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5. The younger sister asked the math student Gavrila to fix the swings in the yard. After the repair, they became a flat board (seat) rigidly attached to two parallel rods, the distance between which is $d$. The rods are mounted on a horizontal axis in a cylindrical hinge, meaning they can rotate relative to this axis.... | 5. Continue the line connecting the bases of the rods until it intersects with the axis of the swings. Between them, there is a constant angle $\alpha$, independent of the angle of deflection of the swings. The normal to the seat at any angle of deflection forms an angle $\pi / 2 + \alpha$ with the axis of the swings (... | \arccos(\frac{\sqrt{^{2}+^{2}}}{\sqrt{1+\mu^{2}}}) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,381 |
3. During the cyclic process with an ideal gas, the recorder outputs $P V$ and $P T$ diagrams of this process. When transferring the graphical materials to the theoretical department, the axis labels were lost. The theorists noticed quadrilaterals on both diagrams, and one of the diagonals of one of them turned out to ... | 3. Let's prove that a rectangle was depicted on the $P V$ diagram. Indeed, all straight lines, except for the vertical ones, in $P V$ coordinates are described by the equation $P=\alpha V+\beta$. In this case, $T=\frac{1}{\nu R} V(\alpha V+\beta)$. This equation describes a straight line in $V T$ coordinates only if $\... | \sqrt{(t_{1}+273)(t_{2}+273)}-273 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,382 |
1.2.1 The time of the aircraft's run from the moment of start until takeoff is 15 seconds. Find the length of the run if the takeoff speed for this aircraft model is 100 km/h. Assume the aircraft's motion during the run is uniformly accelerated. Provide the answer in meters, rounding to the nearest whole number if nece... | Solution. $v=a t, 100000 / 3600=a \cdot 15$, from which $a=1.85\left(\mathrm{~m} / \mathrm{s}^{2}\right)$. Then $S=a t^{2} / 2=208(\mathrm{m})$.)
Answer. 208 m | 208 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,384 |
1.2.3 The takeoff run time of the aircraft from the moment of start until the moment of lift-off is 15 seconds. Find the length of the takeoff run if the lift-off speed for this aircraft model is 100 km/h. Assume the aircraft's motion during the takeoff run is uniformly accelerated. Provide the answer in meters, roundi... | Answer: $208 \mathrm{m}$ | 208\mathrm{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,385 |
2.2.1. A covered football field of rectangular shape with a length of 90 m and a width of 60 m is being designed, which should be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle, the radius of which is equal to the height at which the spotlight is hanging. ... | Solution. Let in rectangle $A B C D$ the diagonals intersect at point $O$. Place the projectors on the ceiling above the points that are the midpoints of segments $A O, B O, C O$ and $D O$, at a height equal to a quarter of the diagonal of the rectangle. Then the first projector will fully illuminate a circle, inside w... | 27.1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,386 |
# 2.2 .3 .
A covered football field of rectangular shape with a length of $100 \mathrm{m}$ and a width of $80 \mathrm{m}$ is being designed, which must be illuminated by four spotlights, each hanging at some point on the ceiling. Each spotlight illuminates a circle, the radius of which is equal to the height at which ... | Answer: $32.1 \mathrm{m}$.
# | 32.1\mathrm{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,387 |
3.3.1 Gavriil got on the train with a fully charged smartphone, and by the end of the trip, his smartphone was completely drained. For half of the time, he played Tetris, and for the other half, he watched cartoons. It is known that the smartphone fully discharges in 3 hours of video watching or in 5 hours of playing T... | Answer: 257 km.
Solution. Let's take the "capacity" of the smartphone battery as 1 unit (u.e.). Then the discharge rate of the smartphone when watching videos is $\frac{1}{3}$ u.e./hour, and the discharge rate when playing games is $\frac{1}{5}$ u.e./hour.
If the total travel time is denoted as $t$ hours, we get the ... | 257 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,388 |
4.2.1 An ideal gas is used as the working substance of a cyclic heat engine. The cycle consists of three stages: isochoric pressure reduction from $3 P_{0}$ to $P_{0}$, isobaric density increase from $\rho_{0}$ to $3 \rho_{0}$, and return to the initial state, where the process in the $P / P_{0}, \rho / \rho_{0}$ axes ... | Solution. Let's redraw the cycle in the $p v$ diagram. There, the line first goes down, then to the left, and then closes along some curve. Since the pressure at all points does not exceed $n P_{0}$, and the volume $V_{0}=m / \rho_{0}$, where $m$ is the mass of the gas, the maximum temperature is $T_{1}=n P_{0} V_{0} /... | \frac{1}{9} | Other | math-word-problem | Yes | Yes | olympiads | false | 11,389 |
5.1.1. A particle moves in the plane $O x y$ of a rectangular coordinate system according to the law (here $x, y-$ are the coordinates of the point in meters, $t$ - time in seconds):
$$
x(t)=3+\sin t \cdot \cos t-\sin t-\cos t ; y(t)=1
$$
In the same plane, a light ray constantly emanates from the origin according to... | Solution.
Let's make the substitution of variables $z=\sin t+\cos t$. Then $z^{2}=1+2 \sin t \cos t$, and the dependence of $x$ on $t$ takes the form
$$
x(t)=3+\frac{z^{2}-1}{2}-z, \quad z=\sin t+\cos t, \quad |z| \leqslant \sqrt{2}
$$
The function $f(z)=3+\frac{z^{2}-1}{2}-z=\frac{z^{2}}{2}-z+\frac{5}{2}=\frac{1}{2... | \in[\frac{2(7-2\sqrt{2})}{41};\frac{1}{2}] | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,390 |
5.1.2. A particle moves in the plane $O x y$ of a rectangular coordinate system according to the law (here $x, y$ are the coordinates of the point in meters, $t$ is time in seconds):
$$
x(t)=3 ; y(t)=3+\sin t \cdot \cos t-\sin t-\cos t
$$
In the same plane, a light ray constantly emanates from the origin according to... | Answer: $c \in\left(0 ; \frac{3}{2}\right),\left(\frac{7+2 \sqrt{2}}{6} ; \infty\right)$ | \in(0;\frac{3}{2}),(\frac{7+2\sqrt{2}}{6};\infty) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,391 |
6.2.1. A vertically oriented construction consisting of two identical masses connected by a spring (see figure) is subjected to an impact, as a result of which the upper mass begins to move with an initial velocity $V_{1}$ at an angle $\beta$ to the horizontal, and the lower mass - with a velocity $V_{2}$ at an angle $... | # Solution.
Thesis 1. The center of mass will always be located at point $A$. Let the position vectors of the points with masses be $r_{1}\left(x_{1} ; y_{1}\right), r_{2}\left(x_{2} ; y_{2}\right)$. Then the position vector of the center of mass is determined by the relation:
$r_{c}=\frac{m r_{1}+m r_{2}}{m+m}=\frac... | \frac{1}{2}(\frac{V_{1}\sin\beta+V_{2}\sin\alpha}{2})^{2} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 11,392 |
1. A tractor is pulling a very long pipe on sled runners. Gavrila walked along the entire pipe at a constant speed in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction at the same speed, the number of steps was 100. What is the length of the pipe if Gavrila's step i... | Answer: 108 m. Solution. Let the length of the pipe be $x$ (meters), and for each step Gavrila takes of length $a$ (m), the pipe moves a distance of $y$ (m). Then, if $m$ and $n$ are the number of steps Gavrila takes in one direction and the other, respectively, we get two equations: ${ }^{x=m(a-y)}, x=n(a+y)$. From th... | 108 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,394 |
2. A little goat is tied to a stake with a 4.7 m long rope, located on a flat meadow next to an old tree. The trunk of the tree has the shape of a circular cylinder with a radius of 0.5 m, and the shortest distance from the stake to the surface of the tree is $1 \mathrm{m}$.
Can the goat walk around the tree and appro... | Answer: No. Solution. When the kid goes around the tree and approaches the stake, the rope will partially lie on the surface of the tree, and the remaining part will consist of two straight segments. The minimum necessary length is the sum of twice the length of the tangent segment drawn from the stake to the tree and ... | No | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,395 |
3. To determine the speed of the current of a wide and turbulent river, Gavrila and Glafira conducted the following experiment. At a certain point on the shore 100 m from the bridge, they set up a siren emitting sound signals at equal intervals. Glafira took another identical siren and positioned herself at the beginni... | Answer: 12.4 km/h = 3.44 m/s. Solution. Introduce a coordinate system, the x-axis of which is directed along the shore, the origin of coordinates at the starting point of Gavrila. The siren on the shore has coordinates (L, 0), Glafira is moving along the line x = -L (in our case L = 50 m). Since the experimenters are a... | 12.4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,396 |
5. Since it is required that the stone does not hit the specified segment, the sought interval of velocities has the form $v>v_{0}$. Let's call $v_{0}-$ the minimum velocity.
By choosing the throwing point and the angle of inclination of the initial velocity to the horizon, one can ensure that the trajectory passes th... | The answer is $V>\sqrt{g\left(2 H+l \frac{1-\sin \alpha}{\cos \alpha}\right)}$ | V>\sqrt{(2H+\frac{1-\sin\alpha}{\cos\alpha})} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,397 |
6. On a windless day, a polar bear found itself on a small piece of ice that had broken off from an iceberg in the middle of still water. Rescuers from a hovering helicopter noted that the animal was walking in a circle with a diameter of 9.5 meters. How surprised they were when they later saw a chain of the bear's foo... | Answer: 11.4 t. Solution. When observed from the helicopter, the diameter of the trajectory was measured in the reference frame associated with the ground, while the chain of tracks shows the diameter of the trajectory relative to the ice floe. These trajectories do not coincide because the ice floe and the animal move... | 11.4 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,398 |
5. The oscillations of the ball consist of two stages: in water and in air. When the ball enters the water, a constant Archimedes' force begins to act on it, which does not change the frequency of the spring pendulum oscillations but shifts the equilibrium position.
Upon entering the water, the spring is stretched by ... | Answer: $\sqrt{\frac{m}{k}}\left[\pi+2 \operatorname{arctg}\left(\frac{v \rho}{g \rho_{0}} \sqrt{\frac{k}{m}}\right)\right]$
In all options, the answer is the same. | \sqrt{\frac{}{k}}[\pi+2\operatorname{arctg}(\frac{v\rho}{\rho_{0}}\sqrt{\frac{k}{}})] | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,399 |
1. Usually, schoolboy Gavriila takes a minute to go up a moving escalator by standing on its step. But if Gavriila is late, he runs up the working escalator and saves 36 seconds this way. Today, there are many people at the escalator, and Gavriila decides to run up the adjacent non-working escalator. How much time will... | Solution. Let's take the length of the escalator as a unit. Let $V$ be the speed of the escalator, and $U$ be Gavrila's speed relative to it. Then the condition of the problem can be written as:
$$
\left\{\begin{array}{c}
1=V \cdot 60 \\
1=(V+U) \cdot(60-36)
\end{array}\right.
$$
The required time is determined from ... | 40 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,401 |
3. The weight is made of an alloy of four metals and weighs 16.3 kg. The weight of the first metal in this alloy is one and a half times greater than that of the second, the weight of the second metal is to the weight of the third as $3: 4$, and the weight of the third to the weight of the fourth - as $5: 6$. Determine... | Solution. Denoting the weight of the second metal by $x$, express the other weights in terms of $x$. The weight of the first metal will be: $1.5 x$; the weight of the third: $\frac{4}{3} x$; the weight of the fourth: $\frac{24}{15} x$. It is convenient to eliminate the denominators. Then the weights of the metals are r... | 4.8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,402 |
4. After adding another tug to the one pushing the barge, they started pushing the barge with double the force. How will the power spent on movement change if the water resistance is proportional to the square of the barge's speed? | Solution. Since the barge is moving at a constant speed, the traction force of the tugboats is balanced by the resistance force. When the traction force is doubled, the resistance force also increases by the same factor, meaning the speed of the barge has increased by a factor of $\sqrt{2}$. Power is the product of for... | 2\sqrt{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,403 |
5. Take a ruler and place its ends flat on the index fingers of your outstretched hands. If you now move your fingers towards each other, they will meet approximately in the middle of the ruler. If, however, you place the middle of the ruler on your index fingers brought together and try to move your fingers apart towa... | Answer: When moving away from the center, one finger always starts a little faster and overtakes the second. Therefore, more of the ruler's weight ends up over the second finger, and due to friction, it cannot move. Conversely, it becomes easier and easier for the first finger. When moving towards the center, if one of... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,404 |
1. The lieutenant is engaged in drill training with the new recruits. Upon arriving at the parade ground, he saw that all the recruits were lined up in several rows, with the number of soldiers in each row being the same and 5 more than the number of rows. After the training session, the lieutenant decided to line up t... | Solution. Let $n$ be the number of rows in the original formation. Then, there were originally $n+5$ soldiers in each row, and in the second formation, there were $n+9$ soldiers in each row. Let the age of the lieutenant be $x$. Then, according to the problem, we get the equation
$$
x=\frac{n(n+5)}{n+9} \Rightarrow x=... | 24 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,406 |
2. Let $I$ be the center of the inscribed circle in triangle $ABC$. Prove that the center of the circle circumscribed about triangle $AIC$ lies on the circle circumscribed about triangle $ABC$. | Solution. Describe a circle around $\triangle A B C$. Since $I$ is the center of the inscribed circle of triangle $A B C$, $B I$ is the bisector of angle $A B C$. Extend $B I$ to intersect the circumcircle again at point $K$, and we will show that point $K$ is the center of the circumcircle of triangle $A I C$. To do t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 11,407 |
3. Among all six-digit natural numbers, the digits of which are arranged in ascending order (from left to right), numbers containing the digit 1 and not containing this digit are considered. Which numbers are more and by how many? | Solution. First, let's calculate how many six-digit natural numbers there are in total, with their digits arranged in ascending order. For this, we will write down all the digits from 1 to 9 in a row. To get six-digit numbers of the considered type, we need to strike out any three digits. Thus, the number of six-digit ... | 28 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,408 |
4. Find the area of the figure, the coordinates $(x ; y)$ of the points of which satisfy the system of inequalities
$$
\left\{\begin{array}{l}
|x|+|y| \geq 1 \\
(|x|-1)^{2}+(|y|-1)^{2} \leq 1
\end{array}\right.
$$ | Solution. The region defined by the first inequality of the system is the entire plane without the square with vertices at points $(1 ; 0),(0 ; 1),(-1 ; 0),(0 ;-1)$, and the second inequality defines a circle centered at the origin with a radius of 1. Therefore, the desired area is the difference between the area of th... | \pi-2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,409 |
5. Solve the equation $x^{2}-x y-6 y^{2}+2 x+19 y=18$ in integers.
Rewrite the equation as
$$
x^{2}-x(y-2)-6 y^{2}+19 y+k=18+k
$$
and we will choose the number $k$ so that the discriminant of the quadratic trinomial on the left side of the equation becomes a perfect square. We have
$$
D=25 y^{2}-80 y+4-4 k
$$
To o... | Answer: $(2 ; 2),(-2 ; 2)$. | (2;2),(-2;2) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,410 |
7. For what values of the parameter $a$ will the minimum value of the function
$$
f(x)=|7 x-3 a+8|+|5 x+4 a-6|+|x-a-8|-24
$$
be the smallest. | Solution. If $7 x>3 a-8$, then with any method of unfolding the remaining two modules, the coefficient of $x$ will be positive, meaning the function will be increasing. If, however, $7 x<3 a-8$, then with any method of unfolding the modules, the coefficient of $x$ will be negative and the function will be decreasing. T... | \frac{82}{43} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,411 |
8. Solve the equation $\sqrt{15 x^{2}-52 x+45} \cdot(3-\sqrt{5 x-9}-\sqrt{3 x-5})=1$. | Solution. Rewrite our equation in the form
$$
\sqrt{3 x-5} \cdot \sqrt{5 x-9} \cdot(3-\sqrt{5 x-9}-\sqrt{3 x-5})=1
$$
Such a transformation is possible because the solution to the original equation exists only for $x>\frac{9}{5}$. Let $\sqrt{3 x-5}=a>0, \sqrt{5 x-9}=b>0$. We have
$$
a+b+\frac{1}{a b}=3
$$
Apply the... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,412 |
2. Let $I$ be the center of the inscribed circle in triangle $ABC$. Prove that the center of the circle circumscribed around triangle $AIC$ lies on the circle circumscribed around triangle $ABC$. | Solution. Describe a circle around $\triangle A B C$. Since $I$ is the center of the inscribed circle of triangle $A B C$, $B I$ is the bisector of angle $A B C$. Extend $B I$ to intersect the circumcircle again at point $K$, and we will show that point $K$ is the center of the circumcircle of triangle $A I C$. To do t... | proof | Geometry | proof | Yes | Yes | olympiads | false | 11,413 |
4. Solve the system of inequalities $\left\{\begin{array}{l}\left(49^{x+1}-50 \cdot 7^{x}+1\right) \cdot \log _{x+\frac{5}{2}}\left|x+\frac{1}{2}\right| \geq 0 \\ 49^{x+1}+\log _{x+\frac{5}{2}}\left|x+\frac{1}{2}\right|+1 \leq 50 \cdot 7^{x}\end{array}\right.$. | Solution. Let $\left(49^{x+1}-50 \cdot 7^{x}+1\right)=a, \log _{x+\frac{5}{2}}\left|x+\frac{1}{2}\right|=b$. Then the original system takes the form
$$
\left\{\begin{array}{l}
\left\{\begin{array}{l}
\left\{\begin{array}{l}
a=0 \\
b \leq 0
\end{array}\right. \\
a+b \leq 0
\end{array}\right. \\
\left\{\begin{array}{l}
... | notfound | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 11,414 |
5. Given a trapezoid, the ratio of its bases is 2. If the trapezoid is rotated $360^{\circ}$ around the larger base, a certain solid figure $\Phi_{1}$ is obtained. If the trapezoid is rotated $360^{\circ}$ around the smaller base, a certain solid figure $\Phi_{2}$ is obtained. Find the ratio of the volumes of the resul... | Solution. Consider trapezoid $ABCD$ with bases $AD=2a, BC=a$. Draw the heights of the trapezoid $BH_{1}=BH_{2}=h$. Denote $AH_{1}=a_{1}, AH_{2}=a_{2}$, where $a_{1}+a_{2}=a$.
When the trapezoid is rotated around the larger base, a cylinder is formed with a generatrix equal to $a$ and a base radius equal to the height ... | \frac{5}{4} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,416 |
6. Find the minimum value of the function $f(x)=\sqrt{x^{2}-8 x+25}+\sqrt{x^{2}-4 x+13}$. | Solution. By completing the square under each of the radicals, we transform the function into the form
$$
f(x)=\sqrt{(x-4)^{2}+9}+\sqrt{(x-2)^{2}+9}
$$
For this expression, a geometric interpretation can be proposed. Consider each of the radicals as the distance from the point with coordinates $(x ; 0)$ to the point ... | 2\sqrt{10} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,417 |
8. Solve the equation $\log _{5}(3 x-4) \cdot \log _{5}(7 x-16) \cdot\left(3-\log _{5}\left(21 x^{2}-76 x+64\right)\right)=1$. | Solution. Rewrite our equation in the form
$$
\log _{5}(3 x-4) \cdot \log _{5}(7 x-16) \cdot\left(3-\log _{5}(3 x-4)-\log _{5}(7 x-16)\right)=1
$$
Such a transformation is possible because the solution to the original equation exists only for $x>\frac{16}{7}$. Let $\log _{5}(3 x-4)=a>0, \log _{5}(7 x-16)=b>0$. Note t... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,418 |
3. For the third question: in the appendix to the task, more than 20 events covered by a standard property insurance policy are listed. The student was able to provide five logically justified examples - 5 points, four examples - 4 points, three examples - 3 points, and so on, no examples - 0 points.
## Task 2
You ar... | # Solution:
To calculate the family's income as of June 1, 2018, we will perform the following steps: | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,419 | |
5. The Ivanovs' income as of the beginning of June:
$$
105000+52200+33345+9350+70000=269895 \text { rubles }
$$ | Answer: 269895 rubles
## Evaluation Criteria:
Maximum score - 20, if everything is solved absolutely correctly, the logic of calculations is observed, and the answer is recorded correctly.
## 20 points, including:
6 points - the final deposit amount is calculated correctly;
4 points - the size of the mother's sala... | 269895 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,420 |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive than a, which means the highest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend
$$
4800 + 1500 - 4800 * 0.2 = 5340 \text{ rubles.}
$$
This is the most profitable... | # Solution:
The average value of the last purchases is $(785+2033+88+3742+1058) / 5 = 1541.2$ rubles. Therefore, an allowable purchase is no more than $1541.2 * 3 = 4623.6$ rubles. With this amount, you can buy $4623.6 / 55 \approx 84$ chocolates.
## Maximum 20 points
20 points - fully detailed solution and correct ... | 84 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,421 |
Task 14. (2 points)
Ivan opened a deposit in a bank for an amount of 100 thousand rubles. The bank is a participant in the state deposit insurance system. How much money will Ivan receive if the bank's license is revoked / the bank goes bankrupt?
a) Ivan will receive 100 thousand rubles and the interest that has been... | # Solution:
In accordance with the federal insurance law Federal Law No. 177-FZ of $23 \cdot 12.2003$ (as amended on 20.07.2020) "On Insurance of Deposits in Banks of the Russian Federation" (with amendments and additions, effective from 01.10.2020), compensation for deposits in a bank where an insurance case has occu... | 100 | Other | MCQ | Yes | Yes | olympiads | false | 11,427 |
4. Excursions (20,000 rubles for the whole family for the entire vacation).
The Seleznev family is planning their vacation in advance, so in January, the available funds for this purpose were calculated. It turned out that the family has 150,000 rubles at their disposal. Mr. Seleznev plans to set aside a certain amoun... | # Solution:
1) Calculate the vacation expenses
Flight expenses $=10200.00$ rubles * 2 flights * 3 people $=61200.00$
Hotel expenses $=6500$ rubles * 12 days $=78000.00$ rubles
Food expenses $=1000.00$ rubles * 14 days * 3 people $=42000.00$ rubles
Excursion expenses $=20000.00$ rubles
Total expenses $=201200.00$ ... | 47825 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,428 |
4. Excursions (20,000.00 rubles for the whole family for the entire vacation).
The Sobolkovs are planning their vacation in advance, so in January, they calculated the funds available for this purpose. It turned out that the family has 150,000.00 rubles at their disposal. Mr. Sobolkov plans to set aside a certain amou... | # Solution:
1) Calculate the vacation expenses
Flight expenses $=10200.00$ rubles * 2 flights * 3 people $=61200.00$
Hotel expenses $=6500$ rubles * 10 days $=65000.00$ rubles
Food expenses $=1000.00$ rubles * 12 days * 3 people $=36000.00$ rubles
Tour expenses $=20000.00$ rubles
Total expenses $=182200.00$ ruble... | 29479.67 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,433 |
4. The company will have a net profit at the end of 2018 in the amount of: (1 $500000-674992)$ - (1 $500000-674$ 992)*0.2 $=660006.4$ rubles
The amount of principal repayments for 2018 will be:
$$
23914 \text { * } 12 \text { - } 74992=211976 \text { rubles }
$$
Dividends per share for 2018:
$(660006.4-211976) / 10... | # Solution:
When evaluating information, it is necessary to consider not only the quantity and nature of reviews, but also the following:
Translated as requested, maintaining the original text's line breaks and format. | 246400 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,438 |
2. Inflation over two years will be:
$$
\left((1+0.015)^{\wedge} 2-1\right)^{*} 100 \% = 3.0225 \%
$$
The real return on a bank deposit, taking into account the extension for the second year, will be $(1.07 * 1.07 /(1+0.030225)-1) * 100 \% = 11.13 \%$
## Evaluation Criteria:
A maximum of 25 points for a correct and... | # Solution:
Possible scenarios:
- ATMs - skimming (installation of an overlay keyboard and card reader), shimming (installation of an intercepting scheme in the card reader), microcameras (to record the PIN code), physical theft of ATMs, hacking the ATM management system.
- Online banking - hacking the bank's website... | 11.13 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,439 |
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 | 11,440 |
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:
$$
(739000-300 \text { 000) / } 160... | Solution:
The information indicated on the packaging is intended to convince the buyer that this product has a significant advantage over similar products from other manufacturers (in this case, significant for healthy eating).
Thus, in sunflower oil as a plant product:
- there cannot be cholesterol (animal fat);
- ... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,441 |
# 2. Inflation over two years will be
$$
\left((1+0.025)^{\wedge 2-1}\right)^{*} 100 \% = 5.0625 \%
$$
The real return on a bank deposit, taking into account the extension for the second year, will be $(1.06 * 1.06 / (1+0.050625) - 1) * 100 = 6.95 \%$
## Evaluation Criteria:
A maximum of 25 points for a correct and... | # Solution:
- employee profiles of the bank - employee awareness programs aimed at typical scenarios of criminal actions. These programs are designed to protect against various social attacks in which an employee discloses confidential information. For example, an employee might disclose information on social networks... | 6.95 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,442 |
5. Additionally: taking into account the creative nature of the assignment, intermediate scores, but not exceeding the maximum, may be given for creative suggestions on the topic under consideration.
## Task 2
Once, on New Year's Eve, December 31, 2017, Ivan Tsarevich was walking home from service. His mood was excel... | # Solution:
Let's calculate how much the family will be able to save monthly from 01.2018 to 05.2018. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,443 |
9. Revenues from 09.2019 to 12.2019 will be:
$$
(55000+45000+10000+17400) * 4=509600 \text { rubles }
$$
Expenses from 09.2019 to 11.2019 will be:
$$
(40000+20000+5000+2000+2000) * 4=276000 \text { rubles }
$$
By 12.31.2019, the family will have saved $1147240+521600-276000=1340840$ rubles and will be able to buy a... | Answer: The family can save up for a car by two years, by 31.12.2019.
## Maximum 30 points
## Criteria: | 1340840 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,444 |
11. Revenues for the period from 11.2019 to 03.2020 will be:
$$
5^{*}(45000+35000+7000+10000+13000)=550000 \text { rubles }
$$
Expenses for the period from 11.2019 to 03.2020 will be:
$$
5 *(30000+10000+5000+4500+9000)=292500 \text { rubles }
$$
By 31.03.2020, the family will have accumulated $849400+550000-292500=... | Answer: In 2 years, the family can save 1,106,900 rubles
## Evaluation Criteria: | 1,106,900 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,446 |
3. (13 points) Money placed in the 1st month will be on deposit for 6 months and will bring a nominal income of $8700\left((1+0.06 / 12)^{\wedge}-1\right)=264.28$ rubles.
Funds placed in the 2nd month will be on deposit for 5 months and will bring a nominal income of $8700\left((1+0.06 / 12)^{\wedge} 5-1\right)=219.69... | # Response:
The following security measures can be implemented:
- Use of two-factor authentication when launching the application, i.e., in addition to login|password, require the input of a one-time password from an SMS or token.
- Use of two-factor authentication for payment services, including conducting payments ... | 921.15 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,447 |
3. Maria Ivanovna decided to use the services of an online clothing store and purchase summer clothing: trousers, a skirt, a jacket, and a blouse. Being a regular customer of this store, Maria Ivanovna received information about two ongoing promotions. The first promotion allows the customer to use an electronic coupon... | Solution:
(a) Maria Ivanovna can make one purchase, using only one of the promotions, or she can "split" the selected items into two purchases, using both promotions in this case.
Let's consider all possible options:
1) One purchase. In this case, Maria Ivanovna can save either 900 rubles by using the "third item fr... | 6265 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,452 |
Task 4.
Eugene 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 banki... | 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 | 11,457 |
Task 4.
Georgy Semenovich decided to take a loan from a banking institution in the amount of 7,500,000 rubles to purchase a studio in Bdensk. The loan terms are as follows: he returns the initial loan amount and 450,000 rubles in interest over 20 months. Determine the annual interest rate of the banking institution on... | Solution: (450000/20)/7500000*12*100
Answer: $3.6 \%$
Criteria:
Correct solution and answer - 20 points | 3.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,461 |
5. Additionally: taking into account the creative nature of the assignment, intermediate scores, but not exceeding the maximum, may be given for creative suggestions on the topic under consideration.
## Task 2
Once, on New Year's Eve, December 31, 2017, Ivan Tsarevich was walking home from service. His mood was excel... | # Solution:
Let's calculate how much the family will be able to save monthly from 01.2018 to 05.2018. | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,462 |
3. Petya is the owner of a cashback card offering $1 \%$ cashback on all purchases. In addition to this, the bank offers Petya the option to choose categories with increased cashback each month, but no more than three. Which categories would be the most profitable to choose, considering the cashback percentage and the ... | Answer: $2 ; 3 ; 4$.
Cashback is the percentage of the purchase amount that is returned to the card after the purchase. Let's calculate the cashback that Petya will receive in each case:
1) $2000 * 0.05 = 100$ rubles;
2) $5000 * 0.03 = 150$ rubles;
3) $3000 * 0.04 = 120$ rubles;
4) $3000 * 0.05 = 150$ rubles;
5) $150... | 2;3;4 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,464 |
Task 11. (16 points)
The Dorokhov family plans to purchase a vacation package to Crimea. The family plans to travel with the mother, father, and their eldest daughter Polina, who is 5 years old. They carefully studied all the offers and chose the "Bristol" hotel. The head of the family approached two travel agencies, ... | # Solution:
Cost of the tour with the company "Globus"
$(3 * 25400) *(1-0.02)=74676$ rubles.
Cost of the tour with the company "Around the World"
$(11400+2 * 23500) * 1.01=58984$ rubles.
Answer: 58984 | 58984 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,468 |
Task 12. (16 points)
The Vasilievs' family budget consists of the following income items:
- parents' salary after income tax deduction - 71000 rubles;
- income from renting out property - 11000 rubles;
- daughter's scholarship - 2600 rubles
The average monthly expenses of the family include:
- utility payments - 84... | # Solution:
family income
$71000+11000+2600=84600$ rubles
average monthly expenses
$8400+18000+3200+2200+18000=49800$ rubles
expenses for forming a financial safety cushion
$(84600-49800) * 0.1=3480$ rubles
the amount the Petrovs can save monthly for the upcoming vacation
$84600-49800-3480=31320$ rubles
## Ans... | 31320 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,469 |
Task 13. (8 points)
Natalia Petrovna has returned from her vacation, which she spent traveling through countries in North America. She has a certain amount of money left in foreign currency.
Natalia Petrovna familiarized herself with the exchange rates at the nearest banks: "Rebirth" and "Garnet." She decided to take... | # Solution:
1) cost of currency at Bank "Vozrozhdenie":
$$
120 * 74.9 + 80 * 59.3 + 10 * 3.7 = 13769 \text{ RUB}
$$
2) cost of currency at Bank "Garant":
$$
120 * 74.5 + 80 * 60.1 + 10 * 3.6 = 13784 \text{ RUB}
$$
Answer: 13784 | 13784 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,470 |
Task 14. (8 points)
To attend the section, Mikhail needs to purchase a tennis racket and a set of tennis balls. Official store websites have product catalogs. Mikhail studied the offers and compiled a list of stores where the items of interest are available:
| Item | Store | |
| :--- | :---: | :---: |
| | Higher Le... | # Solution:
1) cost of purchase in the store "Higher League":
$$
\text { 5600+254=5854 rub. }
$$
1) cost of purchase in the store "Sport-guru": $(2700+200)^{*} 0.95+400=6005$ rub.
## Answer: 5854 | 5854 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,471 |
Task 15. (8 points)
The fast-food network "Wings and Legs" offers summer jobs to schoolchildren. The salary is 25000 rubles per month. Those who work well receive a monthly bonus of 5000 rubles.
How much will a schoolchild who works well at "Wings and Legs" earn per month (receive after tax) after the income tax is d... | Solution:
The total earnings will be 25000 rubles + 5000 rubles $=30000$ rubles
Income tax $13 \%-3900$ rubles
The net payment will be 30000 rubles - 3900 rubles $=26100$ rubles
## Correct answer: 26100
## 2nd Option | 26100 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,472 |
Task 11. (16 points)
One way to save on utility bills is to use the night tariff (from 23:00 to 07:00). To apply this tariff, a multi-tariff meter needs to be installed.
The Romanov family is considering purchasing a multi-tariff meter to reduce their utility bills. The cost of the meter is 3500 rubles. The installat... | # Solution:
2) use of a multi-tariff meter:
$$
3500+1100+(230 * 3.4+(300-230) * 5.2) * 12 * 3=45856 \text { rub. }
$$
3) use of a typical meter
$$
300 * 4.6 * 12 * 3=49680 \text { rub. } \quad \text {. } \quad \text {. }
$$
the savings will be
$49680-45856=3824$ rub.
## Answer: 3824 | 3824 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,473 |
Task 12. (16 points)
The budget of the Petrovs consists of the following income items:
- parents' salary after income tax deduction - 56000 rubles;
- grandmother's pension - 14300 rubles;
- son's scholarship - 2500 rubles
Average monthly expenses of the family include:
- utility payments - 9800 rubles;
- food - 210... | # Solution:
family income
$56000+14300+2500=72800$ rubles.
average monthly expenses
$9800+21000+3200+5200+15000=54200$ rubles.
expenses for forming a financial safety cushion
$(72800-54200) * 0.1=1860$ rubles.
the amount the Petrovs can save monthly for the upcoming vacation
$72800-54200-1860=16740$ rubles.
An... | 16740 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,474 |
Task 13. (8 points)
Maxim Viktorovich returned from a trip to Asian countries. He has a certain amount of money in foreign currency left.
Maxim Viktorovich familiarized himself with the exchange rates at the nearest banks: "Voskhod" and "Elfa". He decided to take advantage of the most favorable offer. What amount wil... | # Solution:
1) cost of currency at "Voskhod" bank:
$110 * 11.7 + 80 * 72.1 + 50 * 9.7 = 7540$ rubles.
2) cost of currency at "Alpha" bank: $110 * 11.6 + 80 * 71.9 + 50 * 10.1 = 7533$ rubles.
Answer: 7540. | 7540 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,475 |
Task 14. (8 points)
Elena decided to get a pet - a budgerigar. She faced the question of where to buy a cage and a bath more cost-effectively.
On the official websites of the stores, product catalogs are posted. Elena studied the offers and compiled a list of stores where the items she is interested in are available:... | # Solution:
2) cost of purchase in the "Zoimir" store: $4500+510=5010$ rubles
3) cost of purchase in the "Zooidea" store: $(3700+680) * 0.95+400=4561$ rubles
## Answer: 4561 | 4561 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,476 |
Task 15. (8 points)
Announcement: "Have free time and want to earn money? Write now and earn up to 2500 rubles a day working as a courier with the service 'Food.There-Here!'. Delivery of food from stores, cafes, and restaurants.
How much will a school student working as a courier with the service 'Food.There-Here!' e... | # Solution:
The total earnings will be (1250 rubles * 4 days) * 4 weeks = 20000 rubles
Income tax 13% - 2600 rubles
The amount of earnings (net pay) will be 20000 rubles - 2600 rubles = 17400 rubles
Correct answer: 17400 | 17400 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,477 |
5. There are several technologies for paying with bank cards: chip, magnetic stripe, paypass, cvc. Arrange the actions performed with a bank card in the order corresponding to the payment technologies.
1 - tap
2 - pay online
3 - swipe
4 - insert into terminal | Answer in the form of an integer, for example 1234.
Answer: 4312 | 4312 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,479 |
# Task 1. (4 points)
The price of a new 3D printer is 625000 rubles. Under normal operating conditions, its resale value decreases by $20 \%$ in the first year, and then by $8 \%$ each subsequent year. After how many years will the resale value of the printer be less than 400000 rubles? | Solution:
Let's calculate the cost of the printer year by year:
1 year $=625000 * 0.8=500000$ rubles
2 year $=500000 * 0.92=460000$ rubles (1 point)
3 year $=460000 * 0.92=423200$ rubles (1 point)
4 year $=423200 * 0.92=397694$ rubles. (1 point)
Answer: in 4 years. ( $\mathbf{1}$ point) | 4 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,480 |
# Task 2. (6 points)
Provide 3 examples of "investing in yourself" and explain why they are beneficial. | Solution:
Successful examples:
- expenses on education (paid education or tutors) allow obtaining better education, getting a better-paid job,
- expenses on sports sections/fitness/health allow saving significant amounts of money on medical services in the future,
- time spent on reading books.
Unsuccessful examples... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,481 |
# Problem 4. (8 points)
Kolya's parents give him pocket money once a month, calculating it as follows: 100 rubles for each A in math, 50 rubles for a B, 50 rubles are deducted for a C, and 200 rubles are deducted for a D. If the amount turns out to be negative, Kolya simply gets nothing. The math teacher gives a grade... | # Solution:
If Kolya received a final grade of 2 for the quarter, then for each 5 he received more than 5 2s, for each 4 - more than 3 2s, and for each 3 - more than 1 2. This means that the number of 2s was greater than the total number of all other grades combined, so Kolya could receive money in at most one of the ... | 250 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,482 |
# Task 5. (8 points)
At the beginning of the 20th century, sellers did not indicate prices on goods; they had to remember all of them. Additionally, the seller could estimate the quality and quantity of the goods by eye and state their price. The buyer would haggle, and if the price was not suitable, they would simply... | # Solution:
- buying becomes easier: the buyer needs fewer actions to purchase a product, as a result, revenue increases
- lower requirements for the seller and store staff, i.e., costs can be reduced;
- additional advertising method (price tags can be of different colors or promotions "everything for 50 rubles")
- wr... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,483 |
# Problem 6. (10 points)
Vasily is planning to graduate from college in a year. Only 270 out of 300 third-year students successfully pass their exams and complete their bachelor's degree. If Vasily ends up among the 30 expelled students, he will have to work with a monthly salary of 25,000 rubles. It is also known tha... | # Solution:
In 4 years after graduating from school, Fedor will earn $25000 + 3000 * 4 = 37000$ rubles (2 points).
The expected salary of Vasily is the expected value of the salary Vasily can earn under all possible scenarios (2 points). It will be 270/300 * $(1 / 5 * 60000 + 1 / 10 * 80$ $000 + 1 / 20 * 25000 + (1 -... | 45025 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,484 |
# Task 2.
Lev Davidovich is a regular bulk buyer in rare tea shops, so he is eligible for a special discount. According to the discount terms, if Lev Davidovich pays before the specified deadline, he can take advantage of the discount; if he pays after, he must pay the full amount specified in the contract. Lev Davido... | # Solution:
Let
A - discount (%)
B - maximum deferment
C - period during which the discount is valid.
Using, for example, the method of simple interest accrual, we will determine whether the proposed discount justifies the costs of paying interest on the loan. For each day, 22/365 percent of the loan amount, which... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,485 |
Task 2.
The son of Lieutenant Schmidt received an inheritance of 500,000 rubles and decided that it would be prudent to invest this money. Consulting a financial advisor with the gift of clairvoyance, the son of Lieutenant Schmidt learned about the following investment opportunities:
1) deposit the money in the "Flyi... | # Solution:
1) With annual capitalization of the deposit, interest is accrued on the amount including already accumulated interest. In this case:
$500000 *(1.07^5)=701275.87$ rubles.
Profit from the deposit: 701275.87-500000=201275.87 rubles.
2) Without capitalization, interest is accrued on a fixed deposit amount.... | 201275.87 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,488 |
# Problem 1. (4 points)
In the run-up to the New Year, a fair is being held at the school where students exchange festive toys. As a result, the following exchange norms have been established:
1 Christmas tree ornament can be exchanged for 2 crackers, 5 sparklers can be exchanged for 2 garlands, and 4 Christmas tree ... | # Solution:
a) 10 sparklers $=4$ garlands = 16 ornaments = $\mathbf{32}$ crackers. (1 point)
b) Convert everything to crackers. In the first case, we have $\mathbf{11}$ crackers. In the second case, 2 sparklers $=4 / 5$ garlands $=16 / 5$ ornaments $=32 / 5$ crackers $=\mathbf{6.4}$ crackers. Answer: 5 ornaments and ... | 32 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,490 |
# Task 2. (6 points)
After reviewing where the family budget is being spent, Vasya found that savings can be made on each item of expenditure without compromising the quality of life. Provide an example of irrational spending and an improvement option for any three items: transportation, communication services, grocer... | # Solution: (2 points for at least one of the examples provided in each group. No more than 2 points for each group) | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,491 |
# Task 3. (8 points)
In the Sidorov family, there are 3 people: the father works as a programmer with an hourly rate of 1500 rubles. The mother works as a hairdresser at home and charges 1200 rubles per haircut, which takes her 1.5 hours. The son tutors in mathematics and earns 450 rubles per academic hour (45 minutes... | # Solution
In this problem, there are 2 possible interpretations, both of which were counted as correct.
In one case, it is assumed that 8 hours are spent on work on average over the month, in the other that no more than 8 hours are spent on work each day.
## First Case:
1) Determine the hourly wage for each family... | 19600 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,492 |
# Problem 4. (10 points)
On December 31 at 16:35, Misha realized he had no New Year's gifts for his entire family. He wants to give different gifts to his mother, father, brother, and sister. Each of the gifts is available in 4 stores: Romashka, Odynachik, Nezabudka, and Lysichka, which close at 20:00. The journey fro... | # Solution:
Notice that in each of the stores, there is a "unique" gift with the lowest price.
If Misha managed to visit all 4 stores, he would spend the minimum amount of $980+750+900+800=3430$ rubles. However, visiting any three stores would take Misha at least $30 * 3+25+30+35=180$ minutes. Considering the additio... | 3435 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,493 |
3. For the third question: in the appendix to the task, more than 20 events covered by a standard property insurance policy are listed. The student was able to provide five logically justified examples - 5 points, four examples - 4 points, three examples - 3 points, and so on, no examples - 0 points.
## Task 2
You ar... | # Solution:
To calculate the family's income as of June 1, 2018, we will perform the following steps: | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,500 | |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive than a, which means the highest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend $4800 + 1500 - 4800 * 0.2 = 5340$ rubles. This is the most profitable way to make t... | # Solution:
The average value of the last purchases is $(785+2033+88+3742+1058) / 5 = 1541.2$ rubles. Therefore, an acceptable purchase would be no more than $1541.2 * 3 = 4623.6$ rubles. With this amount, one can buy $4623.6 / 55 \approx 84$ chocolates.
## Maximum 20 points
20 points - fully detailed solution and c... | 84 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,501 |
1. What was NOT used as money?
1) gold
2) stones
3) horses
4) dried fish
5) mollusk scales
6) all of the above were used | Answer: 6. All of the above were used as money. | 6 | Logic and Puzzles | MCQ | Yes | Yes | olympiads | false | 11,502 |
2. A stationery store is running a promotion: there is a sticker on each notebook, and for every 5 stickers, a customer can get another notebook (also with a sticker). Fifth-grader Katya thinks she needs to buy as many notebooks as possible before the new semester. Each notebook costs 4 rubles, and Katya has 150 rubles... | Answer: 46.
1) Katya buys 37 notebooks for 148 rubles.
2) For 35 stickers, Katya receives 7 more notebooks, after which she has notebooks and 9 stickers.
3) For 5 stickers, Katya receives a notebook, after which she has 45 notebooks and 5 stickers.
4) For 5 stickers, Katya receives the last 46th notebook. | 46 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 11,503 |
3. How much did the US dollar exchange rate change over the 2014 year (from January 1, 2014 to December 31, 2014)? Give the answer in rubles, rounding to the nearest whole number (the answer is a whole number). | Answer: 24. On January 1, 2014, the dollar was worth 32.6587, and on December 31, it was 56.2584.
$56.2584-32.6587=23.5997$. Since rounding was required, the answer is 24.
Note: This problem could have been solved using the internet. For example, the website https://news.yandex.ru/quotes/region/1.html | 24 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,504 |
5. Vanya decided to give Masha a bouquet of an odd number of flowers for her birthday, consisting of yellow and red tulips, so that the number of flowers of one color differs from the number of flowers of the other by exactly one. Yellow tulips cost 50 rubles each, and red ones cost 31 rubles. What is the largest numbe... | Answer: 15.
A bouquet with one more red tulip than yellow ones is cheaper than a bouquet with the same total number of flowers but one more yellow tulip. Therefore, Vanya should buy a bouquet with one more red tulip. The remaining flowers can be paired into red and yellow tulips, with each pair costing 81 rubles. Let'... | 15 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,505 |
6. 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 | 11,506 |
7. What is a sign of a financial pyramid?
1) an offer of income significantly above average
2) incomplete information about the company
3) aggressive advertising
4) all of the above | Answer: 4. All of the above are signs of a financial pyramid. | 4 | Other | MCQ | Yes | Yes | olympiads | false | 11,507 |
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 | 11,511 |
Problem 10. (6 points)
Artyom is planning to get a consumer loan from a bank in the amount of 65 thousand rubles for a term of 1 year. Which of the following interest calculation options, in your opinion, would be the most beneficial for Artyom in the case where the principal is repaid at the end of the loan term in o... | Answer: $b, 2$.
## Comment:
option a. 68380 rub.
option b. 65000 rub. $\times(1+5 \% / 100 \%)=68250$ rub.
option c. 65000 rub. $\times(1+5 \% / 100 \% / 12)^{12}=68325.52$ rub.
option d. 65000 rub. $\times(1+5 \% / 100 \%)=68250$ rub.
option e. 65000 rub. $\times(1+5 \% / 100 \% / 4)^{4}=68311.45$ rub.
option f... | b,2 | Algebra | MCQ | Yes | Yes | olympiads | false | 11,515 |
Problem 12. (6 points)
Victor received a large sum of money as a birthday gift in the amount of 45 thousand rubles. The young man decided to save this part of his savings in dollars on a currency deposit. The term of the deposit agreement was 2 years, with an interest rate of 4.7% per annum, compounded quarterly. On t... | Answer: 873 USD.
## Comment:
45000 RUB / 56.60 RUB $\times(1+4.7\% / 4 \text { quarters })^{2 \text { years } \times 4 \text { quarters }}=873$ USD. | 873 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,516 |
Problem 13. (4 points)
Nikolai, Maxim, Oxana, and Olga decided to bet on who is better at managing their money. The condition was as follows: each would invest the amount of money they could afford at the moment, but at the end of the bet, which would occur in one year, each should have a sum in rubles, obtained as a ... | # Answer: Nikolai.
## Comment:
Nikolai's return $=7.1 \%$ annually, taking into account monthly capitalization, the return at maturity will be $7.34 \%$ annually.
Maxim's return $=(80000$ rubles $/ 58.42$ rubles $) \times(1+3.6 \%) \times 58.61$ rubles $/ 80000$ rubles $\times 100 \%-100 \%=3.94 \%$
Oksana's return... | Nikolai | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,517 |
Problem 16. (5 points)
Maxim Sergeevich 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 Sergeevich saved money for a down payment on a mortgage for about two years, and finally, when a sufficient amount was in his account, he c... | # 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 | 11,518 |
# Task 17-19. (2 points per Task)
Alena opened a multi-currency deposit at "Severny" Bank for 3 years. The deposit involves the deposit of funds in three currencies: euros, dollars, and rubles. At the beginning of the deposit agreement, Alena's account contained 3000 euros, 4000 dollars, and 240000 rubles. The interes... | Answer 17: $3280;
Answer 18: 4040 euros,
Answer 19: 301492 rubles.
## Comment:
1 year
Euros: 3000 euros $\times(1+2.1 \%)=3063$ euros.
Dollars: 4000 dollars $\times(1+2.1 \%)=4084$ dollars.
Rubles: 240000 rubles $\times(1+7.9 \%)=258960$ rubles.
2 year
Euros: (3063 euros - 1000 euros $) \times(1+2.1 \%)=2106$ ... | 3280 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,519 |
Problem 6. (4 points)
Ivan bought a used car from 2010 for 90,000 rubles with an engine power of 150 hp and registered it on January 29. On August 21 of the same year, the citizen sold his car and a month later bought a horse and a cart for 7,500 rubles. The transport tax rate is set at 20 rubles per 1 hp. What amount... | Solution: transport tax $=150 \times 20 \times 8 / 12=2000$ rubles. A horse and a cart are not subject to transport tax. | 2000 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,521 |
Problem 7. (6 points)
Sergei, being a student, worked part-time in a student cafe after classes for a year. Sergei's salary was 9000 rubles per month. In the same year, Sergei paid for his medical treatment at a healthcare facility in the amount of 100000 rubles and purchased medications on a doctor's prescription for... | Solution: the amount of the social tax deduction for medical treatment will be: $100000+20000=$ 120000 rubles. The possible tax amount eligible for refund under this deduction will be $120000 \times 13\% = 15600$ rubles. However, in the past year, Sergey paid income tax (NDFL) in the amount of $13\% \times (9000 \times... | 14040 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,522 |
Problem 8. (2 points)
After graduating from a technical university, Oleg started his own business producing water heaters. This year, Oleg plans to sell 5000 units of water heaters. Variable costs for production and sales of one water heater amount to 800 rubles, and total fixed costs are 1000 thousand rubles. Oleg wa... | Solution: the price of one kettle $=((1000000+0.8 \times 5000)+1500$ 000) / $(1000000$ + $0.8 \times 5000)=1300$ rub. | 1300 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,523 |
Problem 10. (4 points)
To buy new headphones costing 275 rubles, Katya decided to save money on sports activities. Until now, she has been buying a single-visit ticket to the swimming pool, including a visit to the sauna for 250 rubles, to warm up. However, summer has arrived, and the need to visit the sauna has disap... | Solution: one visit to the sauna costs 25 rubles, the price of one visit to the swimming pool is 225 rubles. Katya needs to visit the swimming pool 11 times without going to the sauna in order to save up for buying headphones. | 11 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,524 |
Problem 18. (4 points)
By producing and selling 4000 items at a price of 6250 rubles each, a budding businessman earned 2 million rubles in profit. Variable costs for one item amounted to 3750 rubles. By what percentage should the businessman reduce the production volume to make his revenue equal to the cost? (Provide... | Solution: fixed costs $=-2$ million, gap $+4000 \times 6250-3750 \times 4000$ million $=8$ million.
$6.25 \times Q=3750 Q+8 \text{ million}$.
$Q=3200$ units.
Can be taken out of production $=4000-3200=800$ units, that is, $20\%$. | 20 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,527 |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive than a, which means the highest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend $4800 + 1500 - 4800 * 0.2 = 5340$ rubles. This is the most profitable way to make t... | # Solution:
The average value of the last purchases is $(785+2033+88+3742+1058) / 5 = 1541.2$ rubles. Therefore, an allowable purchase is no more than $1541.2 * 3 = 4623.6$ rubles. With this amount, you can buy $4623.6 / 55 \approx 84$ chocolates.
## Maximum 20 points
20 points - fully detailed solution and correct ... | 84 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,531 |
2. Petya is the owner of a cashback card offering 1% on all purchases. In addition to this, the bank offers Petya the option to choose categories with increased cashback each month, but no more than three. Which categories would be the most profitable to choose, considering the cashback percentage and the expected expe... | Answer: $2 ; 3 ; 4$.
Cashback is the percentage of the purchase amount that is returned to the card after the purchase.
Let's calculate the cashback that Petya will receive in each case:
1) $2000 * 0.05 = 100$ rubles;
2) $5000 * 0.03 = 150$ rubles;
3) $3000 * 0.04 = 120$ rubles;
4) $3000 * 0.05 = 150$ rubles;
5) $15... | 2;3;4 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,532 |
5. Entrepreneurs Vasiliy Petrovich and Pyotr Gennadiyevich opened a clothing factory called "ViP". Vasiliy Petrovich invested 200 thousand rubles, and Pyotr Gennadiyevich invested 350 thousand rubles. The factory proved to be successful, and after a year, Anastasia Alekseevna approached them with an offer to buy part o... | Answer: 1,000,000. Anastasia Alekseevna paid 1,100,000 for her share, which is one-third, so all the shares of the factory are worth 3,300,000. Let the shares of Vasiliy Petrovich be worth X rubles, then the shares of Petr Gennadiyevich are worth $3,300,000 - X$ rubles. It should be that $200 / 350 = X / (3,300,000 - X... | 1,000,000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,534 |
6. Vasya's car was insured under a CASCO policy. During the policy period, an insurance event occurred - Vasya got into an accident with his car. How will this affect the insurance premium when Vasya renews his insurance?
1) the insurance premium will increase
2) the insurance premium will decrease
3) it will not affec... | Answer: 1. The insurance premium is the amount paid by the insured when arranging insurance. Due to the accident in the past, the insurance will cost Vasya more. | 1 | Other | MCQ | Yes | Yes | olympiads | false | 11,535 |
7. Leshа has 10 million rubles. Into what minimum number of banks should he deposit them to receive the full amount through ACB insurance payouts in case the banks cease operations? | Answer: 8. The maximum insurance payout is 1,400,000, which means no more than this amount should be deposited in each bank. | 8 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,536 |
1. In the "Sweet Tooth" store, they are thinking of what promotion to announce before March 8. Manager Vasya suggests reducing the price of a box of candies by $20 \%$ and hopes to sell twice as much of the product as usual. Deputy Director Kolya says that it would be more profitable to increase the price of the same b... | Answer: 6000.
Let the box of candies cost $\mathrm{X}$ rubles.
In Vasya's option, the revenue will be $(10000 / \mathrm{X}) * 2 * 0.8 * \mathrm{X}=16000$ rubles.
In Kolya's option: $(10000 / X) * 4 / 3 * X=13333.(3)$.
Thus, the revenue will be higher in Vasya's option, and it will be 6000 rubles more than usual. | 6000 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,537 |
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