rubles per 1 hp. What amount... | Solution: transport tax $=150 \times 20 \times 8 / 12=2000$ rubles. A horse and a cart are not subject to transport tax. | 2000 | Other | math-word-problem | Yes | Yes | olympiads | false |
Problem 7. (6 points)
Sergei, being a student, worked part-time in a student cafe after classes for a year. Sergei's salary was 9000 rubles per month. In the same year, Sergei paid for his medical treatment at a healthcare facility in the amount of 100000 rubles and purchased medications on a doctor's prescription for... | Solution: the amount of the social tax deduction for medical treatment will be: $100000+20000=$ 120000 rubles. The possible tax amount eligible for refund under this deduction will be $120000 \times 13\% = 15600$ rubles. However, in the past year, Sergey paid income tax (NDFL) in the amount of $13\% \times (9000 \times... | 14040 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 8. (2 points)
After graduating from a technical university, Oleg started his own business producing water heaters. This year, Oleg plans to sell 5000 units of water heaters. Variable costs for production and sales of one water heater amount to 800 rubles, and total fixed costs are 1000 thousand rubles. Oleg wa... | Solution: the price of one kettle $=((1000000+0.8 \times 5000)+1500$ 000) / $(1000000$ + $0.8 \times 5000)=1300$ rub. | 1300 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 10. (4 points)
To buy new headphones costing 275 rubles, Katya decided to save money on sports activities. Until now, she has been buying a single-visit ticket to the swimming pool, including a visit to the sauna for 250 rubles, to warm up. However, summer has arrived, and the need to visit the sauna has disap... | Solution: one visit to the sauna costs 25 rubles, the price of one visit to the swimming pool is 225 rubles. Katya needs to visit the swimming pool 11 times without going to the sauna in order to save up for buying headphones. | 11 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 18. (4 points)
By producing and selling 4000 items at a price of 6250 rubles each, a budding businessman earned 2 million rubles in profit. Variable costs for one item amounted to 3750 rubles. By what percentage should the businessman reduce the production volume to make his revenue equal to the cost? (Provide... | Solution: fixed costs $=-2$ million, gap $+4000 \times 6250-3750 \times 4000$ million $=8$ million.
$6.25 \times Q=3750 Q+8 \text{ million}$.
$Q=3200$ units.
Can be taken out of production $=4000-3200=800$ units, that is, $20\%$. | 20 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive than a, which means the highest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend $4800 + 1500 - 4800 * 0.2 = 5340$ rubles. This is the most profitable way to make t... | # Solution:
The average value of the last purchases is $(785+2033+88+3742+1058) / 5 = 1541.2$ rubles. Therefore, an allowable purchase is no more than $1541.2 * 3 = 4623.6$ rubles. With this amount, you can buy $4623.6 / 55 \approx 84$ chocolates.
## Maximum 20 points
20 points - fully detailed solution and correct ... | 84 | Other | math-word-problem | Yes | Yes | olympiads | false |
7. Leshа has 10 million rubles. Into what minimum number of banks should he deposit them to receive the full amount through ACB insurance payouts in case the banks cease operations? | Answer: 8. The maximum insurance payout is 1,400,000, which means no more than this amount should be deposited in each bank. | 8 | Other | math-word-problem | Yes | Yes | olympiads | false |
2. How much did the US dollar exchange rate change over the 2014 year (from January 1, 2014 to December 31, 2014)? Give your answer in rubles, rounding to the nearest whole number. | Answer: 24. On January 1, 2014, the dollar was worth 32.6587, and on December 31, it was 56.2584.
$56.2584-32.6587=23.5997 \approx 24$.
Note: This problem could have been solved using the internet. For example, the website https://news.yandex.ru/quotes/region/23.html | 24 | Other | math-word-problem | Yes | Yes | olympiads | false |
7. Alexey plans to buy one of two car brands: "A" for 900 thousand rubles or "B" for 600 thousand rubles. On average, Alexey drives 15 thousand km per year. The cost of gasoline is 40 rubles per liter. The cars consume the same type of gasoline. The car is planned to be used for 5 years, after which Alexey will be able... | Answer: 160000.
Use of car brand "A":
$900000+(15000 / 100) * 9 * 5 * 40+35000 * 5+25000 * 5-500000=970000$
Use of car brand "B":
$600000+(15000 / 100) * 10 * 5 * 40+32000 * 5+20000 * 5-350000=810000$
Difference: $970000-810000=160000$ | 160000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
8. A family of 4, consisting of a mom, a dad, and two children, has arrived in city $\mathrm{N}$ for 5 days. They plan to make 10 trips on the subway each day. What is the minimum amount they will have to spend on tickets, given the following tariffs in city $\mathrm{N}$?
| Adult ticket for one trip | 40 rubles |
| :-... | Answer: 5200. The family will spend this amount if the parents buy a three-day pass for themselves, and for the remaining two days, they will buy a one-day pass. For this, they will spend ($900 + 350 * 2$) * $2 = 3200$ rubles.
For the children, it is most cost-effective to buy single-trip tickets for all 5 days, spend... | 5200 | Other | math-word-problem | Yes | Yes | olympiads | false |
Problem 9. (12 points)
Andrey lives near the market, and during the summer holidays, he often helped one of the traders lay out fruits on the counter early in the morning. For this, the trader provided Andrey with a $10 \%$ discount on his favorite apples. But autumn came, and the price of apples increased by $10 \%$.... | # Answer: 99.
## Comment
Solution: the new price of apples at the market is 55 rubles per kg, with a discount of $10 \%$ applied to this price. Thus, the price for 1 kg for Andrei will be 49.5 rubles, and for 2 kg Andrei will pay 99 rubles monthly. | 99 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 10. (12 points)
The Ivanov family owns an apartment with necessary property worth 3 million rubles, a car that is currently valued at 900 thousand rubles on the market, and savings, part of which, amounting to 300 thousand rubles, is placed in a bank deposit, part is invested in securities worth 200 thousand rubl... | # Answer: 2300000
## Comment
Solution: equity (net worth) = value of assets - value of liabilities. Value of assets $=3000000+900000+300000+200000+100000=$ 4500000 rubles. Value of liabilities $=1500000+500000+200000=2200000$ rubles. Net worth $=4500000-2200000=2300000$ rubles
## MOSCOW FINANCIAL LITERACY OLYMPIAD ... | 2300000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 4. (8 points)
Konstantin's mother opened a foreign currency deposit in the "Western" Bank for an amount of 10 thousand dollars for a term of 1 year. Literally 4 months later, the Bank of Russia revoked the license of the "Western" Bank. The exchange rate on the date of the license revocation was 58 rubles 15 k... | # Answer: b.
## Comment
$10000 \times 58.15$ RUB $=581500$ RUB. | 581500 | Other | MCQ | Yes | Yes | olympiads | false |
Problem 9. (12 points)
Ivan, a full-time student, started working at his long-desired job on March 1 of the current year. In the first two months, during the probationary period, Ivan's salary was 20000 rubles. After the probationary period, the salary increased to 25000 rubles per month. In December, for exceeding th... | Answer: 32500.
## Comment
Solution: Personal Income Tax from salary $=(20000 \times 2+25000 \times 8+10000) \times 13\% = 32500$ rubles. The scholarship is not subject to Personal Income Tax. | 32500 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 4. (8 points)
Lena receives 50,000 rubles per month and spends 45,000 rubles per month. She gets her salary on the 6th of each month. Lena has: a deposit in the bank $\mathrm{X}$ with a $1\%$ monthly interest rate with monthly capitalization and the possibility of topping up, but no withdrawals allowed, with... | Solution:
1) Of all the bank's products, the deposit brings Lena the highest income. She can use $50-45=5$ thousand rubles for savings, depositing them immediately after receiving her salary. Over a year, this will bring Lena $5000 * \frac{1.01 * (1.01^{12}-1)}{1.01-1} - 5000 * 12 \approx 4000$.
2) The remaining 45 th... | 4860 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Problem 6. (8 points)
Vasily is planning to graduate from the institute in a year. Only 270 out of 300 third-year students successfully pass their exams and complete their bachelor's degree. If Vasily ends up among the 30 expelled students, he will have to work with a monthly salary of 25,000 rubles. It is also know... | Solution:
Four years after graduating from school, Fedor will earn $25000 + 3000 * 4 = 37000$ rubles (2 points)
The expected salary of Vasily is the expected value of the salary Vasily can earn under all possible scenarios (2 points). It will be 270/300*(1/5*60 000 + 1/10*80 000 + 1/20*25 000 + (1 - 1/5 - 1/10 - 1/20... | 45025 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. How much did the euro exchange rate change over the 2012 year (from January 1, 2012 to December 31, 2012)? Provide the answer in rubles, rounded to the nearest whole number. | Answer: 1 or -1. On January 1, 2012, the euro was worth 41.6714, and on December 31, it was 40.2286. $40.2286-41.6714=-1.4428 \approx-1$.
Note: This problem could have been solved using the internet. For example, the website https://news.yandex.ru/quotes/region/23.html | -1 | Other | math-word-problem | Yes | Yes | olympiads | false |
4. The Petrovs family has decided to renovate their apartment. They can hire a company for a "turnkey renovation" for 50,000 or buy materials for 20,000 and do the renovation themselves, but for that, they will have to take unpaid leave. The husband earns 2000 per day, and the wife earns 1500. How many working days can... | Answer: 8. The combined daily salary of the husband and wife is $2000+1500=3500$ rubles. The difference between the cost of a turnkey repair and buying materials is $50000-20000=30000$.
$30000: 3500 \approx 8.57$, so the family can spend no more than 8 days on the repair. | 8 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
5. In the store "Third is Not Excessive," there is a promotion: if a customer presents three items at the cash register, the cheapest of them is free. Ivan wants to buy 11 items costing $100, 200, 300, \ldots, 1100$ rubles. For what minimum amount of money can he buy these items? | Answer: 4800. It is clear that items should be listed in descending order of price, then the cost of the purchase will be $1100+1000+800+700+500+400+200+100=4800$ rubles. | 4800 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
6. A supermarket discount card that gives a $3 \%$ discount costs 100 rubles. Masha bought 4 cakes for 500 rubles each and fruits for 1600 rubles for her birthday. The seller at the cash register offered her to buy the discount card before the purchase. Should Masha agree?
1) no, they offer these cards to everyone
2) y... | Answer: 2. The cost of Masha's purchase is $4 * 500 + 1600 = 3600$. If Masha buys a discount card, she will spend $100 + 3600 * 0.97 = 3592$. | 3592 | Algebra | MCQ | Yes | Yes | olympiads | false |
7. Vanya decided to give Masha a bouquet of an odd number of flowers for her birthday, consisting of yellow and red tulips, so that the number of flowers of one color differs from the number of flowers of the other by exactly one. Yellow tulips cost 50 rubles each, and red ones cost 31 rubles. What is the largest numbe... | Answer: 15.
A bouquet with one more red tulip than yellow ones is cheaper than a bouquet with the same total number of flowers but one more yellow tulip. Therefore, Vanya should buy a bouquet with one more red tulip. The remaining flowers can be paired into red and yellow tulips, with each pair costing 81 rubles. Let'... | 15 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 4. (8 points)
Irina Mikhailovna opened a foreign currency deposit in the "Western" Bank for an amount of $23,904 for a term of 1 year. The interest rate on the deposit was 5% per annum. Exactly 3 months later, the Bank of Russia revoked the license of the "Western" Bank. The official exchange rate on the date ... | Answer: $b$.
## Comment:
23904 USD $\times 58.15$ RUB $\times(1+5\% / 4)=1407393$ RUB. Since the Deposit Insurance Agency compensates deposits up to 1400000 RUB, this amount will be paid to Irina Mikhailovna. | 1400000 | Algebra | MCQ | Yes | Yes | olympiads | false |
Problem 13. (8 points)
Dar'ya received a New Year's bonus of 60 thousand rubles, which she decided to save for a summer vacation. To prevent the money from losing value, the girl chose between two options for saving the money - to deposit the money at an interest rate of $8.6 \%$ annually for 6 months or to buy dollar... | Answer: the loss incurred from the second option for placing funds is (rounded to the nearest whole number) $\underline{3300 \text{ RUB}}$ | 3300 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 6. (8 points)
Anna Ivanovna bought a car from her neighbor last November for 300,000 rubles with an engine power of 250 hp, and in May she purchased a used rowing catamaran for 6 rubles. The transport tax rate is set at 75 rubles per 1 hp. How much transport tax should Anna Ivanovna pay? (Provide the answer as... | Solution: transport tax $=250 \times 75 \times 2 / 12=3125$ rubles. A rowing catamaran is not a taxable object. | 3125 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 7. (8 points)
The earned salary of the citizen was 23,000 rubles per month from January to June inclusive, and 25,000 rubles from July to December. In August, the citizen, participating in a poetry competition, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs ... | Solution: Personal Income Tax from salary $=(23000 \times 6+25000 \times 6) \times 13\%=37440$ rubles.
Personal Income Tax from winnings $=(10000-4000) \times 35\%=2100$ rubles.
Total Personal Income Tax = 37440 rubles +2100 rubles$=39540$ rubles. | 39540 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
# Task 14. (1 point)
Calculate the amount of personal income tax (NDFL) paid by Sergey for the past year, if he is a Russian resident and during this period had a stable income of 30000 rub./month and a one-time vacation bonus of 20000 rub. In the past year, Sergey sold a car he inherited two years ago for 250000 rub.... | Answer: 10400.
## Comment:
Solution: tax base $=30000 \times 12+20000+250000=630000$ rubles. The amount of the tax deduction $=250000+300000=550000$ rubles. The amount of personal income tax $=13 \% \times(630000-$ $550000)=10400$ rubles. | 10400 | Other | math-word-problem | Yes | Yes | olympiads | false |
Problem 20. (6 points)
Ivan Sergeyevich decided to raise quails. In a year, he sold 100 kg of poultry meat at a price of 500 rubles per kg, and also 20000 eggs at a price of 50 rubles per dozen. The expenses for the year amounted to 100000 rubles. What profit did Ivan Sergeyevich receive for this year? (Provide the an... | Answer: 50000.
Comment:
Solution: revenue $=100 \times 500 + 50 \times 20000 / 10 = 150000$ rubles. Profit $=$ revenue costs $=150000-100000=50000$ rubles. | 50000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 21. (8 points)
Dmitry's parents decided to buy him a laptop for his birthday. They calculated that they could save the required amount in two ways. In the first case, they need to save one-tenth of their salary for six months. In the second case, they need to save half of their salary for one month, and then d... | Answer: 25000.
## Comment:
Solution:
Mom's salary is $x$, then dad's salary is $1.3x$. We set up the equation:
$(x + 1.3x) / 10 \times 6 = (x + 1.3x) / 2 \times (1 + 0.03 \times 10) - 2875$
$1.38x = 1.495x - 2875$
$x = 25000$ | 25000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. When insuring property, the insurance amount cannot exceed its actual value (insurance value) at the time of concluding the insurance contract.
Insurance tariff - the rate of the insurance premium or the insurance premium (insurance premium) expressed in rubles, payable per unit of the insurance amount, which is us... | # Solution:
In accordance with the instruction, the base rate is $0.2\%$ of the insurance amount, apply a reducing factor for the absence of a change in ownership over the past 3 years $(0.8)$ and an increasing factor for the absence of certificates from the PND and ND $(1.3)$.
In total: $0.2 * 0.8 * 1.3=0.208\%$
Th... | 31200 | Other | math-word-problem | Yes | Yes | olympiads | false |
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 m... | # 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... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. In July, Volodya and Dima decided to start their own business producing non-carbonated bottled mineral water called "Dream," investing 1,500,000 rubles, and used these funds to purchase equipment for 500,000 rubles. The technical passport for this equipment indicates that the maximum production capacity is 100,000 b... | # Answer:
a) The norm for 1 bottle of water = initial cost / maximum quantity: $500000 / 100000 = 5$ rubles;
- depreciation in July $5 \times 200 = 1000$ rubles;
- depreciation in March $15000 \times 5 = 75000$ rubles;
- depreciation in September $12300 \times 5 = 61500$ rubles, Total depreciation 137500 (6 points).
... | 372500 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Newlyweds Alexander and Natalia successfully got jobs at an advertising company in April. With their earnings, they want to buy new phones next month: phone "A" for Alexander, which costs 57,000 rubles, and phone "B" for Natalia, costing 37,000 rubles. Will they be able to do this, given the following data?
- Alexa... | Answer:
a) Total expenses: $17000+15000+12000+20000+30000+30000=$ $=124000$ (4 points);
b) Net income: $(125000+61000) \times 13 \% = 24180.186000 - 24180=$ $=161820$ (6 points);
c) Remaining funds: $161820-124000=37820$. The phone can only be bought for Natalia, and to buy a phone for Alexander, it is necessary to ... | 37820 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Let's say you have two bank cards for making purchases: a debit card and a credit card. Today, you decided to buy airline tickets for 20,000 rubles. If you pay for the purchase with a credit card (the credit limit allows it), you will have to return the money to the bank in $\mathrm{N}$ days to avoid going beyond th... | # Solution:
When paying by credit card, the amount of 20,000 rubles will be on your debit card for $\mathrm{N}$ days, which will earn you $\frac{6 \mathrm{~N}}{100 \cdot 12 \cdot 30} \cdot 20000$ rubles in interest on the remaining funds.
You will also receive $20000 \times 0.005 = 100$ rubles in cashback.
When payi... | 31 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Two friends, Arthur and Timur, with the support of their parents, decided to open several massage salons in Moscow. For this purpose, a business plan was drawn up, the economic indicators of which are presented below.
- Form of ownership - LLC
- Number of employees - no more than 50 people
- Planned revenues for th... | # Solution:
Reference information: Chapter 26.2, Part 2 of the Tax Code of the Russian Federation.
- Criteria applicable under the simplified tax system (STS) - Article 346.13;
- Taxable object - Article 346.14;
- Determination of income - Article 346.15;
- Determination of expenses - Article 346.16;
- Recognition of... | 222000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Let's say you have two bank cards for making purchases: a debit card and a credit card. Today, at the beginning of the month, you decided to buy airline tickets for 12,000 rubles. If you pay for the purchase with a credit card (the credit limit allows it), you will have to return the money to the bank in $\mathrm{N}... | # Solution:
When paying by credit card, the amount of 12,000 rubles will be on your debit card for $\mathrm{N}$ days, which will earn you $\frac{6 \mathrm{~N}}{100 \cdot 12 \cdot 30} \cdot 12000$ rubles in interest on the remaining funds. Additionally, you will receive $12000 \times 0.01 = 120$ rubles in cashback.
Wh... | 59 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Twin brothers, Anton Sergeyevich, a civil aviation pilot by profession, and Mikhail Sergeyevich, a neurologist, born on 05.06.1977, decided to go on vacation together and purchase a life and health insurance policy for 2,000,000 rubles. Anton Sergeyevich and Mikhail Sergeyevich had the same height - 187 cm and weigh... | # Solution:
Let's calculate the age. Both men are 40 years old. The base rate is $0.32\%$. By occupation class: Mikhail (doctor) $0.32\% \times 1.02 = 0.3264\%$.
Anton (pilot) $0.32\% \times 1.5 = 0.48\%$.
Calculate the BMI $= 98 / 1.87^2 = 98 / 3.4969 = $ (approximately) 28.025.
Find the increasing coefficient fro... | 3072 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Anna and Ekaterina have opened a cosmetic salon in New Moscow. The enterprise applies the general taxation system. Ekaterina attended a seminar on taxation and learned about the Simplified System of Taxation (USNO). To avoid changing the document flow and control over financial and economic operations, the friends d... | # Solution:
Reference information: Chapter 26.2, Chapter 25, Part 2 of the Tax Code of the Russian Federation.
- Criteria applicable under the simplified tax system (STS) - Article 346.13;
- Taxable object - Article 346.14;
- Determination of income - Article 346.15;
- Determination of expenses - Article 346.16;
- Re... | 172800 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 7. (8 points)
The earned salary of the citizen was 23,000 rubles per month from January to June inclusive, and 25,000 rubles from July to December. In August, the citizen, participating in a poetry contest, won a prize and was awarded an e-book worth 10,000 rubles. What amount of personal income tax needs to b... | Solution: Personal Income Tax from salary $=(23000 \times 6+25000 \times 6) \times 13\%=37440$ rubles.
Personal Income Tax from winnings $=(10000-4000) \times 35\%=2100$ rubles.
Total Personal Income Tax = 37440 rubles +2100 rubles$=39540$ rubles. | 39540 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Task 4.
Every day after lunch, 7 half-eaten pieces of white bread are left on the desks of the second-grade students. If these pieces are put together, they make up half a loaf of bread. How many loaves of bread will the second-grade students save in 20 days if they do not leave these pieces? How much money will the s... | Solution: 1. 0.5 (1/2) * 20 = 10 (loaves); 10 * 35 = 350 rubles; 2. 0.5 (1/2) * 60 = 30 (loaves); 30 * 35 = 1050 rubles.
Themes for extracurricular activities: "Young Economist," "Bread is the Head of Everything," "Saving and Frugality in Our School Canteen," "Journey to the School of the Frugal," "Frugality - the Mai... | 350 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
3. Once, a god sent a little cheese to two ravens. The first raven received 100 g, from which part was taken by a fox. The piece of the second raven turned out to be twice as large as that of the first, but she managed to eat only half as much as the first raven. The portion of cheese that the fox got from the second r... | Solution. Let the first crow eat $x$ grams of cheese. Then the fox received $100-x$ grams of cheese from the first crow. The second crow ate $\frac{x}{2}$ grams of cheese. From the second crow, the fox received $200-\frac{x}{2}$ grams of cheese. This was three times more, so: $200-\frac{x}{2}=3(100-x)$. Solution: $x=40... | 240 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
5. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same... | Solution. Since $A$ with $C$ and $B$ with $D$ meet every 7 minutes, their approach speeds are equal: $V_{A}+V_{C}=V_{B}+V_{D}$. In other words, the speeds of separation of $A, B$ and $C, D$ are equal: $V_{A}-V_{B}=V_{D}-V_{C}$. Therefore, since $A$ and $B$ meet for the first time at the 53rd minute, $C$ and $D$ will al... | 53 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. How many solutions in natural numbers does the equation $(a+1)(b+1)(c+1)=2 a b c$ have? | Solution. Rewrite the equation as $(1+1 / a)(1+1 / b)(1+1 / c)=2$. Due to symmetry, it is sufficient to find all solutions with $a \leqslant b \leqslant c$. Then $(1+1 / a)^{3} \geqslant 2$, which means $a \leqslant(\sqrt[3]{2}-1)^{-1}<4$ and $a \in\{1,2,3\}$. In the case $a=1$, the inequality $2(1+1 / b)^{2} \geqslant... | 27 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
8. The park is a $10 \times 10$ grid of cells. A lamp can be placed in any cell (but no more than one lamp per cell).
a) The park is called illuminated if, no matter which cell a visitor is in, there is a $3 \times 3$ square of 9 cells that contains both the visitor and at least one lamp. What is the minimum number of... | Solution. a) 4. Divide the park into 4 quarters (squares $5 \times 5$), then there must be at least one lamp in each quarter (to illuminate, for example, the corner cells). By placing one lamp in the center of each quarter, we get an example.
b) 10.
Estimate. In each corner square $3 \times 3$ there must be at least ... | 4 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Marina needs to buy a notebook, a pen, a ruler, and a pencil to participate in the Olympiad. If she buys a notebook, a pencil, and a ruler, she will spend 47 tugriks. If she buys a notebook, a ruler, and a pen, she will spend 58 tugriks. If she buys only a pen and a pencil, she will spend 15 tugriks. How much money ... | Solution. If Marina buys all three sets from the condition at once, she will spend $47+58+$ $15=120$ tugriks, and she will buy each item twice, so the full set of school supplies costs $120 / 2=60$ tugriks.
Criteria. Only the answer without explanation - 1 point. If in the solution they try to determine the cost of th... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. There is a rectangular sheet, white on one side and gray on the other. It was folded as shown in the picture. The perimeter of the first rectangle is 20 more than the perimeter of the second rectangle. And the perimeter of the second rectangle is 16 more than the perimeter of the third rectangle. Find the perimeter ... | Solution. From the figure, it can be seen that when folding, the perimeter of the rectangle decreases by twice the short side, so the short side of rectangle-1 is $20 / 2=10$, the short side of rectangle-2 is $16 / 2=8$. Therefore, the long side of rectangle-1 is 18, and the long side of the original sheet is 28. Thus,... | 92 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
5. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). The result was the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible ... | Solution. The number must be divisible by 25, so “$\lambda$A” equals 25, 50, or 75 (00 cannot be, as the letters are different). If “LA” equals 50, then for the other letters (“G”, “V”, “T”, “E”, “M”) there are $A_{8}^{5}$ options; otherwise, for the other letters there are $7 \cdot A_{7}^{4}$ options. In total, $8!/ 6... | 18480 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
8. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same... | Solution. Since $A$ with $C$ and $B$ with $D$ meet every 7 minutes, their approach speeds are equal: $V_{A}+V_{C}=V_{B}+V_{D}$. In other words, the speeds of separation of $A, B$ and $C, D$ are equal: $V_{A}-V_{B}=V_{D}-V_{C}$. Therefore, since $A$ and $B$ meet for the first time at the 53rd minute, $C$ and $D$ will al... | 53 | Other | math-word-problem | Yes | Yes | olympiads | false |
5. A few years ago, in the computer game "Minecraft," there were 11 different pictures (see the figure): one horizontal with dimensions $2 \times 1$, and two each with dimensions $1 \times 1$, $1 \times 2$ (vertical), $2 \times 2$, $4 \times 3$ (horizontal), and $4 \times 4$. In how many ways can all 11 pictures be pla... | Answer: 16896.
Solution. We will say that two pictures are in different columns if no block of the first picture is in the same column as any block of the second. It is clear that the $4 \times 4$ pictures are in different columns from each other and from the $4 \times 3$ pictures in any arrangement. Thus, the $4 \tim... | 16896 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
6. In the Thrice-Tenth Kingdom, there are 17 islands, each inhabited by 119 people. The inhabitants of the kingdom are divided into two castes: knights, who always tell the truth, and liars, who always lie. During the census, each person was first asked: "Excluding you, are there an equal number of knights and liars on... | Answer: 1013.
## Solution.
1) Consider the first question. A "yes" answer would be given by either a knight on an island with exactly 60 knights, or a liar if the number of knights is different. A "no" answer would be given by either a liar on an island with 59 knights, or a knight if the number of knights is differe... | 1013 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
4. Find all real solutions to the system of equations
$$
\left\{\begin{array}{l}
\sqrt{x-997}+\sqrt{y-932}+\sqrt{z-796}=100 \\
\sqrt{x-1237}+\sqrt{y-1121}+\sqrt{3045-z}=90 \\
\sqrt{x-1621}+\sqrt{2805-y}+\sqrt{z-997}=80 \\
\sqrt{2102-x}+\sqrt{y-1237}+\sqrt{z-932}=70
\end{array}\right.
$$
(L. S. Korechkova, A. A. Tessl... | Answer: $x=y=z=2021$.
Solution. First, we prove that the solution is unique if it exists. Let $\left(x_{1}, y_{1}, z_{1}\right)$ and $\left(x_{2}, y_{2}, z_{2}\right)$ be two different solutions and, without loss of generality, $x_{1} \leqslant x_{2}$. Then there are four possible cases: $y_{1} \leqslant y_{2}$ and $z... | 2021 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. In each cell of a $10 \times 10$ table, a natural number was written. Then, each cell was shaded if the number written in it was less than one of its neighbors but greater than another neighbor. (Two numbers are called neighbors if they are in cells sharing a common side.) As a result, only two cells remained unshad... | Solution. Answer: 20. This value is achieved if the unshaded cells are in opposite corners and contain the numbers 1 and 19.
Evaluation.
1) The cells that contain the minimum and maximum numbers are definitely not shaded. This means that the minimum and maximum each appear exactly once, and they are in the unshaded c... | 20 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
2. How many five-digit numbers are divisible by their last digit?
将上面的文本翻译成英文，请保留源文本的换行和格式，直接输出翻译结果。 | 2. The total number of five-digit numbers is $-99999-9999=90000$, and among them, there are an equal number of numbers ending in $0,1, \ldots, 9$, that is, 9000 numbers of each type.
Let $n_{i}$, where $i=0,1, \ldots, 9$, be the number of numbers ending in $i$ that are divisible by $i$. Then
$n_{0}=0$ (a number canno... | 42036 | Number Theory | proof | Yes | Yes | olympiads | false |
1. In one move, you can either add one of its digits to the number or subtract one of its digits from the number (for example, from the number 142 you can get $142+2=144, 142-4=138$ and several other numbers).
a) Can you get the number 2021 from the number 2020 in several moves?
b) Can you get the number 2021 from th... | Solution. a) Yes, for example, like this: $20 \mathbf{2 0} \rightarrow 20 \mathbf{1 8} \rightarrow \mathbf{2 0 1 9} \rightarrow 2021$.
b) Yes. For example, by adding the first digit (one), we can reach the number 2000; by adding the first digit (two), we can reach 2020; then see part a.
Criteria. Part a) 3 points, b)... | 2021 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. In a rectangular grid 303 cells long and 202 cells wide, two diagonals were drawn and all cells through which they passed were painted. How many cells were painted?
(O. A. Pyayve, A. A. Tseler)
Notice that each diagonal intersects 101 such rectangles (passing through their vertices), and in each of them, it passes through 4 cells. Thus, the two diagonals, it seems, pass thro... | 806 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. There are 28 students in the class. On March 8th, each boy gave each girl one flower - a tulip, a rose, or a daffodil. How many roses were given, if it is known that there were 4 times as many roses as daffodils, but 3 times fewer than tulips?
(A. A. Tesler) | Solution. Let the number of narcissus be $x$, then the number of roses is $4x$, and the number of tulips is $12x$, so the total number of flowers is $17x$. The number of flowers is the product of the number of boys and the number of girls. Since 17 is a prime number, one of these quantities must be divisible by 17, mea... | 44 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
5. Once Valera left home, walked to the cottage, painted 11 fence boards there, and returned home 2 hours after leaving. Another time, Valera and Olga went to the cottage together, painted 9 fence boards (without helping or hindering each other), and returned home together 3 hours after leaving. How many boards will Ol... | Solution. The strange result (working together for a longer time, the characters managed to do less work) is explained by the different times spent walking, since the speed of "joint" walking is equal to the lower of the two walkers' speeds. The second time, Valery's working time decreased, which means the travel time ... | 0 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. Once, a god sent a little cheese to two ravens. The first raven received 100 g, from which a part was taken by a fox. The piece of the second raven turned out to be twice as large as that of the first, but she managed to eat only half as much as the first raven. The portion of cheese that the fox got from the second... | Solution. Let the first crow eat $x$ grams of cheese. Then the fox got $100-x$ grams of cheese from the first crow. The second crow ate ${ }_{2}^{x}$ grams of cheese. From the second crow, the fox received $200-\frac{x}{2}$ grams of cheese. This was three times more, so: $200-\frac{x}{2}=3(100-x)$. Solution: $x=40$. Th... | 240 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. A natural number $n>5$ is called new if there exists a number that is not divisible by $n$, but is divisible by all natural numbers less than $n$. What is the maximum number of consecutive numbers that can be new? | Solution. Answer: 3.
Example: the number 7 is new (60 is divisible by the numbers from 1 to 6, but not by 7);
the number 8 is new (420 is divisible by the numbers from 1 to 7, but not by 8);
the number 9 is new (840 is divisible by the numbers from 1 to 8, but not by 9).
Evaluation: every fourth number has the form... | 3 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same... | Solution. Since $A$ with $C$ and $B$ with $D$ meet every 7 minutes, their approach speeds are equal: $V_{A}+V_{C}=V_{B}+V_{D}$. In other words, the speeds of separation of $A, B$ and $C, D$ are equal: $V_{A}-V_{B}=V_{D}-V_{C}$. Therefore, since $A$ and $B$ meet for the first time at the 53rd minute, $C$ and $D$ will al... | 53 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
8. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). The result was the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible ... | Solution. The letter A must equal 0. The remaining 6 letters are non-zero digits with a sum that is a multiple of 3. Note that each remainder when divided by 3 appears three times. By enumeration, we find all sets of remainders whose sum is a multiple of three: 000111, 000222, 111222, 001122. We count the 6-element sub... | 21600 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
2. The pond has a square shape. On the first frosty day, the part of the pond within 10 meters of the nearest shore froze. On the second day, the part within 20 meters froze, on the third day, the part within 30 meters, and so on. On the first day, the area of open water decreased by $19 \%$. How long will it take for ... | Solution. It is not hard to understand that a pond of $200 \times 200$ fits, for which the answer is - in 10 days (since each day the side decreases by 20 meters). There are no other options, as the larger the side of the pond, the smaller the percentage that will freeze on the first day.
More rigorously: let the side... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. How many ways are there to cut a $10 \times 10$ square into several rectangles along the grid lines such that the sum of their perimeters is 398? Ways that can be matched by rotation or flipping are considered different. | Solution: 180 ways.
If the entire square is cut into 100 unit squares, the sum of the perimeters will be $4 \times 100=400$. Therefore, we need to reduce this sum by 2, which is achieved by keeping one internal partition intact (in other words, the square is cut into 98 squares and 1 domino). There are a total of 180 ... | 180 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
4. A rectangle $11 \times 12$ is cut into several strips $1 \times 6$ and $1 \times 7$. What is the minimum total number of strips? | Solution. Answer: 20. The example is shown in the figure.
Evaluation: we will paint every seventh diagonal so that 20 cells are shaded (see figure). Each strip contains no more than one cell, so there are no fewer than 20 strips.
. However, it is forbidden to place a package inside... | Solution. 8. Example: ((6)(2)) ((3)(4)) ((1)4) (there are other examples).
There cannot be more than 8 packages. Indeed, then the sum of the number of candies in the packages (or rather, the number of incidences of candies to packages) is not less than $1+2+\ldots+9=45$. But there are 20 candies, so at least one of th... | 8 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
1. There is a rectangular sheet, white on one side and gray on the other. It was folded as shown in the picture. The perimeter of the first rectangle is 20 more than the perimeter of the second rectangle. And the perimeter of the second rectangle is 16 more than the perimeter of the third rectangle. Find the area of th... | Solution. From the figure, it can be seen that when folding, the perimeter of the rectangle decreases by twice the short side, so the short side of rectangle-1 is $20 / 2=10$, the short side of rectangle-2 is $16 / 2=8$. Therefore, the long side of rectangle-1 is 18, and the long side of the original sheet is 28. Thus,... | 504 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, the same letters correspond to the same digits). The result was the word "GUATEMALA". How many different numbers could Egor have initially written if his number was divisible ... | Solution. For a number to be divisible by 8, "АЛА" must be divisible by 8, with "А" -
the expression in parentheses is clearly divisible by 8, so it is sufficient to require that ("А" + 2 * "... | 67200 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
4. Four cars $A, B, C$, and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counterclockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the start of the race, $A$ meets $C$ for the first time, and at the same... | Solution. $A$ and $C$ meet every 7 minutes, while $A$ and $B$ meet every 53 minutes. Therefore, all three will meet at a time that is a multiple of both 7 and 53, which is $7 \cdot 53 = 371$ minutes. At the same time, $B$ and $D$ also meet every 7 minutes, so on the 371st minute, car $D$ will be at the same point as th... | 371 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1. The pond has a rectangular shape. On the first frosty day, the part of the pond within 10 meters of the nearest shore froze. On the second day, the part within 20 meters froze, on the third day, the part within 30 meters, and so on. On the first day, the area of open water decreased by 20.2%, and on the second day, ... | Solution. First method. Let the sides of the pond be $a$ and $b$ meters, then
$(a-20)(b-20)=(1-0.202) a b, (a-40)(b-40)=(1-0.388) a b$,
from which $20(a+b)-400=0.202 a b, 40(a+b)-1600=0.388 a b$, that is, $800=0.016 a b, a b=5000$ and further $a+b=525$. It turns out that the sides are 400 and 125 meters.
Answer: on ... | 7 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. Here is a problem from S. A. Rachinsky's problem book (late 19th century): "How many boards 6 arshins long and 6 vershoks wide are needed to cover the floor of a square room with a side of 12 arshins?" The answer to the problem is: 64 boards. Determine from these data how many vershoks are in an arshin. | Solution. The area of the room is $12 \cdot 12=144$ square arshins. Therefore, the area of each board is $144 / 64=2.25$ square arshins. Since the length of the board is 6 arshins, its width should be $2.25 / 6=3 / 8=6 / 16$ arshins. Thus, 6 vershoks make up $6 / 16$ arshins, meaning 1 vershok is $1 / 16$ arshin.
Anot... | 16 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. On a glade, two firs, each 30 meters tall, grow 20 meters apart from each other. The branches of the firs grow very densely, and among them are some that are directed straight towards each other, and the length of each branch is half the distance from it to the top. A spider can crawl up or down the trunk (strictly ... | Solution. From the diagram, it can be seen that the branches of the firs intersect at a height of no more than 10 meters from the ground. Indeed, at this height, the distance to the treetop is 20 meters, so the length of each branch is $20 / 2 = 10$ meters, and the total length of the branches of the two firs is equal ... | 60 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
5. Nikita has a magic jar. If you put $n$ candies in the jar and close it for an hour, the number of candies inside will increase by the sum of the digits of $n$. For example, if there were 137 candies, it would become $137+1+3+7=148$. What is the maximum number of candies Nikita can get in 20 hours and 16 minutes, if ... | Solution. We need to strive to have as many candies as possible at the end of each hour. However, this does not mean that we should always put all the candies in the jar. The greatest sum of digits (i.e., the greatest increase in the number of candies) is achieved with a number where all digits (except the first) are n... | 267 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
6. How many numbers from 1 to 999 without the digit "0" are written in the Roman numeral system exactly one symbol longer than in the decimal system?
(P. D. Mulyenko)
Reference. To write a number in Roman numerals, you need to break it down into place value addends, write each place value addend according to the tabl... | Solution. Note that, regardless of the digit place and other digits of the number, a decimal digit $a$ is written as:
- one Roman numeral when $a=1$ and $a=5$,
- two Roman numerals when $a$ is $2,4,6,9$,
- three Roman numerals when $a=3$ and $a=7$,
- four Roman numerals when $a=8$.
Thus, in suitable numbers, only the... | 68 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
5. In a spring math camp, between 50 and 70 children arrived. On Pi Day (March 14), they decided to give each other squares if they were just acquaintances, and circles if they were friends. Andrey calculated that each boy received 3 circles and 8 squares, while each girl received 2 squares and 9 circles. And Katya not... | Answer: 60.
Solution. Let the number of boys be $m$, and the number of girls be $-d$. Then $3 m + 9 d = 8 m + 2 d$ (the number of circles equals the number of squares). Transforming, we get $5 m = 7 d$, which means the number of boys and girls are in the ratio $7: 5$. Therefore, the total number of children is divisib... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
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