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# 20. Problem 20
How many employees can the company insure with the given insurance company if it is willing to pay an insurance premium of 5000000 rubles? The average cost of outpatient care (without hospitalization) for one person at a clinic, with which the insurance company has a contract, is 18000 rubles. The ave... | Answer in the form of a number shouldbewrittenwithoutspaces,withoutunitsofmeasurementandanycharacters. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,729 |
5.
Expenses for medical services provided to a child (under 18 years old) of the taxpayer by medical organizations
Correct answers: Pension contributions under a non-state pension agreement concluded by the taxpayer with a non-state pension fund in their own favor, Expenses for medical services provided to a child (... | Answer and write it in rubles as an integer without spaces and units of measurement.
Answer: $\qquad$
Correct answer: 16250
Question 12
Score: 5.00
Insert the missing terms from the drop-down list.
Under the insurance contract, one party
insured; insurance premium; beneficiary; insurer; insurance amount
undertak... | 16250 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,731 |
5.
the higher the risk of a financial instrument, the higher its return
Correct answers: the higher the reliability of a financial instrument, the higher its return, a financial instrument can be reliable, profitable, and liquid at the same time, risk is not related to the return of a financial instrument
Question ... | Answer write in rubles as an integer without spaces and units of measurement.
Answer: $\qquad$
Correct answer: 1378
Question 16
Score: 5.00
Establish the correspondence between specific taxes and their types.
| personal income tax | federal tax; local tax; regional tax; |
| :---: | :---: |
| land tax | federal ta... | 1378 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,732 |
# 20. Problem 20
On March 1 of the current year, an investor bought a share with a nominal value of 1 ruble for 100 rubles on the stock market, and on November 1 of the same year, sold it for 115 rubles. In April, a dividend of 5 rubles was paid on the share. Determine the real annualized return in percentage for the ... | Roundtothenearestwholenumber,recordwithoutspaces,unitsandanycharacters.
Translate the above text into English, please retain the source text's line breaks and format, and output the translation result directly. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,742 | |
# 15. Problem 15
What amount of mortgage credit in rubles will a bank client receive if their initial payment of 1800000 rubles amounted to $30 \%$ of the cost of the purchased property? | Answer in the form of a number should bewrittenwithoutspaces,withoutunitsofmeasurementandanycharacters.
## Answer: 4200000
# | 4200000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,743 |
# 15. Problem 15
Calculate the annual return received by the investor from a conservative investment portfolio. According to the data presented in the table, assume there are 360 days in a year.
| No. | Name of Securities | Number of Securities, units | Cost of Security at the Time of Purchase, RUB | Cost of Security... | Write the answer as a number without spaces, without units of measurement and any signs (if the answer is a fractional number, it must be written using a decimal fraction, with accuracy to the second digit after the decimal point, for example "2.17").
# | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,754 |
# 17. Problem 17
Commercial Bank "Zero" accrues interest on deposits with capitalization annually at a rate of $5.5\%$ per annum. The depositor has deposited 1,000,000 rubles into the bank. What will the deposit amount be after 3 years? | Answer in the form of a number shouldbewrittenwithoutspaces, withoutunits of measurement and any signs
(rounding if necessary to integer values)
# | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,756 | |
# 18. Problem 18
The owner of an apartment rents it out for five years and receives 240,000 rubles from tenants at the end of each year. The owner does not spend the rental income but saves it in a term deposit with an annual interest rate of $10\%$ (capitalization occurs once at the end of the year). What amount will... | Write the answer as a number without spaces, without units of measurement, and without any symbols (rounding to the nearest whole number if necessary).
# | 159383 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,757 |
# 19. Problem 19
The owner of an apartment insured it for 3,750,000 rubles (the actual cost of the apartment is 7,500,000 rubles). The actual damage amounted to 2,750,000 rubles. The insurance compensation under the system of proportional liability was 1,350,000 rubles. Determine the amount of the absolute deductible ... | Write the answer as a number without spaces, without units of measurement and any signs.
# | 1100000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,758 |
11. Correct. The coefficients for unlimited insurance are lower than when including drivers with less than 3 years of experience in the policy.
## Evaluation Criteria:
10-11 manifestations of financial illiteracy or financial literacy are listed and explained: 20 points
7-9 manifestations of financial illiteracy or ... | # Solution:
A) The maximum number of points that can be obtained - **8** points
The size of the annuity payment $=0.01(0.01+1)^{\wedge} 12 /\left(1.01^{\wedge} 12-1\right)=106618.57$ rubles.
The total amount of payments with annuity payments $=106618.57 * 12=1279$ 422.84 rubles. (4 points)
(A fully correct solution... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,760 |
# 16. Problem 16
Olga Alekseevna owns two apartments. The area of the first apartment is $60 \mathrm{~m}^{2}$, its cadastral value is 6 million rubles, and the current market value is 7 million rubles. The area of the second apartment is $42 \mathrm{~m}^{2}$, its cadastral value is 4.2 million rubles, and the current ... | Give the answer in percentages, but without the "%" sign; to separate the fractional part of the number, use a comma. For example, if your answer is 0.95%, write in the answer simply 0,95.
# | 0,95 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,763 |
# 18. Problem 18
In the country of Neverwell, the annual inflation rate was $1900 \%$. What real return would a deposit in the local currency in a local bank, placed at $100 \%$ per annum, bring to a citizen of Neverwell? | Give the answer in percentages, but without the "%" sign and with a plus sign if the answer is positive, or with a minus sign if the answer is negative. For example, if your answer is 15%, write +15 in your answer. If your answer is -5%, write -5 in your answer. | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,765 | |
5.
A depository stores securities, clients' money, and other material valuables
Question 7
Score: 7.00
Find the correspondence between the term and the statement so that all 5 pairs are correct. All 5 terms must be used.
The service may involve changing the terms of an existing loan
According to federal legislat... | Answer in rubles, without spaces and units of measurement. Round the answer to the nearest whole number according to rounding rules.
Answer:
Question 9
Score: 3.00
Select all possible features of an authentic ruble banknote.
Select one or more answers: | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,767 |
5.
watermark
Question 10
Score: 7.00
Vladimir has saved 16,500 rubles to buy a gaming console as a birthday gift for his brother, which amounts to 220 US dollars at the current exchange rate. The birthday is only a month away, and Vladimir is comparing three
Financial Literacy 11th Grade Day 1
alternatives: 1) ... | Answer in rubles, without spaces and units of measurement. Round the answer to the nearest whole number according to rounding rules.
Answer: $\qquad$
What services can currently be provided remotely? Select all appropriate options.
## \ulcorner | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,768 |
6.
when transferring money abroad
Question 17
Score: 7.00
Last year, a beauty salon offered a $20 \%$ discount on a facial massage when purchasing a membership for 30000 rubles. This year, it was decided to change the loyalty program, and when purchasing a membership for 30000 rubles, an additional $20 \%$ of this... | Answer:
Question 18
Score: 3.00
Select all correct statements regarding digital financial assets $(DFA)$.
Select one or more answers: | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,770 |
5.
The Bank of Russia will ensure the conversion of the "digital ruble" into another form of money (rubles) at a one-to-one ratio.
Question 19
Score: 7.00
Angelika owns a commercial property with an area of 30 square meters. She wanted to organize a music club there and give piano lessons. However, a friend offere... | Provide the answer in rubles, without spaces and units of measurement. Round the answer to the nearest whole number.
Answer:
Question 20
Score: 7.00
Financial Literacy 11th Grade Day 1
Ivan and Petr, twin brothers, went on vacation to the sea together and purchased two different travel insurance policies, which al... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,771 |
5.
reducing transaction time
## Correct answers:
using licensed software, using a personal computer instead of a public one, using antivirus programs
Question 3
Balya: 7.00
Mr. Vshokoladov earned X rubles per month throughout 2021. In addition, during this year, he won 2000000 rubles in a lottery. What is $X$ if... | Answer in rubles, without spaces and units of measurement.
Answer:
Correct answer: 600000
Question 4
Score: 3.00
Select all correct continuations of the statement.
2022 Higher Trial - qualifying stage
To file a petition to recognize a citizen as bankrupt...
## Select one or more answers:
\ulcorner | 600000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,772 |
5.
A custodian stores securities, clients' money, and other material assets
Correct answers: A paid investment advisor consults and provides recommendations to the client on investment management, A trustee manages the client's property in their own name
Find the correspondence between the term and the statement so... | Answer in rubles, without spaces and units of measurement. Round the answer to the nearest whole number according to rounding rules.
Answer:
Correct answer: 13806
question 9
Score: 3.00
Select all possible features of an authentic ruble banknote.
Select one or more answers:
$\Gamma$ | 13806 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,773 |
5.
watermark
Correct answers: raised relief of certain text fragments, watermark, inscriptions and ornaments
Question 10
Score: 7.00
Vladimir has saved 16,500 rubles to buy a gaming console as a birthday gift for his brother, which amounts to 220 US dollars at the current exchange rate. The birthday is not until ... | Answer in rubles, without spaces and units of measurement. Round the answer to the nearest whole number according to rounding rules.
Answer: $\qquad$
Correct answer: 76
Question 11
Score: 3.00
What services can currently be provided remotely? Select all appropriate options. | 76 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,774 |
5.
the driver's marital status
Correct answers: bonus-malus coefficient, engine power, driver's age
Question 14
Score: 7.00
Maria Ivanovna has retired. She did not have a funded pension, only a social insurance pension, and her individual pension coefficient amount is 120. In addition, Maria Ivanovna has a bank d... | Provide the answer in rubles, without spaces and units of measurement.
Answer:
The correct answer is: 19930
Question 15
Score: 7.00
Insert the missing words from the list below (not all provided words will be needed!):
Paying
credit; preferential; higher; cash withdrawal; service; blocking; bonus; debit; freeze;... | 19930 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,775 |
6.
when transferring money abroad
Correct answers: when using the Fast Payment System for amounts up to 100 thousand rubles per month, when transferring funds between one's own accounts in the same bank
Question 17
Score: 7.00
Last year, a beauty salon offered a $20 \%$ discount on facial massage when purchasing ... | Answer:
The correct answer: 1
Question 18
Score: 3.00
Select all true statements regarding digital financial assets $(DFA)$.
Select one or more answers: | 1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,776 |
5.
The Bank of Russia will ensure the conversion of the "digital ruble" into another form of money (rubles) at a one-to-one ratio.
Correct answers: Stablecoins, backed by cash or gold, are an example of a CBDC., The Bank of Russia will ensure the conversion of the "digital ruble" into another form of money (rubles) ... | Provide the answer in rubles, without spaces and units of measurement. Round the answer to the nearest whole number.
Answer: $\qquad$
Correct answer: 1321412
Question 20
Score: 7.00
Financial Literacy 11th Grade Day 1
Ivan and Petr, twin brothers, went on a vacation by the sea together and purchased two different ... | 1321412 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,777 |
1. Over three years, Marina did not invest funds from her Individual Investment Account (IIS) into financial instruments and therefore did not receive any income from the IIS. However, she still acquired the right to an investment tax deduction for contributing personal funds to the IIS (Article 291.1 of the Tax Code o... | Answer: $\mathbf{3 , 5 5 \%}$ | 3.55 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,782 |
Task 1. How not to lose money? (15 points).
Alexander Ivanovich has 300,000 rubles, for which he is considering three possible investment options: investing in his neighbor's business, buying a security, and purchasing a plot of land on the Moon. Each investment option will either bring a 20% profit (50 chances out of... | Solution.
a) The chances of incurring a loss when investing in a security are 50 out of 100 (or, in other words, one chance in 2), which directly follows from the condition.
b) Let's consider all possible combinations of options:
| Combination | Security | Neighbor's Business | Investment Result |
| :--- | :--- | :-... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,784 |
# 15. Problem 15
Full score -8
Grisha owns a room with an area of 9 m $^{2}$ in a communal apartment (its cadastral value is 1 million rubles, and the current market value is 1.5 million rubles), as well as a residential house with an area of $90 \mathrm{~m}^{2}$ (the cadastral value of the house is 1.8 million ruble... | Give the answer in rubles and write it without spaces, units of measurement, and any signs.
# | 2700 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,786 |
# 16. Problem 16
Full score - 8
Dima wants to buy a car on the secondary market. To find a car and check its technical characteristics, he needs to spend three working days, taking leave at his own expense. If he buys a car without checking, he will have to spend approximately $20 \%$ of the car's cost on repairs. Di... | Give the answer in rubles and write it without spaces, units of measurement, and any signs.
# | 140000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,787 |
# 19. Problem 19
Full score -8
In the "turbulent 90s," Grandpa Zakhar deposited money in the bank for a year at an annual interest rate of $200\%$ and received a negative real return of $-80\%$ on this deposit. Calculate the inflation rate for that year? | Answer in percentages and write without spaces or any signs (without the sign "\%"). For example, if your answer is $15 \%$, write in the answer simply 15.
## 20. Problem 20
Full score - 8
Igor works at a bank and earns 500000 rubles per year. He opened an Individual Investment Account (IIS) with a special tax regim... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,790 |
5.
the higher the risk of a financial instrument, the higher its return
Correct answers: the higher the reliability of a financial instrument, the higher its return, a financial instrument can be reliable, profitable, and liquid at the same time, risk is not related to the return of a financial instrument
Question ... | Answer write in rubles as an integer without spaces and units of measurement.
Answer: $\qquad$
Correct answer: 1378
Question 16
Score: 5.00
Establish the correspondence between specific taxes and their types.
| personal income tax | federal tax; local tax; regional tax; |
| :---: | :---: |
| land tax | federal ta... | 1378 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,791 |
# 15. Problem 15
Calculate the annual return received by an investor from a conservative investment portfolio. According to the data presented in the table, assume there are 360 days in a year.
| No. | Name of Securities | Number of Securities, units | Cost of Security at the Time of Purchase, RUB | Cost of Security ... | Answer in the form of a number should be written without spaces, without units of measurement and any signs (if the answer is a fractional number, it should be written using a decimal fraction, with accuracy to the second digit after the decimal point, for example "2.17").
Answer: 9.96
# | 9.96 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,792 |
2. Take out a mortgage for the missing amount from the bank for 2 years at an annual rate of $13\%$ with interest payments at the end of each year and the principal repayment in equal installments also at the end of each year. At the beginning of 2021, buy an apartment and rent it to friends for 34,000 rubles per month... | # Solution and Evaluation Criteria:
Translate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.
However, it seems there was a misunderstanding in your request. Here is the translation of the provided text:
# Solution and Evaluation ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,793 |
4. (7 points) The inflation index for the deposit period (year) will be $\mathrm{j}=\left(1+\mathrm{i}_{1}\right) \times\left(1+\mathrm{i}_{2}\right)$
Where: $\mathrm{i}_{1}$ and $\mathrm{i}_{2}$ are the inflation rates for the first and second halves of the deposit period, respectively, in fractions;
$\mathrm{j}=\le... | Answer: 1. - 350600 rubles, 2. Sasha - 268003 rubles, Kolya - 267875 rubles, 3. Sasha has 128 rubles more, 4. Sasha's real income - 8865 rubles. Kolya's real income - 8741 rubles.
## Checking criteria:
Point: | Sasha'\real\income\-\8865\rubles,\Kolya'\real\income\-\8741\rubles | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,798 |
1. Express \( z \) from the second equation and substitute into the first:
\[
\begin{aligned}
& z=a\left(x+2 y+\frac{5}{2}\right) \rightarrow x^{2}+y^{2}+2 x-y+a\left(x+2 y+\frac{5}{2}\right)=0 \rightarrow \\
& \rightarrow\left(x+\frac{a+2}{2}\right)^{2}+\left(y+\frac{2 a-1}{2}\right)^{2}=\frac{(a+2)^{2}}{4}+\frac{(2 ... | Answer: 1) $a=1$; 2) $x=-\frac{3}{2}, y=-\frac{1}{2} ; z=0$ | 1,-\frac{3}{2},-\frac{1}{2},0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,799 |
Task 1. Dad and Mom are cooking pancakes on the stove, while their children Petya and Vasya are eating them. Dad can cook 70 pancakes in an hour, and Mom can cook 100 pancakes. Petya, if he tries hard, can eat 10 pancakes in 15 minutes, and Vasya can eat twice as much. After how much time will there be no less than 20 ... | Answer: 24 min
Solution. Let the baking rate of pancakes be $p_{1}=\frac{170}{60}$ pancakes/min, and the eating rate of pancakes be $p_{2}=\frac{10+20}{15}=2$ pancakes/min.
In $k$ minutes, there will be no less than $q=k\left(p_{1}-p_{2}\right)$ pancakes left on the table: $k \cdot \frac{5}{6} \geq 20 \rightarrow k \... | 24 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,801 |
Problem 2. In a square of grid paper containing an integer number of cells, a hole in the shape of a square, also consisting of an integer number of cells, was cut out. How many cells did the large square contain if 209 cells remained after the cutout? | Answer: 225 cells
Solution. The side of the larger square contains $n$ sides of a cell, and the side of the smaller square contains $m$ sides of a cell. Then $n^{2}-m^{2}=209 \rightarrow(n-m)(n+m)=209=11 \cdot 19$.
Case 1. $\left\{\begin{array}{c}n+m=209 \\ n-m=1\end{array} \rightarrow\left\{\begin{array}{c}n=105 \\ ... | 225 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,802 |
Problem 3. A line passing through any point $D$ on the median $B M$ of triangle $A B C$ and parallel to side $A B$, intersects the line passing through vertex $C$ and parallel to $B M$, at point $E$. Prove that $B E=A D$. | Solution. Additional construction: through point $M$ we draw line $MN$, parallel to $DE$. Figure $MDEN$ is a parallelogram (by construction).
Triangles $ABM$ and $MNC$ are equal (by two equa... | proof | Geometry | proof | Yes | Yes | olympiads | false | 2,803 |
1. For what values of $x$ does the expression $\cos ^{2}(\pi \cos x)+\sin ^{2}(2 \pi \sqrt{3} \sin x)$ take its minimum value | 1. Answer: $x= \pm \frac{\pi}{3}+\pi t, t \in Z$. Solution. Let's check if the sine and cosine in the given expression can simultaneously be zero for some values of $x$. If so, the expression takes its minimum value at these values of $x$.
$$
\left\{\begin{array} { c }
{ \operatorname { c o s } ( \pi \operatorname { ... | \\frac{\pi}{3}+\pi,\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,804 |
2. To fight against the mice, the cat Leopold must drink animalin daily. The cat has a bottle of animalin solution in water (a colorless transparent liquid) with a volume of $V=0.5$ l and a volumetric concentration of animalin $C_{0}=40 \%$. Every day, Leopold drinks $v=50$ ml of the solution, and to avoid being notice... | 2. Let's find the change in the concentration of the absinthe solution after the cat drinks one portion. By definition of volume concentration,
$$
C_{0}=\frac{v_{O}}{V}
$$
where $C_{0}$ is the volume concentration of the solution, $v_{o}$ is the initial volume of absinthe in the bottle, and $V$ is the total volume of... | 23.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,805 |
3. A cyclic process $1-2-3-4-1$ is carried out with an ideal gas, consisting of two isotherms (2-3 and 4-1) and two isochors ($1-2$ and $3-4$; see figure). It is known that the ratio of temperatures on the isotherms 2-3 and 4-1 is $T_{2-3} / T_{4-1}=3$, and that during the isothermal expansion, the gas received twice a... | 3. Let the gas receive an amount of heat $Q$ during the isothermal expansion segment. Then, according to the condition, the gas receives an amount of heat $Q / 2$ during the isochoric heating segment. Therefore, the total amount of heat received by the gas from the heater during the cycle is
$$
Q_{n}=\frac{3}{2} Q
$$
... | \frac{4}{9} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,806 |
4. The electrical circuit shown in the figure contains a very large number of identical resistors and identical voltmeters. It is known that the readings of voltmeters $V_{1}$ and $V_{2}$ are $U=4$ V and $U^{\prime}=6$ V (although it is unknown which reading corresponds to which voltmeter). Find the readings of voltmet... | 4. Since the current flowing through the voltmeter $V_{1}$ is greater than the current flowing through the voltmeter $V_{2}$, the resistances of the voltmeters are the same, and each voltmeter shows the voltage across it, the value $U^{\prime}=6$ V is the reading of the voltmeter $V_{1}$, $U=4$ V is the reading of the ... | 15\mathrm{~V} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,807 |
5. An integer N is given as input. The task is to write a program that finds a decomposition of the number N into a sum of squares of natural numbers, such that the number of terms in this decomposition is minimal. For example, N=11:
\[
\begin{aligned}
& 11=3^{2}+1^{2}+1^{2} \\
& 11=2^{2}+2^{2}+1^{2}+1^{2}+1^{2} \\
& ... | # 5.
\{pascal $\}$
program task 11 ;
var
$\mathrm{N}:$ integer;
m_len:array[1..10000] of integer; //array for recording the length of the minimum decomposition
m_val:array[1..10000] of integer; //array for saving the addends of the minimum decomposition
//declare the recursive function for calculating the result... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,808 |
1. (mathematics) The probabilities of finding Kuzia the flea at vertices $A, B, C$ and $D$ of the square $A B C D$ are $p_{1}, p_{2}, p_{3}$ and $p_{4}$ respectively $\left(p_{1}+p_{2}+p_{3}+p_{4}=1\right)$. The flea jumps to the nearest vertex of the square, and which of the two it jumps to is completely random and eq... | Solution. Let $P_{A}(n), P_{B}(n), P_{C}(n), P_{D}(n)$ be the probabilities of finding the flea at vertices $A, B, C$, and $D$ after $n$ jumps. Then
$$
P_{A}(1)=\frac{1}{2}\left(p_{2}+p_{4}\right)=P_{C}(1), P_{B}(1)=\frac{1}{2}\left(p_{1}+p_{3}\right)=P_{D}(1)
$$
Figure 1 shows the probabilities of events after the f... | \frac{1}{2}(p_{1}+p_{3}) | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,809 |
2. (mathematics) Thirteen circles of the same radius $r=2 \sqrt{2-\sqrt{3}}$ are cut out of colored paper and laid out on a plane so that their centers lie on one straight line. The distance between the centers of adjacent circles is the same and equals 2. Find the perimeter of the figure on the plane formed by these c... | Answer: $P=44 \pi \sqrt{2-\sqrt{3}}$
## Solution:
The circles intersect since $r>1$. In the figure, three of the thirteen circles are depicted.
$$
A C=2, A B=B C=r=2 \sqrt{2-\sqrt{3}}, \squ... | 44\pi\sqrt{2-\sqrt{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,810 |
3. (physics) A wire is made into a right-angled triangle with legs of length $l=1$ m. From vertices A and B, two bugs start crawling along the legs simultaneously with speeds $v_{A}=5$ cm/s and $v_{B}=2 v_{A}=10$ cm/s (see figure). What is the minimum time after the start of their movement when the distance between the... | Solution. Let the time that has passed since the beetles started moving be $t$. Then beetle
A will travel a distance of $v_{A} t$, beetle B - a distance of $v_{B} t=2 v_{A} t$. Therefore, the distance $S$ between the beetles at this moment will be
$$
S=\sqrt{\left(l-v_{A} t\right)^{2}+4 v_{A}^{2} t^{2}}
$$
We will f... | 4\mathrm{~} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 2,811 |
5. (informatics) In a square matrix of size $N \times N$, all cells are filled with numbers from 1 to 5. A connected component in the matrix is defined as a set of cells that are filled with the same number, and between any two cells in the set, a path can be constructed. Cells can only be connected in a path if they a... | # Solution.
program prog11;
var n,i,j,k,l:integer;
a,b:array[0..11,0..11] of integer;
m,mc,mmax:integer;
begin
{initialize the array}
for i:=0 to 11 do
for j:=0 to 11 do
```
a[i,j]:=0;
{read input data}
readln(N);
for i:=1 to N do
begin
for j:=1 to N do
read(a[i,j]);
readln;
end;
mmax:=0;{c... | 12 | Other | other | Yes | Yes | olympiads | false | 2,813 |
5. The star "tau" in the constellation Cetus has a planetary system. On the third planet from the star, there is a very unusual gravity: the acceleration due to gravity is $g=10 \mathrm{~m} / \mathrm{c}^{2}$ up to a height of $h=10$ m from the surface, but above that, it is half as much. An astronaut throws a body stra... | 5. Since the height of the body's rise on Earth is $H$, the initial velocity of the thrown body is
$$
v_{0}=\sqrt{2 g H}
$$
On the planet from the "tau Ceti" system, the body will move to a height $h$ with acceleration $g$, and then with acceleration $g / 2$. Therefore, above this boundary, the body will rise to a he... | 30\mathrm{~} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,818 |
6. A sphere of radius $R$, made of a material with a density equal to that of water, floats in water, touching its surface (see figure). Find the force with which the water acts on the lower half of the sphere. The density of water $\rho$ is known.
 There are scales with two pans, 4 weights of 2 kg each, 3 weights of 3 kg each, and two weights of 5 kg each. In how many different ways can a 12 kg load be balanced on the scales, if the weights are allowed to be placed on both pans? | Answer: 7 ways
Solution: Let $x$ be the number of 2 kg weights used in weighing, $y$ be the number of 3 kg weights, and $z$ be the number of 5 kg weights. Then the equilibrium condition is g... | 7 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,820 |
2. (mathematics) Segments $A B$ and $C D$, with lengths 3 and 4 respectively, are chords of a circle with radius 5. Points $M$ and $N$ are located on them such that $A M: M B=1: 2, \quad C N: N D=1: 3$. Find the smallest and largest possible lengths of segment $M N$. | Answer: $d_{\min }=\sqrt{23}-\sqrt{22} \quad d_{\max }=\sqrt{23}+\sqrt{22}$
Solution. The lengths of segments $A B=a, C D=b, \quad A M: M B=p: q$ $C N: N D=m: n$. Points $P, Q$ are the midpoints of segments $A B$ and $D C$. We will prove that points $M$ and $N$ lie on two concentric circles for any position of segment... | d_{\}=\sqrt{23}-\sqrt{22}\quadd_{\max}=\sqrt{23}+\sqrt{22} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,821 |
3. (physics) In a vessel, there is a mixture of equal masses of nitrogen $N_{2}$ and helium He at a pressure $p$. The absolute temperature of the gas is doubled, during which all nitrogen molecules dissociate into atoms. Find the pressure of the gas mixture at this temperature. The molar masses of the gases are $\mu_{\... | Solution. Let the mass of helium and nitrogen in the vessel be $m$. Then Dalton's law for the initial gas mixture gives
$$
p=\frac{v_{\mathrm{He}} R T}{V}+\frac{v_{N_{2}} R T}{V}
$$
where
$$
v_{N_{2}}=\frac{m}{\mu_{N_{2}}}=v ; \quad v_{\mathrm{He}}=\frac{m}{\mu_{\mathrm{He}}}=7 v
$$
are the amounts of substance of ... | \frac{9}{4}p | Other | math-word-problem | Yes | Yes | olympiads | false | 2,822 |
4. (physics) Two bodies with masses $m$ and $2 m$ are connected by ropes. One rope is thrown over a pulley, the axis of which is attached to the ceiling. Another rope has the same pulley placed on it, to the axis of which a body with mass $4 m$ is attached (see figure). Find the tension force in the rope connecting the... | Solution. Since the blocks do not move (only rotate), the acceleration of the body with mass $4 m$ is zero. Therefore, the tension force in the lower rope connecting the bodies $m$ and $2 m$ is
$$
T_{u}=2 \mathrm{mg}
$$
Newton's second law for the bodies with masses $m$ and $2 m$ in projections on the axis directed v... | \frac{10}{3} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,823 |
5. A sphere of radius $R$, made of a material with a density equal to that of water, floats in water, touching its surface (see figure). Find the force with which the water acts on the lower half of the sphere. The density of water $\rho$ is known.
. By what factor does the maximum resistance of the square differ from the ... | 6. Let's find the resistance of a square for some arrangement of contacts and investigate this value for maximum and minimum. Obviously, for any arrangement of contacts on the square, the sum of the resistances of its upper and lower branches is equal to the resistance of the wire from which it is made, i.e.,
. At some moment, the ring and the body were given velocities $v$ and $2 v$, directed horizontally along... | 6. At the moment of rising to the maximum height, the body moves horizontally. Therefore, from the condition of the inextensibility of the rod, we conclude that the speed is the same (their projections on the direction of the rod should be the same, and the speeds of the body and the ring at the moment of the body's ri... | \frac{9v^{2}}{4(1-\sin\alpha)} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,840 |
1. Subtract the second equation from the first:
$x^{2}-2 x+y^{4}-8 y^{3}+24 y^{2}-32 y=-17 \rightarrow(x-1)^{2}+(y-2)^{4}=0 \rightarrow\left\{\begin{array}{l}x=1 \\ y=2\end{array}\right.$
Then $z=x^{2}+y^{4}-8 y^{3}=1+16-64=-47$ | Answer: the only solution is $x=1, y=2, z=-47$. | -47 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,841 |
1. Find all integer values of $x$ that are solutions to the equation $\sin (\pi(2 x-1))=\cos (\pi x / 2)$. | 1. Answer: $x=4 t \pm 1, t \in Z$. Solution.
$\sin (\pi(2 x-1))=\sin (\pi(1-x) / 2) \rightarrow\left\{\begin{array}{l}2 x-1=(1-x) / 2+2 m \rightarrow 5 x / 2-2 m=3 / 2 \\ 2 x-1=(x+1) / 2+2 n \rightarrow 3 x / 2-2 n=3 / 2\end{array}\right.$
The equation $5 x-4 m=3$ in integers has the following solutions $\left\{\begi... | 4\1,\inZ | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,842 |
3. A sealed horizontal cylindrical vessel of length $l$ is divided into two parts by a movable partition. On one side of the partition, there is 1 mole of oxygen, and on the other side, there is 1 mole of helium and 1 mole of oxygen, with the partition being in equilibrium. At some point in time, the partition becomes ... | 3. From the condition of equilibrium of the partition (equality of pressures to the right and left of it), we find that initially it is located at a distance of $2 l / 3$ and $l/3$ from the ends of the vessel. After helium is uniformly distributed throughout the vessel, its partial pressures to the right and left of th... | \frac{}{6} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,843 |
4. A small weightless ring is hinged to a light rod of length $2 l$. At the midpoint and end of the rod, point masses of $m$ and $2 m$ are fixed. The ring is placed on a smooth horizontal rod. Initially, the rod was held horizontally and then released. Find the speed of the ring at the moment when the rod passes the ve... | 4. Since the external forces on the system of bodies and the spoke act only in the vertical direction (the gravitational forces of the bodies and the reaction force of the rod), the projection of the momentum of the system of bodies on the horizontal direction is conserved. Therefore, if the lower body has a speed $v$ ... | \sqrt{\frac{125}{3}} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,844 |
1. Express z from the first equation and substitute into the second:
$x^{2}-2 x+y^{2}-2 \sqrt{3} y=-4 \rightarrow(x-1)^{2}+(y-\sqrt{3})^{2}=0 \rightarrow\left\{\begin{array}{c}x=1 \\ y=\sqrt{3}\end{array} \rightarrow z=x^{2}+y^{2}+2 x=6\right.$ | Answer: $x=1, y=\sqrt{3}, z=6$ | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,846 |
Task 1. The administration divided the region into several districts based on the principle: the population of a large district exceeds $8 \%$ of the region's population and for any large district, there are two non-large districts with a combined population that is larger. Into what minimum number of districts was the... | Answer: 8 districts.
Solution. The number of "small" districts is no less than 2 according to the condition, and their population does not exceed $8 \%$ of the total population of the region. We will show that the number of districts in the region is no less than 8. If the number of districts in the region is no more ... | 8 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,847 |
Problem 2. The number $x$ satisfies the condition $\frac{\sin 3 x}{\sin x}=\frac{5}{3}$. Find the value of the expression $\frac{\cos 5 x}{\cos x}$ for such $x$ | Answer: $-\frac{11}{9}$.
Solution.
$$
\begin{aligned}
& \sin x \neq 0 \rightarrow \frac{\sin 3 x}{\sin x}=\frac{3 \sin x-4 \sin ^{3} x}{\sin x}=3-4 \sin ^{2} x \rightarrow 3-4 \sin ^{2} x=\frac{5}{3} \rightarrow \sin ^{2} x=\frac{1}{3} \rightarrow \\
& \rightarrow \cos ^{2} x=\frac{2}{3} \rightarrow \cos 2 x=\frac{1}... | -\frac{11}{9} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,848 |
Problem 3. Point $M$ is located on side $CD$ of a square such that $CM: MD=1: 3$. Line $AM$ intersects the circumcircle of the square at point $E$. The area of triangle $ACE$ is 14. Find the side length of the square. | Answer: 10.
Solution.
Triangles $A M D$ and $C M E$ are similar with a similarity coefficient $k=5$. Then
$$
C E=\frac{4 x}{5}, M E=\frac{3 x}{5} \rightarrow A E=5 x+\frac{3 x}{5}=\frac{28... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,849 |
Problem 1. Seven students in the class receive one two every two days of study, and nine other students receive one two every three days each. The rest of the students in the class never receive twos. From Monday to Friday, 30 new twos appeared in the journal. How many new twos will appear in the class journal on Satur... | Answer: 9
Solution. Over the period from Monday to Saturday (six days), in the journal, there will be 3 new twos from each student of the first group (seven people) and 2 new twos from each of the 9 students of the second group. The total number of new twos for the school week is $7 \cdot 3 + 9 \cdot 2 = 39$. Then, on... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 2,850 |
Problem 2. Prove that for any positive numbers $a$ and $b$ the following inequality holds
$$
\sqrt{\frac{a^{2}}{b}}+\sqrt{\frac{b^{2}}{a}} \geq \sqrt{a}+\sqrt{b}
$$ | Solution. Let $b \geq a$. Divide the inequality by $\sqrt{a}$:
$$
\begin{aligned}
& \sqrt{\frac{a}{b}}+\sqrt{\frac{b^{2}}{a^{2}}} \geq 1+\sqrt{\frac{b}{a}} \rightarrow \sqrt{\frac{b}{a}}\left(\sqrt{\frac{b}{a}}-1\right) \geq 1-\sqrt{\frac{a}{b}} \rightarrow \sqrt{\frac{b}{a}} \cdot \frac{1}{\sqrt{a}}(\sqrt{b}-\sqrt{a}... | proof | Inequalities | proof | Yes | Yes | olympiads | false | 2,851 |
Problem 3. Through the vertex $B$ of an equilateral triangle $ABC$, a line $L$ is drawn, intersecting the extension of side $AC$ beyond point $C$. On line $L$, segments $BM$ and $BN$ are laid out, each equal in length to the side of triangle $ABC$. The lines $MC$ and $NA$ intersect at a common point $D$ and intersect t... | # Answer: 1.
Solution. Triangles $M B C$ and $A B N$ are isosceles, therefore
$$
\begin{gathered}
\angle B M C = \angle B C M = \alpha \rightarrow \angle N B C = 2 \alpha, \angle B A N = \angle B N A = \beta \rightarrow \angle A B M = 2 \beta \\
2 \alpha + 2 \beta + 60^{\circ} = 180^{\circ} \rightarrow \alpha + \beta... | 1 | Geometry | proof | Yes | Yes | olympiads | false | 2,852 |
1. (mathematics) Find the values of $a$ for which the coordinates ( $x ; y$ ) of any point in the square $2 \leq x \leq 3,3 \leq y \leq 4$ satisfy the inequality $(3 x-2 y-a)\left(3 x-2 y-a^{2}\right) \leq 0$. | Answer: $a \in(-\infty ;-4]$
Solution: Points with coordinates $(x ; y)$ of the square $2 \leq x \leq 3, 3 \leq y \leq 4 A B C D$ lie in the strip
$$
(3 x-2 y-a)\left(3 x-2 y-a^{2}\right) \leq 0
$$
between the lines $L_{1}: 3 x-2 y-a=0$ and $L_{2}: 3 x-2 y-a^{2}=0$ (see figure), if this strip contains all the vertic... | \in(-\infty;-4] | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 2,856 |
2. (mathematics) An arbitrary point $E$ inside the square $A B C D$ with side 1 is connected to its vertices by segments. The points $P, Q, F$ and $T$ are the points of intersection of the medians of triangles $B C E, C D E, D A E$ and $A B E$ respectively. Find the area of quadrilateral PQFT. | Answer: $S=\frac{2}{9}$
Solution: Let's introduce the following notations (see the figure): $K, L$ - midpoints of segments $A E$ and $E C, G$ - midpoint of segment $B E$. $K L$ is parallel to the diagonal $A C$ and is equal in length to half of it. Triangles $B P T$ and $B L K$ are similar with a similarity coefficien... | \frac{2}{9} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,857 |
3. (physics) Two weightless glasses with different amounts of water are balanced on a lever of unequal-arm scales with a length of $L$. A mass of water $m$ is transferred from one glass to the other, and the scales lose their balance. To restore balance, the fulcrum of the lever needs to be moved by a distance $l$ rela... | Solution. Let the mass of water in the left glass be $m_{1}$, in the right glass be $m_{2}$, and the lever arms of the glasses be $l_{1}$ and $L-l_{1}$, respectively. Then the condition
for e... | m_{1}+m_{2}=\frac{L}{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,858 |
4. (physics) From a piece of metal with mass $M$, a rectangular parallelepiped is cast. Then, a source of constant voltage is connected to different pairs of mutually opposite faces of the parallelepiped. It turned out that the ratio of the thermal powers dissipated in the parallelepiped when the source is connected to... | Solution. Let the edge lengths of the parallelepiped be $a$ (the smallest), $b$ (the middle), and $c$ (the largest). Then the resistance of the parallelepiped when electrical contacts are connected to the faces with sides $a$ and $b$ is
$$
R_{a b}=\frac{\rho_{0} a b}{c}
$$
where $\rho_{0}$ is the resistivity of the m... | =(\frac{M}{4\rho})^{1/3},b=\sqrt{2}(\frac{M}{4\rho})^{1/3},=2\sqrt{2}(\frac{M}{4\rho})^{1/3} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,859 |
5. (informatics) On a chessboard, a white queen and a black pawn are positioned, given by coordinate pairs ( $R_{1}, C_{1}$ ) and ( $R_{2}, C_{2}$ ) respectively, where $R$ is the row number, and $C$ is the column number (as marked in the figure below). The task is to check if the queen can attack the pawn. The example... | # Solution.
program prog9;
var r1,c1,r2,c2:integer;
begin
{read input data}
readln(r1,c1,r2,c2);
if $(r 1=\mathrm{r} 2)$ {check horizontal}
or ( $\mathbf{c} 1=\mathrm{c} 2)\{$ check vertical $\}$
or $((\mathrm{r} 1+\mathrm{c} 1)=(\mathrm{r} 2+\mathrm{c} 2))\{$ check diagonal\}
or $((\mathrm{r} 1-\mathrm{c} 1)=(... | notfound | Other | other | Yes | Yes | olympiads | false | 2,860 |
1.
Notations:
$C$ - the point on the highway where Kolya got off his bicycle, $A C=10 t$
$D$ - the point on the highway where Vasya was at time $t, A D=5 t$
$E$ - the point where Petya pic... | Answer: 1) $t=\frac{16}{13}$ hour; 2) $T_{\min }=\frac{38}{13}$ hour | \frac{38}{13} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,861 |
5. Two loads with masses $m$ and $2 m$ are connected by three threads as shown in the figure. The thread $\mathrm{AB}$ is horizontal, and the thread CD forms an angle $\alpha$ with the vertical. Find the tensions in the threads AB and CD. The thread AB is cut. Find the accelerations of the loads immediately after this.... | 5. First, let's find the forces acting in all threads in equilibrium. On the body of mass \( m \), the force of gravity \( m \vec{g} \) and three tension forces of the threads \( \vec{T}_{A B}, \vec{T}_{C D} \), and the tension force of the lower thread, equal to \( 2 m \vec{g} \) (see the figure). Therefore, the equil... | a_{}=3\sin\alpha,\quada_{2}=0 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,868 |
6. A body was thrown from the surface of the earth at some angle to the horizon. It is known that the velocity vector of the body is directed at an angle $\alpha=30^{\circ}$ to the horizon at times $t_{1}$ and $t_{2}$ after the throw. Find the maximum height of the body's rise above the ground and the distance from the... | 6. Since the trajectory of the body is symmetric with respect to the highest point, the time of ascent $t_{\text {pod }}$ of the body to the maximum height lies exactly halfway between the moments $t_{1}$ and $t_{2}$:
$$
t_{\text {pod }}=\frac{t_{2}+t_{1}}{2}
$$
and the total time of motion $t_{\text {poli }}$ is twi... | S=\frac{\sqrt{3}}{2}(t_{2}^{2}-t_{1}^{2}),\quad=\frac{(t_{2}+t_{1})^{2}}{8} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,869 |
6. At the ends of a weightless, inextensible rod, a small body and a massive ring of the same mass are attached. The ring can move without friction along a horizontal rigid spoke in a gravitational field (see figure). At some moment, the ring and the body were given velocities $v$ and $2 v$, directed horizontally along... | 6. At the moment of reaching the maximum height, the body moves horizontally. Therefore, from the condition of the inextensibility of the rod, we conclude that the speed is the same (their projections on the direction of the rod should be the same, and the speeds of the body and the ring at the moment the body reaches ... | \frac{9v^{2}}{4(1-\sin\alpha)} | Other | math-word-problem | Yes | Yes | olympiads | false | 2,875 |
1. Two circles touch each other and the sides of two adjacent angles, one of which is $60^{\circ}$. Find the ratio of the radii of the circles. | 1. Answer: $r: R=1: 3$ Solution. The centers of the circles lie on the bisectors $A K$ and $A N$ of the adjacent angles,
$\alpha=\square B A C=60^{\circ}$
$\square K A N=90^{\circ}, \square C A K=\square A Q O=30^{\circ}, A M=h$. $r=h \tan \frac{\alpha}{2}, R=h \cot \frac{\alpha}{2} \rightarrow r: R=\tan^2 \frac{\alp... | r:R=1:3 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,876 |
2. In a garden partnership, they decided to make a pond using water from a spring. All the water from the spring was directed into the pond using a pipeline, and the pond began to fill with water. As a result, after \( t = 16 \) days, the pond was filled to \( 2 / 3 \) of its volume. At this moment, it was noticed that... | 2. Let the volume of the pond be $V$, and the volume of water flowing out of the source per unit time be $w$. Since before the leak was sealed, three-quarters of the water from the spring entered the pond, we have
$$
\frac{3}{4} w t = \frac{2}{3} V \quad \Rightarrow \quad \frac{9}{8} t = \frac{V}{w}
$$
![](https://cd... | t_1=\frac{3}{8} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,877 |
3. Two people simultaneously step onto an escalator from opposite ends, which is moving downward at a speed of $u=1.5 \mathrm{~m} / \mathrm{s}$. The person moving downward has a speed of $v=3 \mathrm{~m} / \mathrm{s}$ relative to the escalator, while the person moving upward has a speed of $2 v / 3$ relative to the esc... | 3. The speeds of the people relative to the ground are
$$
\text { descending } v_{c n}=v+u \text {, ascending } v_{\text {nоо }}=\frac{2}{3} v-u \text {. }
$$
Therefore, the time it takes for the people to meet is
$$
t=\frac{l}{v_{c n}+v_{n o d}}=\frac{3 l}{5 v}
$$
And since the ascending person will travel a dista... | 10\mathrm{~} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,878 |
4. A body of mass $m$ is attached to one end of a weightless, inextensible string, the other end of which is attached to the ceiling. The string is passed over a pulley of mass $M$, which is concentrated on its axis. At the initial moment, the body is held and then released. Find the accelerations of the body and the p... | 4. It is clear that the bodies will not fall freely with the acceleration due to gravity, and the string will be taut. Indeed, if the pulley falls with an acceleration of $g$, then on the side where the body is located, such a length of the string is "stretched" that the acceleration of the body should equal $2g$, whic... | a_{1}=\frac{2(2+M)}{4+M},\quada_{2}=\frac{(2+M)}{4+M} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 2,879 |
5. Two segments on the co-
ordinate line are defined by the coordinates of their two endpoints each. The task is to write a program that calculates the length of the intersection of these se... | 5.
{pascal}
program task_9;
var
x1, x2, x3, x4 : double;
t1, t2: double;
begin
read(x1,x2,x3,x4); // read input data
// determine the larger of the start coordinates
t1 := x1;
if x3 > t1 then
t1 := x3;
// determine the smaller of the end coordinates
t2 := x2;
if x4 < t2 then
t2 := x4;
// output the res... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,880 |
7. [6 points] Find the number of pairs of integers $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \geqslant 110+x-5^{110} \\
y \leqslant \log _{5} x
\end{array}\right.
$$ | The answer should be presented in the form of an algebraic sum of no more than three terms.
## VARIANT 16 | notfound | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 2,883 |
7. [6 points] Find the number of pairs of integers $(x ; y)$ that satisfy the system of inequalities
$$
\left\{\begin{array}{l}
y \geqslant 70+x-4^{70} \\
y \leqslant \log _{4} x
\end{array}\right.
$$ | The answer should be presented in the form of an algebraic sum of no more than three terms.
Translate the text above into English, please keep the original text's line breaks and format, and output the translation result directly. | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 2,884 | |
11. Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$, and $AL$ be the angle bisector of triangle $ABC$. Denote by $P$ the point of intersection of the extension of the altitude $BH$ of triangle $ABC$ with the circle $\omega$. It is known that $BP = CP$, and param1. Find $\angle ALC$. Provide your answ... | 11. Let $\omega$ be the circumcircle of an acute-angled triangle $ABC$, and $AL$ be the angle bisector of triangle $ABC$. Denote by $P$ the point of intersection of the extension of the altitude $BH$ of triangle $ABC$ with the circle $\omega$. It is known that $BP = CP$, and param1. Find $\angle ALC$. Give your answer ... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,885 | |
15. What is the smallest integer value that the sum of the roots of the equation param 1 can take?
| param1 | Answer |
| :---: | :---: |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+6=0$ | |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+8=0$ | |
| $x^{2}-\left(a^{2}+4\right) x+2 a^{2}+7=0$ | |
| $x^{2}-\left(a^{2}+4\right) x+2 a^... | 15. What is the smallest integer value that the sum of the roots of the equation param 1 can take?
| param1 | Answer |
| :---: | :---: |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+6=0$ | 7 |
| $x^{2}-\left(a^{2}+2\right) x+a^{2}+8=0$ | 8 |
| $x^{2}-\left(a^{2}+4\right) x+2 a^{2}+7=0$ | 8 |
| $x^{2}-\left(a^{2}+4\right) x+2... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,889 | |
16. The last digit of a six-digit number was moved to the beginning (for example, $456789 \rightarrow$ 945678), and the resulting six-digit number was added to the original number. Which numbers from the interval param 1 could have resulted from the addition? In the answer, write the sum of the obtained numbers.
| par... | 16. The last digit of a six-digit number was moved to the beginning (for example, $456789 \rightarrow$ 945678), and the resulting six-digit number was added to the original number. Which numbers from the interval param 1 could have resulted from the addition? In the answer, write the sum of the obtained numbers.
| par... | 1279267 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,890 |
17. We took natural numbers param 1 and wrote these raparam 2 along the top side of the table param3 (one number above each column), and wrote these same raparam2 along the left side of the table (one number to the left of each row). In the cells of the table, we wrote the products of the corresponding numbers (a "mult... | 17. We took natural numbers param 1 and wrote these param 2 along the top side of the table param3 (one number above each column), and wrote these same param2 along the left side of the table (one number to the left of each row). In the cells of the table, we wrote the products of the corresponding numbers (a "multipli... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 2,891 |
18. According to the results of a volleyball tournament held in a round-robin format (i.e., each team played one game against each other), it turned out that the top three teams won against each of the other teams, and the sum of points scored by the top three teams was param1 less than the sum of points scored by the ... | 18. According to the results of a volleyball tournament held in a round-robin format (i.e., each team played one game against each other), it turned out that the top three teams won against each of the other teams, and the sum of points scored by the top three teams was param1 less than the sum of points scored by the ... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,892 |
19. Find param1, if param2 for all real $x$, and param3.
| param1 | param2 | param3 | Answer |
| :--- | :--- | :--- | :--- |
| $f(1000)$ | $f(x+3)=f(x)+x-7$ | $f(1)=1$ | |
| :---: | :---: | :---: | :---: |
| $f(700)$ | $f(x+3)=f(x)+2 x+5$ | $f(1)=2$ | |
| $f(1000)$ | $f(x+3)=f(x)-x+10$ | $f(1)=15$ | |
| $f(850)$ ... | 19. Find param1, if param 2 for all real $x$, and param3.
| param1 | param2 | param3 | Answer |
| :--- | :--- | :--- | :--- |
| $f(1000)$ | $f(x+3)=f(x)+x-7$ | $f(1)=1$ | 163837 |
| :---: | :---: | :---: | :---: |
| $f(700)$ | $f(x+3)=f(x)+2 x+5$ | $f(1)=2$ | 163801 |
| $f(1000)$ | $f(x+3)=f(x)-x+10$ | $f(1)=15$ | -... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,893 |
1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{4}+11 x+23$. | Answer: 22.
Solution. Let's find those values of $x$ for which $y$ is positive: $-\frac{x^{2}}{4}+11 x+23>0 \Leftrightarrow-\frac{1}{4}(x+2)(x-46)>0$, from which $-2<x<46$. On this interval, there are 45 natural values of $x: x=1, x=2, \ldots, x=45$. In this interval, $y$ takes integer values only for even $x$ - a tot... | 22 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,895 |
2. Solve the inequality $\frac{\sqrt{x}-2}{1-\sqrt{x+1}} \geq 1+\sqrt{x+1}$. | Answer: $x \in(0 ; 1]$.
Solution. The domain of the inequality is determined by the conditions $x \geq 0,1-\sqrt{x+1} \neq 0$, from which we get that $x>0$. Note that on the domain, the denominator of the fraction is negative, so we can multiply both sides of the inequality by it, changing the inequality sign. Then
$... | x\in(0;1] | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 2,896 |
3. In the number $2 * 0 * 1 * 6 * 0 * 2 *$, each of the 6 asterisks needs to be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can be repeated) so that the resulting 12-digit number is divisible by 45. In how many ways can this be done? | Answer: 13122.
Solution. For a number to be divisible by 45, it is necessary and sufficient that it is divisible by 5 and by 9. To ensure divisibility by 5, we can choose 0 or 5 as the last digit (2 ways).
To ensure divisibility by nine, we proceed as follows. We will choose four digits arbitrarily (this can be done ... | 13122 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,897 |
4. Find all values of the parameter $a$, for each of which the equation $a|x-1|+\frac{x^{2}-7 x+12}{3-x}=0$ has exactly one solution. | Answer: $a \in\left[-1 ;-\frac{1}{2}\right) \cup\left(-\frac{1}{2} ; 1\right)$.
Solution. Given the condition $x \neq 3$, the equation is equivalent to $a|x-1|=x-4$. The graph of the right side of the equation is the line $y=x-4$. The graph of the left side of the equation is a "V" shape with its vertex at the point $... | \in[-1;-\frac{1}{2})\cup(-\frac{1}{2};1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,898 |
5. Solve the system of equations $\left\{\begin{array}{l}x+\sqrt{x+2 y}-2 y=\frac{7}{2}, \\ x^{2}+x+2 y-4 y^{2}=\frac{27}{2}\end{array}\right.$. | Answer: $\left(\frac{19}{4} ; \frac{17}{8}\right)$.
Solution. Let $\sqrt{x+2 y}=u, x-2 y=v$. Then the system takes the form
$$
\left\{\begin{array} { l }
{ u + v = \frac { 7 } { 2 } } \\
{ u ^ { 2 } v + u ^ { 2 } = \frac { 2 7 } { 2 } }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
v=\frac{7}{2}-u \\
u^{2}\lef... | (\frac{19}{4};\frac{17}{8}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,899 |
6. Point $K$ lies on side $A B$ of triangle $A B C$ with an angle of $120^{\circ}$ at vertex $C$. Circles are inscribed in triangles $A K C$ and $B K C$ with centers $O$ and $Q$ respectively. Find the radius of the circumcircle of triangle $O Q C$, if $O K=6, K Q=7$. | Answer: $\sqrt{\frac{85}{3}}$.
Solution. The center of the circle inscribed in an angle lies on the bisector of this angle, so rays $K O$ and $K Q$ are the bisectors of angles $A K C$ and $B K C$. Since the angle between the bisectors of adjacent angles is a right angle, $\angle O K Q=90^{\circ}$, and then by the Pyth... | \sqrt{\frac{85}{3}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,900 |
1. Find the number of points in the $x O y$ plane that have natural coordinates $(x, y)$ and lie on the parabola $y=-\frac{x^{2}}{4}+3 x+\frac{253}{4}$. | Answer: 11.
Solution. Let's find those values of $x$ for which $y$ is positive: $-\frac{x^{2}}{4}+3 x+\frac{253}{4}>0 \Leftrightarrow-\frac{1}{4}(x+11)(x-23)>0$, from which $-11<x<23$. On this interval, there are 22 natural values of $x: x=1, x=2, \ldots, x=22$. During this interval, $y$ takes integer values only for ... | 11 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,901 |
2. Solve the inequality $\frac{\sqrt{2-x}-2}{1-\sqrt{3-x}} \geq 1+\sqrt{3-x}$. | Answer: $x \in[1 ; 2)$.
Solution. The domain of the inequality is determined by the conditions $x \leq 2,1-\sqrt{3-x} \neq 0$, from which we get that $x<2$. Note that on the domain, the denominator of the fraction is negative, so we can multiply both sides of the inequality by it, changing the inequality sign. Then
$... | x\in[1;2) | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 2,902 |
3. In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks needs to be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can repeat) so that the resulting 10-digit number is divisible by 18. In how many ways can this be done? | Answer: 3645.
Solution. For a number to be divisible by 18, it is necessary and sufficient that it is divisible by 2 and by 9. To ensure divisibility by 2, we can choose the last digit from the available options as $0, 2, 4, 6$ or 8 (5 ways).
To ensure divisibility by nine, we proceed as follows. Choose three digits ... | 3645 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 2,903 |
4. Find all values of the parameter $a$, for each of which the equation $a|x+1|+\frac{x^{2}-5 x+6}{2-x}=0$ has exactly one solution | Answer: $a \in\left[-1 ;-\frac{1}{3}\right) \cup\left(-\frac{1}{3} ; 1\right)$.
Solution. Given the condition $x \neq 2$, the equation is equivalent to $a|x+1|=x-3$. The graph of the right-hand side of the equation is the line $y=x-3$. The graph of the left-hand side of the equation is a "V" shape with its vertex at t... | \in[-1;-\frac{1}{3})\cup(-\frac{1}{3};1) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,904 |
5. Solve the system of equations $\left\{\begin{array}{l}y+\sqrt{y-3 x}+3 x=12, \\ y^{2}+y-3 x-9 x^{2}=144\end{array}\right.$. | Answer: $(-24 ; 72),\left(-\frac{4}{3} ; 12\right)$.
Solution. Let $\sqrt{y-3 x}=u, y+3 x=v$. Then the system takes the form
$$
\left\{\begin{array} { l }
{ u + v = 1 2 } \\
{ u ^ { 2 } v + u ^ { 2 } = 1 4 4 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
v=12-u \\
u^{2}(12-u)+u^{2}=144
\end{array}\right.\righ... | (-24,72),(-\frac{4}{3},12) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,905 |
6. Point $P$ lies on side $B C$ of triangle $A B C$ with an angle of $60^{\circ}$ at vertex $A$. Incircles are inscribed in triangles $A P B$ and $A P C$ with centers $D$ and $T$ respectively. Find the radius of the circumcircle of triangle $A D T$, if $P D=7, P T=4$. | Answer: $\sqrt{65}$.
Solution. The center of the circle inscribed in an angle lies on the bisector of this angle, so rays $P T$ and $P D$ are the bisectors of angles $C P A$ and $B P A$. Since the angle between the bisectors of adjacent angles is a right angle, $\angle D P T=90^{\circ}$, and then by the Pythagorean th... | \sqrt{65} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 2,906 |
1. Find the number of points in the $x O y$ plane having natural coordinates $(x, y)$ and lying on the parabola $y=-\frac{x^{2}}{4}+9 x+19$. | Answer: 18.
Solution. Let's find those values of $x$ for which $y$ is positive: $-\frac{x^{2}}{4}+9 x+19>0 \Leftrightarrow-\frac{1}{4}(x+2)(x-38)>0$, from which $-2<x<38$. On this interval, there are 37 natural values of $x: x=1, x=2, \ldots, x=37$. In this interval, $y$ takes integer values only for even $x$ - a tota... | 18 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 2,907 |
2. Solve the inequality $\frac{\sqrt{x}-6}{2-\sqrt{x+4}} \geq 2+\sqrt{x+4}$. | Answer: $x \in(0 ; 4]$.
Solution. The domain of the inequality is determined by the conditions $x \geq 0, 2-\sqrt{x+4} \neq 0$, from which we get that $x>0$. Note that on the domain, the denominator of the fraction is negative, so we can multiply both sides of the inequality by it, changing the inequality sign. Then
... | x\in(0;4] | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 2,908 |
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