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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 | 11,538 |
3. Vasya's grandmother loves to pick berries in the forest and then make jam from them. Every weekend she goes by train, spending 200 rubles on tickets and collects 5 kilograms of berries. On the market, you can buy a kilogram of berries for 150 rubles, and sugar for 54 rubles per kilogram. From one kilogram of berries... | Answer: 1.
Option 1 - berry picking in the forest. From 1 kilogram of berries collected by the grandmother, 1.5 kilograms of jam can be made. The cost of 1.5 kilograms of jam in this case consists of transportation expenses and the purchase of sugar: $(200 / 5 + 54) = 94$ rubles.
Option 2 - buying berries. To make 1.... | 1 | Algebra | MCQ | Yes | Yes | olympiads | false | 11,539 |
4. The newlyweds, the Batterykins, have a combined monthly income of 150,000 rubles. Monthly expenses (food, utilities, mortgage payments, etc.) amount to 115,000 rubles. At the beginning of the year, the family's savings amounted to 45,000 rubles. The family plans to buy new kitchen furniture worth 127,000 rubles usin... | Answer: April
The family's net monthly income is $150000-115000=35000$ rubles.
The missing amount of funds for the purchase of furniture is $127000-45000=82000$ rubles.
The number of months required to reach this amount is $82000 / 35000=2.3$. Rounding up, we get 3 months. According to the condition of the problem, ... | April | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,540 |
5. In the "6 out of 45" lottery, a participant makes a bet by selecting any 6 numbers from 1 to 45 (the order of selection does not matter, but all numbers must be different). During the draw, a random winning combination of 6 numbers is determined. A "jackpot" is the event where a participant, by making a bet, guesses... | Answer:
1) from 0.01 to 0.1
2) from 0.001 to 0.01
3) from 0.0001 to 0.001
4) from 0.00001 to 0.0001
Answer: 4. In the lottery, there are $C_{45}^{6}=\frac{45 \cdot 44 \cdot 43 \cdot 42 \cdot 41 \cdot 40}{1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot 6}=8145060$ possible combinations of numbers. The probability of a "jackpo... | 0.0000123 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,541 |
6. Petya saw a bank advertisement: "Deposit 'Super-Income' - up to $10 \%$ annually!". He became interested in this offer and found it on the bank's website. The conditions turned out to be slightly different, as reflected in the table.
| Interest period | from 1 to 3 months | from 4 to 6 months | from 7 to 9 months |... | Answer: 8.8%.
The effective interest rate implies the capitalization of interest, that is, adding it to the deposit amount. Let's say Petya deposited X rubles into the described deposit, then by the end of the year, he will have
$X *(1+0.1 / 4) *(1+0.08 / 4) *(1+0.08 / 4) *(1+0.08 / 4) \approx X * 1.0877$.
That is, ... | 8.8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,542 |
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 | 11,543 |
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 | 11,544 |
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 | 11,545 |
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 | 11,546 |
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 | 11,547 |
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 | 11,549 |
# Task 2. (6 points)
Name three reasons why getting a driver's license at 18 might be more advantageous than at 30? | # Solution:
- cheaper any type of car insurance (OSAGO, KASKO)
- there is an opportunity to rent a car during travels, the rental cost is lower
- having a driver's license is sometimes a criterion for employment
- it is easier to get a loan, the interest rates on it may be lower
- over time, the process of obtaining a... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,550 |
# 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 | 11,551 |
# 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 | 11,552 |
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 | 11,553 |
2. Vasya's grandmother loves to pick berries in the forest and then make jam from them. Every weekend she goes by train, spending 200 rubles on tickets and collects 5 kilograms of berries. At the market, you can buy a kilogram of berries for 150 rubles, and sugar for 54 rubles per kilogram. From one kilogram of berries... | Answer: 1.
Option 1 - berry picking in the forest. From 1 kilogram of berries collected by the grandmother, 1.5 kilograms of jam can be made. The cost of 1.5 kilograms of jam in this case consists of transportation expenses and the purchase of sugar: $(200 / 5 + 54) = 94$ rubles.
Option 2 - buying berries. To make 1.... | 1 | Algebra | MCQ | Yes | Yes | olympiads | false | 11,554 |
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 | 11,556 |
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 | 11,557 |
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 | 11,558 |
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 | 11,559 |
8. Petya saw a bank advertisement: "Deposit 'Super-Income' - up to $10 \%$ annually!". He became interested in this offer and found it on the bank's website. The conditions turned out to be slightly different, as reflected in the table.
| Interest period | from 1 to 3 months | from 4 to 6 months | from 7 to 9 months |... | Answer: 8.8%.
The effective interest rate implies the capitalization of interest, that is, adding it to the deposit amount. Let's say Petya deposited $X$ rubles into the described deposit, then by the end of the year, he will have $X * (1+0.1 / 4) * (1+0.08 / 4) * (1+0.08 / 4) * (1+0.08 / 4) \approx X * 1.0877$.
That... | 8.8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,560 |
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. | 1,400,000 | Algebra | MCQ | Yes | Yes | olympiads | false | 11,562 |
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 | 11,563 |
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 | 11,566 |
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 | 11,567 |
# 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 | 11,569 |
Problem 15. (1 point)
The monthly expenses of the Mikhailov family are as follows: utility payments - 5250 rubles, food purchases - 15000 rubles, household chemicals and personal hygiene items - 3000 rubles, clothing and shoe purchases - 15000 rubles, car loan payments - 10000 rubles, transportation costs - 2000 ruble... | Answer: 65000, 82500.
## Comment:
Solution:
Current expenses $=(5250+15000+3000+15000+000+2000+5000+1500+2000+$ $3000)=61750$ rubles.
Required income $=61750 / 0.95=65000$ rubles.
Savings after 10 months, taking into account their increase $=5000 \times 10+3250 \times 10=82500$ rubles. | 65000,82500 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,570 |
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 | 11,571 |
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 | 11,572 |
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 | 11,573 |
4. The company will have a net profit at the end of 2018 in the amount of: $(2500000-1576250)-(2500000-1576250) * 0.2=739000$ rubles. The amount of principal repayments on the loan for 2018 will be:
$$
25000 * 12 \text { = } 300000 \text { rubles }
$$
Dividends per share for 2018:
$$
(739000-300 \text { 000) / } 160... | Solution:
The information indicated on the packaging is intended to convince the buyer that this product has a significant advantage over similar products from other manufacturers (in this case, significant for healthy eating).
Thus, in sunflower oil as a plant product:
- there cannot be cholesterol (animal fat);
- ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,574 |
Problem 16. (5 points)
Maxim Sergeyevich works as a financial manager in a large industrial company. He has long dreamed of acquiring a large plot of land with a house on it. Maxim Sergeyevich saved money for a down payment on a mortgage for about two years, and finally, when a sufficient amount was in his account, he... | # Answer: Maxim Sergeevich.
## Comment:
The return on the stock portfolio is higher than the interest payments on the loan, therefore, the loan will be a cheaper source of funds for buying a house compared to selling the stock portfolio.
# | MaximSergeevich | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,576 |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive, which means the largest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend
$$
\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 | 11,579 |
2. Mikhail Trofimov and Valentin Pavlogradsky are entrepreneurs. Mikhail's business is AO "New Year's Joy": a company for manufacturing and selling gifts and New Year's items. Valentin is the founder and general director of LLC "Festival Comes to You," which specializes in organizing celebrations, morning shows, and co... | # Answer:
According to paragraph 1 of Article 9 of Federal Law 173-FZ "On Currency Regulation and Currency Control", currency transactions between residents are prohibited. The law lists exceptions to this rule, but the case of friends does not fall under this list.
4 points - indicating that the tax authority is cor... | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,580 |
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 | 11,581 |
4. The Petrovs family decided to buy an apartment with a mortgage. The total cost of the apartment was 13,000,000 rubles, and the family did not have enough funds, so they decided to take a loan of 8,000,000 rubles. One of the bank's conditions for issuing the loan was the purchase of a comprehensive mortgage insurance... | # Solution:
$8000000+0.095 \times 8000000=8760000$ - final loan amount, insurance sum
$8760000 \times 0.09\%=7884$ - cost of property insurance
$8760000 \times 0.27\%=23652$ - cost of title insurance
$8760000 \times 0.4 \times 0.17\%=5956.8$ - cost of life and health insurance for Maria Sergeyevna
$8760000 \times ... | 47481.20 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,582 |
6. Imagine you have two bank cards: a debit card and a credit card. Today, you decided to buy a new phone for 10,000 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 exactly 1 month later to avoid going beyond the grace period, during whi... | # Solution:
If you pay with a credit card, your expenses next month will amount to 10,000 rubles. Next month, you will also receive income in the form of cashback of 50 rubles, and while you keep 10,000 rubles on your debit card, interest will be accrued in the amount of 50 rubles. Thus, by spending 10,000 rubles in a... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,583 |
6. Gru and the Minions plan to earn money by mining cryptocurrencies. For mining, they chose Ethereum (Ethereum) as one of the most stable and promising currencies. For this, they bought a system unit for 9499 rubles and two video cards for 31431 rubles each. The power consumption of the system unit is 120 W, and each ... | # Answer:
With two video cards, you can earn $2 \times 0.00877 \times 27790.37 = 487.44$ rubles per day. The power consumption of one system unit and two video cards is $120 + 125 \times 2 = 370$ watts per hour, or 8.88 kWh per day, which amounts to 47.77 rubles spent on electricity per day. Capital expenses amount to... | 164.58 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,584 |
2. Once, a misfortune befell Lyudmila Alekseevna Lerpushkina, a former teacher living on a pension. She was standing in front of the cash register at the store "Sotochka" and, intending to pay for her purchases, accidentally opened her wallet: the zipper got stuck on several banknotes.
"Oh no! Oh dear!" - Lyudmila Ale... | # Response:
Banknotes and coins of the Bank of Russia, which have the force of legal tender in the Russian Federation (including those being withdrawn from circulation), are considered legal tender, provided they do not contain signs of counterfeiting, and are not damaged or have damage of the following nature: dirty,... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,585 |
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 | 11,586 |
4. The Ivanov family decided to buy an apartment with a mortgage. The total cost of the apartment was 7,000,000 rubles, and the family did not have enough funds, so it was decided to take a loan for 4,000,000 rubles. One of the conditions of the bank when issuing the loan was the purchase of a comprehensive mortgage in... | # Solution:
$4000000 + 0.101 \times 4000000 = 4404000$ - final loan amount, insurance sum
$4404000 \times 0.09\% = 3963.6$ - cost of property insurance
$4404000 \times 0.27\% = 11890.8$ - cost of title insurance
$4404000 \times 0.2 \times 0.17\% = 1497.36$ - cost of life and health insurance for Svetlana Markovna
... | 24045.84 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,587 |
6. You have decided to buy yourself a New Year's gift for 8,000 rubles. You have the option to pay with one of two bank cards: a debit card or a credit card. If you pay with the credit card (the credit limit allows it), you will have to return the money to the bank exactly half a month later to avoid going beyond the g... | # Solution:
If you pay with a credit card, your expenses in half a month will amount to 8,000 rubles. In a month, you will also receive income in the form of cashback in the amount of 40 rubles, and while you keep 8,000 rubles on your debit card, interest will be accrued in the amount of 20 rubles for half a month. Th... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,588 |
6. Joker and Harley Quinn plan to earn money by mining cryptocurrencies. For mining, they chose Ethereum (Ethereum) as one of the most stable and promising currencies. For this, they bought a system unit for 9499 rubles and two video cards for 20990 rubles each. The power consumption of the system unit is 120 W, and ea... | # Answer:
With two video cards, you can earn $2 \times 0.00630 \times 27790.37 = 350.16$ rubles per day. The power consumption of one system unit and two video cards is $120 + 185 \times 2 = 490$ watts per hour, or 11.76 kWh per day, which amounts to 63.27 rubles spent on electricity per day. Capital expenses amount t... | 179.44 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,589 |
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 | 11,591 |
4. Stanislav Petrovich, born on 05.06.1975, height 183 cm, weight 125 kg, an electrician by profession, decided to purchase life and health insurance for 5,000,000 rubles. He visited the insurance company's website, looked at the rates, and without consulting or filling out an application, concluded that the cost of th... | # Solution:
Let's calculate the age. The man is 42 years old. The base rate is $0,35 \%$. As an electrician, the rate is $0,35 \times 1,15=0,4025$.
Calculate the BMI $=125 / 1,83^{2}=125 / 3,3489=$ (approximately) $37,326$.
According to the table, we find the increasing coefficient. It is 1.7.
$0,4025 \times 1,7=0,... | 34212.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,592 |
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 | 11,593 |
6. In recent years, the number of online shopping purchases has significantly increased. Cybercriminals are taking advantage of this by creating "banking" trojans and using various social engineering techniques to steal financial information. Such techniques include sending phishing emails, creating phishing websites, ... | # Solution:
a) The friend violated several of the following security measures when making purchases in an online store:
- Do not download files received from unverified sources, do not follow unreliable links. Do not open suspicious emails and immediately block their sender.
- Do not believe in winning contests and l... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,594 |
2. In some stores, we can observe the following situation. Products of the same category are placed next to each other, but:
- one of the products («A») is better than the others in certain parameters, but its price is clearly overpriced (which makes the probability of its purchase quite low);
- another product («B»),... | Answer: 4 points for a brief answer, +3 points for justifying the answer with product "A", +3 points for justifying the answer with product "B"
This technique allows increasing sales of mid-price category products.
An expensive product is not considered by marketers as an actual "product for sale." It serves as a kin... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,595 |
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 | 11,596 |
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 | 11,597 |
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 | 11,598 |
6. In recent years, the volume of cash transactions using bank cards has significantly increased. Criminals take advantage of this by creating a copy of the magnetic strip when paying for goods and services in stores, installing skimming devices (overlaying card readers and PIN entry panels on ATMs) and shimming device... | # Solution:
a) The friend violated several of the following security measures when withdrawing cash from an ATM:
- Keep your PIN code for the bank card confidential.
- If possible, use cards with a microchip - it is much harder to read data from them.
- For withdrawing cash and making payments, choose ATMs located in... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,599 |
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 | 11,600 |
# Task 5. (10 points)
In Daniel Defoe's novel "Robinson Crusoe," upon finding money in one of the chests from the sunken ship, Robinson Crusoe exclaimed: "Worthless trash, what good are you to me now? I would gladly give up a whole pile of gold for any of these cheap knives. I have no use for you. So off to the bottom... | # Solution:
Stranded on an uninhabited island, Robinson found himself alone. In these conditions, he did not need money for exchanges because there were no exchanges. Thus, money ceased to be a medium of exchange. (4 points)
Among other functions of money, the following can be highlighted:
| notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,601 |
Task 3.
There is a big event in the Denisov family! Their son, Mikhail, is having a birthday, and he will turn 18 years old. Six months before the celebration, the parents approached Mikhail with a request to help choose the most suitable option for preparing the birthday celebration. In addition to the family, 10 mor... | # Solution:
Option 1: Expenses - 25000 rubles.
Option 2: Expenses - 10 * 3000 = 30000 rubles and 5 people (the Denisov family) 5 * 3000 = 15000 rubles. Total: 45000 rubles.
Option 3: Expenses - 10 * 5000 = 50000 rubles and 5 people (the Denisov family) 5 * 5000 = 25000 rubles. Total: 75000 rubles. Income: Before tax... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,607 |
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 | 11,608 |
Task 5. Student Alexander from 5th grade received an SMS message from an unknown sender: "Alexander, your mobile number participated in a prize draw. Details on claiming your prize can be obtained by calling the phone number $+7(* * *)$ ********". Should Alexander call this number? (Answer) Provide two arguments to sup... | Solution: There is no need to call the given number. The offer is tempting, but it is a scam. Let's recall the definition of fraud: "Fraud (this type of crime) involves the acquisition of property belonging to third parties, or the registration of rights to it, through deceit or by gaining the trust of the victim. A di... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,609 |
Task 4.
Yevgeny Petrovich decided to take a loan from a banking institution in the amount of 3,000,000 rubles to purchase a one-bedroom apartment in Andronovka. The loan terms are as follows: he returns the initial loan amount and 150,000 rubles in interest over 8 months. Determine the annual interest rate of the bank... | Solution: (150000/8)/3000000*12*100
Answer: $7.5 \%$
Criteria:
Correct solution and answer - 20 points
# | 7.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,612 |
4. The company will have a net profit at the end of 2018 in the amount of: $(2500000-1576250)-(2500000-1576250) * 0.2=739000$ rubles. The amount of principal repayments on the loan for 2018 will be:
$$
25000 * 12 \text { = } 300000 \text { rubles }
$$
Dividends per share for 2018:
$$
(739000-300 \text { 000) / } 160... | Solution:
The information indicated on the packaging is intended to convince the buyer that this product has a significant advantage over similar products from other manufacturers (in this case, significant for healthy eating).
Thus, in sunflower oil as a plant product:
- there cannot be cholesterol (animal fat);
- ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,613 |
1. Place the digits from 1 to 9 in the figures such that each digit appears in one square, one circle, and one triangle, and the equation is correct:
$\triangle \cdot O \cdot \square+\Delta \cdot O \cdot \square+\Delta \cdot O \cdot \square+\Delta \cdot O \cdot \square+\Delta \cdot O \cdot \square+\Delta \cdot O \cdot... | Solution. Unfortunately, it is impossible to obtain 2019. We apologize for the error in the problem. As compensation for this problem, up to 2 points were awarded to those who provided examples of obtaining the numbers 2018 and 2020.
Remark. The maximum value that can be obtained is $9 \cdot 9 \cdot 9 + 8 \cdot 8 \cdo... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,614 |
2. Seven children met. Some of them gave gifts to some others (one could not give more than one gift to another). Could it have turned out that everyone received an equal number of gifts, even though they all gave different numbers of gifts (including possibly someone giving no gifts at all)? | Solution. Everyone gave a different number of gifts, and no one gave a gift to themselves, so all quantities from 0 to 6 were given. In total, 21 gifts were given. Each child received 3 gifts.
Example of who gave gifts to whom:
1 -st: 5 -th, 6 -th, 7 -th;
2 -nd: 5 -th, 6 -th, 7 -th;
$. Solution: $x=40... | 240 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,616 |
1. Is there a year in the 21st century whose number can be represented as $\frac{a+b \cdot c \cdot d \cdot e}{f+g \cdot h \cdot i \cdot j}$, where $a, b, c, d, e, f, g, h, i, j$ are pairwise distinct digits? | Solution. Yes, for example, $2022=\frac{6+4 \cdot 7 \cdot 8 \cdot 9}{1+0 \cdot 2 \cdot 3 \cdot 5}$. Similarly, 2018, 2020, and 2021 can be represented. | 2022=\frac{6+4\cdot7\cdot8\cdot9}{1+0\cdot2\cdot3\cdot5} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 11,619 |
2. Petya divided a circle into 7 parts with three straight lines and wrote 7 different integers in them so that the sums of the numbers on either side of each line were the same. One of the numbers is zero. Prove that some number is negative. | Solution. Let's consider three cases of the zero's position (a picture for each case). In the pictures, the sum of the yellow sectors equals the sum of the pink ones (since yellow + orange = pink + orange = half the sum of all numbers). We see that the first two cases are impossible (there are duplicate numbers), and i... | proof | Number Theory | proof | Yes | Yes | olympiads | false | 11,620 |
3. A chess championship is being held at a rural club: each participant must play one game with each other. The club has only one board, so two games cannot take place simultaneously. According to the championship regulations, at any time, the number of games already played by different participants should not differ b... | Solution. Not always. For example, let there be 6 players in the championship, and the first matches were played in the following order: $12,34,56,13,24$. Now, the players who have played the fewest matches are 5 and 6, so a match between them should be arranged, but it has already taken place. | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,621 |
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 | 11,623 |
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 | 11,624 |
7. Let's call a natural number useful if it does not contain any zeros or identical digits in its decimal representation, and the product of all its digits is divisible by their sum. Find the two largest consecutive (i.e., differing by 1) useful numbers. | Solution. The numbers 9875213 and 9875214 are useful. We will prove that there do not exist larger consecutive useful numbers. The sums of the digits of consecutive numbers are also consecutive (otherwise, we would have a transition across a place value, meaning a 0 at the end). However, the maximum possible sums of th... | 98752139875214 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 11,625 |
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 | 11,626 |
1. Three lines divide a circle into 7 parts. Is it possible to distribute seven consecutive natural numbers, one in each area, so that the sums of the numbers on either side of each line are equal? | Solution. Yes, for example, like this:
 | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,627 |
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 | 11,628 |
3. A research spacecraft is entering an asteroid belt that could damage the ship's hull, leading to depressurization. All corridors between rooms are equipped with airtight doors. The captain has a droid assistant that can close (but not reopen) doors in the corridors it travels through. Will the droid be able to close... | Solution. There are a total of 23 corridors and 14 compartments on the spaceship. Each visit by the droid to one compartment covers two corridors: the one it entered through and the one it left through. Therefore, in all compartments, except possibly two, there must be an even number of exits (two compartments can serv... | proof | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,629 |
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 | 11,630 |
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 | 11,631 |
7. On the island of knights, who always tell the truth, and liars, who always lie, a school was opened. All $2 N$ students lined up in pairs (i.e., in two columns). The first two people said: "I am taller than two people: my partner in the pair and the person behind me." The last two said: "I am also taller than two pe... | Solution. A) In each pair, there is no more than one knight, so there are no more than $N$ knights (the example is achieved by placing $N$ taller students in a checkerboard pattern).
B) If it turns out that all students are of the same height, then everyone is lying.
Criteria. Correctly done part a) - 5 points. Only ... | N | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,633 |
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 | 11,634 |
1. In the cells of a $5 \times 5$ square, natural numbers from 1 to 5 are placed such that in each column, each row, and each of the two main diagonals, all numbers are distinct. Can the sum of the numbers in the cells shaded in the figure be equal to 19?
(L. S. Korechkova)
![](https://cdn.mathpix.com/cropped/2024_05... | Solution. Let it be. To get the sum of 19 in the shaded cells, there must be three digits "5" and one digit "4" ( $19=5+5+5+4$ ). With any arrangement of the digits "5" on the main diagonal (running from the bottom left corner to the top right), a five can only be placed in the top right corner, but then there is no pl... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,635 |
3. Solve the equation: $[20 x+23]=20+23 x$. Recall that $[a]$ denotes the integer part of the number, that is, the greatest integer not exceeding $a$.
(l. S. Koroleva) | Solution. Let both sides of the equation be denoted by $n$, which is an integer. Then $x=\frac{n-20}{23}$ and $n \leqslant 20 x+23<n+1$. This means that
$$
n \leqslant \frac{20(n-20)}{23}+23<n+1
$$
which is
$$
n \leqslant 43<n+7 \frac{2}{3}
$$
It follows that $x=1-\frac{k}{23}$ for any integer $0 \leqslant k \leqsl... | \frac{16}{23},\frac{17}{23},\frac{18}{23},\frac{19}{23},\frac{20}{23},\frac{21}{23},\frac{22}{23},1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,637 |
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 | 11,638 |
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 | 11,639 |
2. Given a triangle $ABC$. $O_{1}$ is the center of its inscribed circle; $O_{2}$ is the center of the circle that touches side $BC$ and the extensions of the other two sides of triangle $ABC$. On the arc $BO_{2}$ of the circumcircle of triangle $O_{1}O_{2}B$, a point $D$ is marked such that angle $BO_{2}D$ is half of ... | # Solution.
Note that angles $O_{1} B O_{2}$ and $O_{1} C O_{2}$ are right angles (as angles between the bisectors of adjacent angles), so $B, O_{1}, C, O_{2}$ lie on the same circle, and $\angle B C D=\angle B O_{2} D=\frac{1}{2} \angle B A C$. But angle $B C M$ is also equal to $\frac{1}{2} \angle B A C$ (since it s... | proof | Geometry | proof | Yes | Yes | olympiads | false | 11,640 |
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 | 11,641 |
5. A race on an unpredictable distance is held as follows. On a circular running track 1 kilometer long, two points $A$ and $B$ are chosen randomly (using a spinning arrow), after which the athletes run from $A$ to $B$ along the shorter arc. Find the median value of the length of this arc, that is, such an $m$ that the... | Solution. Let's choose the origin and the positive direction on the path, then the pairs $(A, B)$ can be identified with pairs of numbers from $[0,1)$ (when measured in kilometers). The probability that $(A, B)$ belongs to some subset of $[0,1) \times[0,1)$ is equal to the area of this subset. The length of the shorter... | 250\mathrm{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,642 |
6. The magical clock, in addition to the usual pair of hands, has a second pair that is symmetrical to the first at every moment relative to the vertical axis. It is impossible to determine which hands are real from a photograph of the clock. Furthermore, just like with ordinary clocks, it is impossible to distinguish ... | Answer: 3 or 100.
Solution. First, note that for each photograph, the set of possible times has the form $\{t ; 12-t ; 12+t ; 24-t\}(0<t<6)$.
Consider the following photographs: $A$ - the one taken closest to 6:00; $B$ - the one taken closest to 12:00; $C$ - the one taken closest to 18:00.
If all these photographs a... | 3or100 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,643 |
2. Petya and Vasya are playing a game. They have a strip of 9 cells. Each turn, a player writes any digit in any free cell. They take turns, with Petya starting first. If the resulting number is a perfect square at the end of the game, Petya wins; otherwise, Vasya wins. They consider that the number can start with one ... | Solution. Vasya has a winning strategy. Moreover, he has a strategy that ensures a win already after the first move. Let's describe it.
If Petya, on his first move, places a digit not in the last cell, then Vasya places in the last cell one of the digits that cannot be the last digit of a square of a natural number, f... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 11,644 |
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 | 11,646 |
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 | 11,648 |
3. Points $H, K$, and $M$ lie on the sides $BC, AC$, and $AB$ of triangle $ABC$, respectively, where $AH$ is the altitude. Prove that $AH$ is the bisector of angle $KHM$ if and only if $AH$, $BK$, and $CM$ intersect at one point. | 3. See the figure at the end of the file.
We will use Ceva's theorem, which states that the segments $A H, B K$, and $C M$ intersect at one point if and only if
$$
\frac{A K}{K C} \cdot \frac{C H}{H B} \cdot \frac{B M}{M A}=1
$$
Let's introduce the following notations: $\angle M H A=\alpha_{1}, \angle K H A=\alpha_{... | proof | Geometry | proof | Yes | Yes | olympiads | false | 11,649 |
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 | 11,653 |
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 | 11,655 |
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 | 11,656 |
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 | 11,657 |
6. Let's call a number slender if all the digits in its decimal representation are distinct and are in ascending order. Which are there more of: four-digit or five-digit slender numbers?
(V. P. Fedorov) | Solution. Let's match each four-digit number to a five-digit number consisting of all other non-zero digits in ascending order (for example, the number 1378 corresponds to 24569). Note that this results in a one-to-one correspondence, so there are an equal number of such numbers.
Criteria. Understanding that there is ... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,658 |
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 | 11,659 |
3. Vitya has a white board consisting of 16 cells in the form of a $4 \times 4$ square, from which he wants to cut out 4 white three-cell corners. Petya, on the other hand, wants to prevent him by coloring some cells red. What is the minimum number of cells he will have to color? (A corner is a figure shown in the pict... | Solution. If you cut out one cell (any) from the board, the remaining part can be divided into five corners. If you cut out another one, four of these five corners will remain. Therefore, one or two cut-out cells are not enough to prevent Vitya. The placement of three cells (for example, in a row along one of the main ... | 3 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,660 |
5. Does there exist a natural number $x$ such that among the statements " $x+1$ is divisible by 19 ", " $x+2$ is divisible by 18 ", " $x+3$ is divisible by 17 ", ..., " $x+17$ is divisible by 3 ", " $x+18$ is divisible by 2 ", exactly half are true? | Solution. Note that the conditions can be replaced by the following: « $x+20$ is divisible by 19 », « $x+20$ is divisible by 18 », « $x+20$ is divisible by 17 », … « $x+20$ is divisible by 3 », « $x+20$ is divisible by 2 ». Thus, $x+20$ must be divisible by half of the numbers $2,3, \ldots, 19$. For example, $x+20=3 \c... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 11,663 |
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 | 11,666 |
5. In a row, the squares of the first 2022 natural numbers are written: $1,4,9, \ldots, 4088484$. For each written number, except the first and the last, the arithmetic mean of its left and right neighbors was calculated and written below it (for example, under the number 4, $\frac{1+9}{2}=5$ was written). For the resu... | Solution. Let's look at an arbitrary number in the row $x^{2}$. Under it, it will be written
$$
\frac{(x-1)^{2}+(x+1)^{2}}{2}=\frac{x^{2}-2 x+1+x^{2}+2 x+1}{2}=\frac{2 x^{2}+2}{2}=x^{2}+1
$$
Thus, each time the number increases by one. Initially, there are 2022 numbers, and each time their count decreases by 2, so th... | 10231311025154 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,668 |
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 | 11,669 |
7. On an island of knights, who always tell the truth, and liars, who always lie, a school was opened. All $2 N$ students of different heights lined up in pairs (i.e., in two columns). The first two people said: "I am taller than two people: my partner and the person behind me." The last two said: "I am also taller tha... | Solution. A) In each pair, there is no more than one knight, so there are no more than $N$ knights (the example is achieved by placing $N$ taller students in a checkerboard pattern).
B) Since all students are of different heights, the tallest of them is definitely taller than their neighbors, so they are a knight, tha... | N | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,670 |
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