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18. Traffic Light (from 10th grade. 2 points). A traffic light at a pedestrian crossing allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for the green light for a pedestrian who approaches the intersection.
, and therefore the conditional expected waiting time $T$ in this case is 0: $\mathrm{E}(T \mid G)=0$. With probability $\frac{2}{3}$, the pedestrian falls on the red signal and has to wait (event $\ba... | 40 | Other | math-word-problem | Yes | Yes | olympiads | false |
13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).
Upon entering the station, the Scientist boards the first train that arrives. It i... | Solution. If the Scientist boarded trains of different directions with equal probabilities, the average travel time in one direction and the average travel time in the other would be the same. Therefore, the probabilities are not equal.
Let $p$ be the probability that the Scientist boards a train going clockwise. Then... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).
Upon entering the station, the Scientist boards the first train that arrives. It i... | Solution. If the Scientist boarded trains of different directions with equal probabilities, the average travel time in one direction and the average travel time in the other would be the same. Therefore, the probabilities are not equal.
Let $p$ be the probability that the Scientist boards a train going clockwise. Then... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
13. Ring Line (from 8th grade. 3 points). On weekdays, the Absent-Minded Scientist travels to work on the ring line of the Moscow metro from the station "Taganskaya" to the station "Kievskaya", and back in the evening (see the diagram).
Upon entering the station, the Scientist boards the first train that arrives. It i... | Solution. If the Scientist boarded trains of different directions with equal probabilities, the average travel time in one direction and the average travel time in the other would be the same. Therefore, the probabilities are not equal.
Let $p$ be the probability that the Scientist boards a train going clockwise. Then... | 3 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
18. Traffic Light (from 10th grade. 2 points). A traffic light at a pedestrian crossing allows pedestrians to cross the street for one minute and prohibits crossing for two minutes. Find the average waiting time for the green light for a pedestrian who approaches the intersection.
, and therefore the conditional expected waiting time $T$ in this case is 0: $\mathrm{E}(T \mid G)=0$. With probability $\frac{2}{3}$, the pedestrian falls on the red signal and has to wait (event $\ba... | 40 | Other | math-word-problem | Yes | Yes | olympiads | false |
4. Stem-and-leaf plot. (From 6th grade, 2 points). To represent whole numbers or decimal fractions, a special type of diagram called a "stem-and-leaf plot" is often used. Such diagrams are convenient for representing people's ages. Suppose that in the studied group, there are 5 people aged 19, 34, 37, 42, and 48. For t... | Solution. The digits from 0 to 5, representing decades of years, can be placed immediately (Fig. 4a). It is clear that less than 10 years have passed, otherwise there would be no digits in line "1".
If 7 or more years had passed, then the person who is 13 years old would have moved to line "2", and there would be fewe... | 6 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
18. The figure shows a track scheme for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they want, returning to the starting point.
The young dr... | # Solution.
Let $M_{n}$ be the number of all possible routes of duration $n$ minutes. Each such route consists of exactly $n$ segments (a segment is the segment $A B, B A$ or the loop $B B$).
Let $M_{n, A}$ be the number of such routes ending at $A$, and $M_{n, B}$ be the number of routes ending at $B$.
A point $B$ ... | 34 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
11. Toll Road (from 8th grade. 3 points). The cost of traveling along a section of a toll road depends on the class of the vehicle: passenger cars belong to the first class, for which the travel cost is 200 rubles, and light trucks and minivans belong to the second class, for which the cost is 300 rubles.
At the entra... | Solution. We will construct both graphs in the same coordinate system. Draw a vertical line $x=h$ through the point of intersection of the graphs. This value of $h-$ is the one we are looking for.
. Ivan the Tsarevich is learning to shoot a bow. He put 14 arrows in his quiver and went to shoot at pine cones in the forest. He knocks down a pine cone with a probability of 0.1, and for each pine cone he knocks down, the Frog-Princess gives him 3 more arrows.... | Solution. First method. Let Ivan have $n$ arrows at the present moment. Let $X_{0}$ be the random variable "the number of shots needed to reduce the number of arrows by one." Ivan makes a shot. Consider the random variable - the indicator $I$ of a successful shot. $I=0$, if the shot is unsuccessful (probability of this... | 20 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
4. Traffic Lights (from 9th grade. 2 points). Long Highway intersects with Narrow Street and Quiet Street (see fig.). There are traffic lights at both intersections. The first traffic light allows traffic on the highway for $x$ seconds, and for half a minute on Narrow St. The second traffic light allows traffic on the ... | Solution. First method. We will measure time in seconds. The probability of passing the intersection with Narrow St. without stopping is $\frac{x}{x+30}$. The probability of passing the intersection with Quiet St. without stopping is $\frac{120}{x+120}$. Since the traffic lights operate independently of each other, the... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
18. The diagram shows a track layout for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they like, returning to point $A$.
The young driver, Yu... | # Solution.
Let $M_{n}$ be the number of all possible routes of duration $n$ minutes. Each such route consists of exactly $n$ segments (a segment is the segment $A B, B A$ or the loop $B B$).
Let $M_{n, A}$ be the number of such routes ending at $A$, and $M_{n, B}$ be the number of routes ending at $B$.
A point $B$ ... | 34 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
11. Toll Road (from 8th grade. 3 points). The cost of traveling along a section of a toll road depends on the class of the vehicle: passenger cars belong to the first class, for which the travel cost is 200 rubles, and light trucks and minivans belong to the second class, for which the cost is 300 rubles.
At the entra... | Solution. We will construct both graphs in the same coordinate system. Draw a vertical line $x=h$ through the point of intersection of the graphs. This value of $h-$ is the one we are looking for.
. Long Highway intersects with Narrow Street and Quiet Street (see fig.). There are traffic lights at both intersections. The first traffic light allows traffic on the highway for $x$ seconds, and for half a minute on Narrow St. The second traffic light allows traffic on the ... | Solution. First method. We will measure time in seconds. The probability of passing the intersection with Narrow St. without stopping is $\frac{x}{x+30}$. The probability of passing the intersection with Quiet St. without stopping is $\frac{120}{x+120}$. Since the traffic lights operate independently of each other, the... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
7. Solution. Suppose for clarity of reasoning that when a bite occurs, the Absent-Minded Scholar immediately pulls out and re-casts the fishing rod, and does so instantly. After this, he waits again. Consider a 6-minute time interval. During this time, on average, there are 3 bites on the first fishing rod and 2 bites ... | Answer: 1 minute 12 seconds.
Evaluation Criteria
| Correct and justified solution | 3 points |
| :--- | :---: |
| It is shown that on average there are 5 bites in 6 minutes, or an equivalent statement is proven | 1 point |
| The solution is incorrect or missing (in particular, only the answer is given) | 0 points | | 1 | Other | math-word-problem | Yes | Yes | olympiads | false |
# 8. Solution.
a) Suppose there are 9 numbers in the set. Then five of them do not exceed the median, which is the number 2. Another four numbers do not exceed the number 13. Therefore, the sum of all numbers in the set does not exceed
$$
5 \cdot 2 + 4 \cdot 13 = 62
$$
Since the arithmetic mean is 7, the sum of the ... | Answer: a) no; b) 11.
Scoring criteria
| Both parts solved correctly or only part (b) | 3 points |
| :--- | :---: |
| The correct estimate of the number of numbers in part (b) is found, but no example is given | 2 points |
| Part (a) is solved correctly | 1 point |
| The solution is incorrect or missing (in particula... | 11 | Other | math-word-problem | Yes | Yes | olympiads | false |
7. Solution. For clarity, let's assume that when a bite occurs, the Absent-Minded Scholar immediately reels in and casts the fishing rod again, and does so instantly. After this, he waits again. Consider a 5-minute time interval. During this time, on average, there are 5 bites on the first fishing rod and 1 bite on the... | Answer: 50 seconds.
## Grading Criteria
| Solution is correct and well-reasoned | 3 points |
| :--- | :---: |
| It is shown that on average there are 5 bites in 6 minutes, or an equivalent statement is proven | 1 point |
| Solution is incorrect or missing (including only the answer) | 0 points | | 50 | Other | math-word-problem | Yes | Yes | olympiads | false |
# 8. Solution.
a) Suppose the set contains 7 numbers. Then four of them are not less than the median, which is the number 10. Another three numbers are not less than one. Then the sum of all numbers in the set is not less than
$$
3+4 \cdot 10=43
$$
Since the arithmetic mean is 6, the sum of the numbers in the set is... | Answer: a) no; b) 9.
## Grading Criteria
| Both parts solved correctly or only part (b) | 3 points |
| :--- | :---: |
| Correct estimate of the number of numbers in part (b), but no example | 2 points |
| Part (a) solved correctly | 1 point |
| Solution is incorrect or missing (including only the answer) | 0 points | | 9 | Other | math-word-problem | Yes | Yes | olympiads | false |
18. The figure shows a track scheme for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they want, returning to the starting point.
The young dr... | # Solution.
Let $M_{n}$ be the number of all possible routes of duration $n$ minutes. Each such route consists of exactly $n$ segments (a segment is the segment $A B, B A$ or the loop $B B$).
Let $M_{n, A}$ be the number of such routes ending at $A$, and $M_{n, B}$ be the number of routes ending at $B$.
A point $B$ ... | 34 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
11. Toll Road (from 8th grade. 3 points). The cost of traveling along a section of a toll road depends on the class of the vehicle: passenger cars belong to the first class, for which the travel cost is 200 rubles, and light trucks and minivans belong to the second class, for which the cost is 300 rubles.
At the entra... | Solution. We will construct both graphs in the same coordinate system. Draw a vertical line $x=h$ through the point of intersection of the graphs. This value of $h-$ is the one we are looking for.
. Long Highway intersects with Narrow Street and Quiet Street (see fig.). There are traffic lights at both intersections. The first traffic light allows traffic on the highway for $x$ seconds, and for half a minute on Narrow St. The second traffic light allows traffic on the ... | Solution. First method. We will measure time in seconds. The probability of passing the intersection with Narrow St. without stopping is $\frac{x}{x+30}$. The probability of passing the intersection with Quiet St. without stopping is $\frac{120}{x+120}$. Since the traffic lights operate independently of each other, the... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
11. Toll Road (from 8th grade. 3 points). The cost of traveling along a section of a toll road depends on the class of the vehicle: passenger cars belong to the first class, for which the travel cost is 200 rubles, and light trucks and minivans belong to the second class, for which the cost is 300 rubles.
At the entra... | Solution. We will construct both graphs in the same coordinate system. Draw a vertical line $x=h$ through the point of intersection of the graphs. This value of $h-$ is the one we are looking for.
. Long Highway intersects with Narrow Street and Quiet Street (see fig.). There are traffic lights at both intersections. The first traffic light allows traffic on the highway for $x$ seconds, and for half a minute on Narrow St. The second traffic light allows traffic on the ... | Solution. First method. We will measure time in seconds. The probability of passing the intersection with Narrow St. without stopping is $\frac{x}{x+30}$. The probability of passing the intersection with Quiet St. without stopping is $\frac{120}{x+120}$. Since the traffic lights operate independently of each other, the... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
11. Toll Road (from 8th grade. 3 points). The cost of traveling along a section of a toll road depends on the class of the vehicle: passenger cars belong to the first class, for which the travel cost is 200 rubles, and light trucks and minivans belong to the second class, for which the cost is 300 rubles.
At the entra... | Solution. We will construct both graphs in the same coordinate system. Draw a vertical line $x=h$ through the point of intersection of the graphs. This value of $h-$ is the one we are looking for.
. Long Highway intersects with Narrow Street and Quiet Street (see fig.). There are traffic lights at both intersections. The first traffic light allows traffic on the highway for $x$ seconds, and for half a minute on Narrow St. The second traffic light allows traffic on the ... | Solution. First method. We will measure time in seconds. The probability of passing the intersection with Narrow St. without stopping is $\frac{x}{x+30}$. The probability of passing the intersection with Quiet St. without stopping is $\frac{120}{x+120}$. Since the traffic lights operate independently of each other, the... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
18. The figure shows a track scheme for karting. The start and finish are at point $A$, and the kart driver can make as many laps as they want, returning to the starting point.
The young dr... | # Solution.
Let $M_{n}$ be the number of all possible routes of duration $n$ minutes. Each such route consists of exactly $n$ segments (a segment is the segment $A B, B A$ or the loop $B B$).
Let $M_{n, A}$ be the number of such routes ending at $A$, and $M_{n, B}$ be the number of routes ending at $B$.
A point $B$ ... | 34 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
2. Ninth-grader Gavriil decided to weigh a basketball, but he only had 400 g weights, a light ruler with the markings at the ends worn off, a pencil, and many weightless threads at his disposal. Gavriil suspended the ball from one end of the ruler and the weight from the other, and balanced the ruler on the pencil. The... | 2. Let the distances from the pencil to the ball and to the weight be $l_{1}$ and $l_{2}$ respectively at the first equilibrium. Denote the magnitude of the first shift by $x$, and the total shift over two times by $y$. Then the three conditions of lever equilibrium will be:
$$
\begin{gathered}
M l_{1}=m l_{2} \\
M\le... | 600 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. The distances from three points lying in a horizontal plane to the base of a television tower are 800 m, 700 m, and 500 m, respectively. From each of these three points, the tower is visible (from base to top) at a certain angle, and the sum of these three angles is $90^{\circ}$. A) Find the height of the te... | Solution. Let the given distances be denoted by $a, b$, and $c$, the corresponding angles by $\alpha, \beta$, and $\gamma$, and the height of the tower by $H$. Then $\operatorname{tg} \alpha=\frac{H}{a}, \operatorname{tg} \beta=\frac{H}{b}, \operatorname{tg} \gamma=\frac{H}{c}$. Since $\frac{H}{c}=\operatorname{tg} \ga... | 374 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Problem 3. All students in the class scored different numbers of points (positive integers) on the test, with no duplicate scores. In total, they scored 119 points. The sum of the three lowest scores is 23 points, and the sum of the three highest scores is 49 points. How many students took the test? How many points did... | Solution. Let's denote all the results in ascending order $a_{1}, a_{2}, \ldots, a_{n}$, where $n$ is the number of students. Since $a_{1}+a_{2}+a_{3}=23$ and $a_{n-2}+a_{n-1}+a_{n}=49$, the sum of the numbers between $a_{3}$ and $a_{n-2}$ is $119-23-49=47$.
Since $a_{1}+a_{2}+a_{3}=23$, then $a_{3} \geq 9$ (otherwise... | 10 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
Problem 4. One mole of an ideal gas undergoes a closed cycle, in which:
$1-2$ - isobaric process, during which the volume increases by 4 times;
$2-3$ - isothermal process, during which the pressure increases;
$3-1$ - a process in which the gas is compressed according to the law $T=\gamma V^{2}$.
Find how many times... | Solution. Let the initial volume and pressure be denoted as $\left(V_{0} ; P_{0}\right)$. Then $V_{2}=4 V_{0}$.
From the Mendeleev-Clapeyron law, we have three relationships:
$$
P_{0} V_{0}=R T_{1}, P_{0} V_{2}=R T, P_{3} V_{3}=R T
$$
Dividing the third relationship by the second, we get: $\frac{P_{3}}{P_{0}}=\frac{... | 2 | Other | math-word-problem | Yes | Yes | olympiads | false |
Problem 3. All students in the class scored a different number of points (positive integers) on the test, with no duplicate scores. In total, they scored 119 points. The sum of the three lowest scores is 23 points, and the sum of the three highest scores is 49 points. How many students took the test? How many points di... | Solution. Let's denote all the results in ascending order $a_{1}, a_{2}, \ldots, a_{n}$, where $n$ is the number of students. Since $a_{1}+a_{2}+a_{3}=23$ and $a_{n-2}+a_{n-1}+a_{n}=49$, the sum of the numbers between $a_{3}$ and $a_{n-2}$ is $119-23-49=47$.
Since $a_{1}+a_{2}+a_{3}=23$, then $a_{3} \geq 9$ (otherwise... | 10 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
1. If the cold water tap is opened, the bathtub fills up in 5 minutes and 20 seconds. If both the cold water tap and the hot water tap are opened simultaneously, the bathtub fills up to the same level in 2 minutes. How long will it take to fill the bathtub if only the hot water tap is opened? Give your answer in second... | Solution. According to the condition: $\frac{16}{3} v_{1}=1,\left(v_{1}+v_{2}\right) 2=1$, where $v_{1}, v_{2}$ are the flow rates of water from the cold and hot taps, respectively. From this, we get: $v_{1}=3 / 16, v_{2}=5 / 16$.
Then the time to fill the bathtub from the hot tap is $\frac{16}{5}$. Answer: 3 minutes ... | 192 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. A weight with a mass of 200 grams stands on a table. It was flipped and placed on the table with a different side, the area of which is 15 sq. cm smaller. As a result, the pressure on the table increased by 1200 Pa. Find the area of the side on which the weight initially stood. Give your answer in sq. cm, rounding t... | Solution. After converting to SI units, we get: $\frac{2}{S-1.5 \cdot 10^{-3}}-\frac{2}{S}=1200$.
Here $S-$ is the area of the original face.
From this, we get a quadratic equation: $4 \cdot 10^{5} S^{2}-600 S-1=0$.
After substituting the variable $y=200 S$, the equation becomes: $10 y^{2}-3 y-1=0$, the solution of ... | 25 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. The villages of Arkadino, Borisovo, and Vadimovo are connected by straight roads. A square field adjoins the road between Arkadino and Borisovo, one side of which completely coincides with this road. A rectangular field adjoins the road between Borisovo and Vadimovo, one side of which completely coincides with this ... | Solution. The condition of the problem can be expressed by the following relation:
$r^{2}+4 p^{2}+45=12 q$
where $p, q, r$ are the lengths of the roads opposite the settlements Arkadino, Borisovo, and Vadimovo, respectively.
This condition is in contradiction with the triangle inequality:
$r+p>q \Rightarrow 12 r+12... | 135 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. Alloy $A$ of two metals with a mass of 6 kg, in which the first metal is twice as much as the second, placed in a container with water, creates a pressure force on the bottom of $30 \mathrm{N}$. Alloy $B$ of the same metals with a mass of 3 kg, in which the first metal is five times less than the second, placed in a... | Solution. Due to the law of conservation of mass, in the resulting alloy, the mass of each metal is equal to the sum of the masses of these metals in the initial alloys. Thus, both the gravitational forces and the forces of Archimedes also add up. From this, it follows that the reaction force will be the sum of the rea... | 40 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. From a square steel sheet with a side of 1 meter, a triangle is cut off from each of the four corners so that a regular octagon remains. Determine the mass of this octagon if the sheet thickness is 3 mm and the density of steel is 7.8 g/cm ${ }^{3}$. Give your answer in kilograms, rounding to the nearest whole numbe... | Answer: $46.8(\sqrt{2}-1) \approx 19$ kg.
Solution. A regular octagon must have equal angles and sides. Therefore, four equal triangles with angles $45^{\circ}, 45^{\circ}$, and $90^{\circ}$ are cut off. If the legs of this triangle are equal to $x$, then the hypotenuse is $x \sqrt{2}$ - this will be the side of the o... | 19 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank the pond dry in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it dry in 30 days. How many days would it take for one cow... | Answer: In 75 days.
Solution. Let the pond have a volume of $a$ (conditional units). These units can be liters, buckets, cubic meters, etc. Let one cow drink $b$ (conditional units) of water per day, and the springs add $c$ (conditional units) of water per day. Then the first condition of the problem is equivalent to ... | 75 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. Gavriila was traveling in Africa. On a sunny and windy day, at noon, when the rays from the Sun fell vertically, the boy threw a ball from behind his head at a speed of 5 m/s against the wind at an angle to the horizon. After 1 second, the ball hit him in the stomach 1 m below the point of release. Determine the gre... | Answer: 75 cm.
Solution. In addition to the force of gravity, a constant horizontal force $F=m \cdot a$ acts on the body, directed opposite. In a coordinate system with the origin at the point of throw, the horizontal axis $x$ and the vertical axis $y$, the law of motion has the form:
$$
\begin{aligned}
& x(t)=V \cdo... | 75 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. To lift a load, it is attached to the hook of a crane using slings made of steel cable. The calculated mass of the load is $M=20$ t, the number of slings $n=3$. Each sling forms an angle $\alpha=30^{\circ}$ with the vertical. All slings carry the same load during the lifting of the cargo. According to safety require... | Answer: 26 mm
Solution. For each of the $n$ lower tie-downs, the force of the cargo weight is $\frac{P}{n}$. Then the tension force in the tie will be $N=\frac{P}{n \cdot \cos \alpha}$. Therefore, the strength of the rope must be $Q \geq k T=\frac{k P}{n \cdot \cos \alpha}$.
Since the strength of the rope $Q$ is dete... | 26 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. On the shores of a circular island (viewed from above), there are cities $A, B, C$, and $D$. The straight asphalt road $A C$ divides the island into two equal halves. The straight asphalt road $B D$ is shorter than road $A C$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also h... | Answer: 450 sq. km. Solution. The condition of the problem means that a quadrilateral $ABCD$ is given, in which angles $B$ and $D$ are right (they rest on the diameter), $AB = BC$ (both roads are dirt roads, and the cyclist travels them in the same amount of time), $BD = 15 \frac{\text{km}}{\text{hour}} \cdot 2$ hours ... | 450 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. One mole of a monatomic ideal gas undergoes a cyclic process $a b c a$. The diagram of this process in the $P-T$ axes represents a curvilinear triangle, the side $a b$ of which is parallel to the $T$ axis, the side $b c$ - a segment of a straight line passing through the origin, and the side $c a$ - an arc of a para... | Answer: 664 J. Solution. Process $a b$ is an isobar, process $b c$ is an isochore, process $c a$ is described by the equation $T=P(d-k P)$, where $d, k$ are some constants. It is not difficult to see that in such a process, the volume turns out to be a linear function of pressure, that is, in the $P V$ axes, this cycli... | 664 | Other | math-word-problem | Yes | Yes | olympiads | false |
5. For moving between points located hundreds of kilometers apart on the Earth's surface, people in the future will likely dig straight tunnels through which capsules will move without friction, solely under the influence of Earth's gravity. Let points $A, B$, and $C$ lie on the same meridian, and the distance from $A$... | Answer: 42 min. Solution. Let point $O$ be the center of the Earth. To estimate the time of motion from $A$ to B, consider triangle $A O B$. We can assume that the angle $\alpha=90^{\circ}-\angle A B O$ is very small, so $\sin \alpha \approx \alpha$. Since the point in the tunnel $A B$ is attracted to the center by the... | 42 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2.1. Gavriila found out that the front tires of the car last for 20000 km, while the rear tires last for 30000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 2.1. Gavriila found out that the front tires of the car last for 20000 km, while the rear tires last for 30000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{24000\}$. | 24000 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
2.2. Gavriila found out that the front tires of the car last for 24000 km, while the rear tires last for 36000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 2.2. Gavriila found out that the front tires of the car last for 24000 km, while the rear tires last for 36000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{28800\}$. | 28800 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2.3. Gavriila found out that the front tires of the car last for 42,000 km, while the rear tires last for 56,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 2.3. Gavriila found out that the front tires of the car last for 42000 km, while the rear tires last for 56000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{48000\}$. | 48000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2.4. Gavriila found out that the front tires of the car last for 21,000 km, while the rear tires last for 28,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 2.4. Gavriila found out that the front tires of the car last for 21000 km, while the rear tires last for 28000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{24000\}$. | 24000 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
3.2. Two identical cylindrical vessels are connected at the bottom by a small-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \, \text{cm} \). At what level will the water stabili... | 3.2. Two identical cylindrical vessels are connected at the bottom by a small-section pipe with a valve. While the valve was closed, water was poured into the first vessel, and oil into the second, so that the level of the liquids was the same and equal to \( h = 40 \, \text{cm} \). At what level will the water stabili... | 34 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. On the shores of a circular island (viewed from above), there are cities $A, B, C$, and $D$. A straight asphalt road $A C$ divides the island into two equal halves. A straight asphalt road $B D$ is shorter than road $A C$ and intersects it. The speed of a cyclist on any asphalt road is 15 km/h. The island also has s... | Answer: 450 sq. km. Solution. The condition of the problem means that a quadrilateral $ABCD$ is given, in which angles $B$ and $D$ are right (they rest on the diameter), $AB=BC$ (both roads are dirt roads, and the cyclist travels them in the same amount of time), $BD=15 \frac{\text { km }}{\text { h }} 2$ hours $=30$ k... | 450 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Scientists have found a fragment of an ancient manuscript on mechanics. It was a piece of a book, the first page of which was numbered 435, and the last page was written with the same digits but in some other order. How many sheets were in this fragment? | Solution. Since the sheet has 2 pages and the first page is odd, the last page must be even. Therefore, the last digit is 4. The number of the last page is greater than the first. The only possibility left is 534. This means there are 100 pages in total, and 50 sheets.
Answer: 50.
Criteria: 20 points - correct (not n... | 50 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
3. Usually, schoolboy Gavriil takes a minute to go up a moving escalator by standing on its step. But if Gavriil is late, he runs up the working escalator and thus saves 36 seconds. Today, there are many people at the escalator, and Gavriil decides to run up the adjacent non-working escalator. How much time will such a... | Solution. Let's take the length of the escalator as a unit. Let $V$ be the speed of the escalator, and $U$ be the speed of Gavrila 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 ... | 40 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. The engines of a rocket launched vertically upward from the Earth's surface, providing the rocket with an acceleration of $20 \mathrm{~m} / \mathrm{c}^{2}$, suddenly stopped working 40 seconds after launch. To what maximum height will the rocket rise? Can this rocket pose a danger to an object located at an altitude... | Answer: a) 48 km; b) yes. Solution. Let $a=20$ m/s², $\tau=40$ s. On the first segment of the motion, when the engines were working, the speed and the height gained are respectively: $V=a t, y=\frac{a t^{2}}{2}$. Therefore, at the moment the engines stop working: $V_{0}=a \tau, y_{0}=\frac{a \tau^{2}}{2}$ - this will b... | 48 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. The engines of a rocket launched vertically upward from the Earth's surface, providing the rocket with an acceleration of $30 \mathrm{~m} / \mathrm{c}^{2}$, suddenly stopped working 30 seconds after launch. To what maximum height will the rocket rise? Can this rocket pose a danger to an object located at an altitude... | Answer: a) 54 km; b) yes. Solution. Let $a=30 \mathrm{m} / \mathrm{s}^{2}, \tau=30$ s. On the first segment of the motion, when the engines were working, the speed and the height gained are respectively: $V=a t, y=\frac{a t^{2}}{2}$. Therefore, at the moment the engines stop: $V_{0}=a \tau, y_{0}=\frac{a \tau^{2}}{2}$ ... | 54 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. A mobile railway platform has a horizontal bottom in the form of a rectangle 10 meters long and 4 meters wide, loaded with sand. The surface of the sand has an angle of no more than 45 degrees with the base plane (otherwise the sand grains will spill), the density of the sand is 1500 kg/m³. Find the maximum mass of ... | Answer: 52 t. Solution. The calculation shows that the maximum height of the sand pile will be equal to half the width of the platform, that is, 2 m. The pile can be divided into a "horizontally lying along the platform" prism (its height is 6 m, and the base is an isosceles right triangle with legs $2 \sqrt{2}$ and hy... | 52000 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. The engines of a rocket launched vertically upward from the Earth's surface, providing the rocket with an acceleration of $20 \mathrm{~m} / \mathrm{c}^{2}$, suddenly stopped working 50 seconds after launch. To what maximum height will the rocket rise? Can this rocket pose a danger to an object located at an altitude... | Answer: a) 75 km; b) yes. Solution. Let $a=20$ m/s$^2$, $\tau=50$ s. During the first part of the motion, when the engines were working, the speed and the height gained are respectively: $V=a t$, $y=\frac{a t^{2}}{2}$. Therefore, at the moment the engines stop: $V_{0}=a \tau$, $y_{0}=\frac{a \tau^{2}}{2}$ - this will b... | 75 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. A mobile railway platform has a horizontal bottom in the form of a rectangle 8 meters long and 5 meters wide, loaded with grain. The surface of the grain has an angle of no more than 45 degrees with the base plane (otherwise the grains will spill), the density of the grain is 1200 kg/m³. Find the maximum mass of gra... | Answer: 47.5 t. Solution. The calculation shows that the maximum height of the grain pile will be half the width of the platform, that is, 2.5 m. The pile can be divided into a "horizontally lying along the platform" prism (its height is 3 m, and the base is a right-angled isosceles triangle with legs $\frac{5 \sqrt{2}... | 47500 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. The engines of a rocket launched vertically upward from the Earth's surface, providing the rocket with an acceleration of $30 \mathrm{~m} / \mathrm{c}^{2}$, suddenly stopped working 20 seconds after launch. To what maximum height will the rocket rise? Can this rocket pose a danger to an object located at an altitude... | Answer: a) 24 km; b) yes. Solution. Let $a=30 \mathrm{m} / \mathrm{s}^{2}, \tau=20$ s. During the first part of the motion, when the engines were working, the speed and the height gained are respectively: $V=a t, y=\frac{a t^{2}}{2}$. Therefore, at the moment the engines stop: $V_{0}=a \tau, y_{0}=\frac{a \tau^{2}}{2}$... | 24 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. A mobile railway platform has a horizontal bottom in the form of a rectangle 8 meters long and 4 meters wide, loaded with sand. The surface of the sand has an angle of no more than 45 degrees with the base plane (otherwise the sand grains will spill), the density of the sand is 1500 kg/m³. Find the maximum mass of s... | Answer: 40 t. Solution. The calculation shows that the maximum height of the sand pile will be equal to half the width of the platform, that is, 2 m. The pile can be divided into a "horizontally lying along the platform" prism (its height is 4 m, and the base is an isosceles right triangle with legs $2 \sqrt{2}$ and hy... | 40000 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
4. A ten-liter bucket was filled to the brim with currants. Gavrila immediately said that there were 10 kg of currants in the bucket. Glafira thought about it and estimated the weight of the berries in the bucket more accurately. How can this be done if the density of the currant can be approximately considered equal t... | Solution. In approximate calculations, the sizes of the berries can be considered the same and much smaller than the size of the bucket. If the berries are laid out in one layer, then in the densest packing, each berry will have 6 neighbors: the centers of the berries will be at the vertices of equilateral triangles wi... | 7 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1. The time of the aircraft's run from the moment of start until the moment of 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 num... | Answer: 208
$v=a t, 100000 / 3600=a \cdot 15$, from which $a=1.85\left(\mathrm{m} / \mathrm{s}^{2}\right)$. Then $S=a t^{2} / 2=208$ (m). | 208 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Hot oil at a temperature of $100^{\circ} \mathrm{C}$ in a volume of two liters is mixed with one liter of cold oil at a temperature of $20^{\circ} \mathrm{C}$. What volume will the mixture have when thermal equilibrium is established in the mixture? Heat losses to the external environment can be neglected. The coeff... | Answer: 3
Let $V_{1}=2$ L be the volume of hot oil, and $V_{2}=1$ L be the volume of cold oil. Then we can write $V_{1}=U_{1}\left(1+\beta t_{1}\right), V_{2}=U_{2}\left(1+\beta t_{2}\right)$, where $U_{1}, U_{2}$ are the volumes of the respective portions of oil at zero temperature; $t_{1}=100^{\circ} \mathrm{C}, t_{... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. 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 Tetr... | Answer: 257
Let's assume the "capacity" of the smartphone battery is 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 equation $\fr... | 257 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. Grandma baked 19 pancakes. The grandchildren came from school and started eating them. While the younger grandson eats 1 pancake, the older grandson eats 3 pancakes, and during this time, grandma manages to cook 2 more pancakes. When they finished, there were 11 pancakes left on the plate. How many pancakes ... | Solution. In one "cycle", the grandsons eat $1+3=4$ pancakes, and the grandmother bakes 2 pancakes, which means the number of pancakes decreases by 2. There will be ( $19-11$ ) $/ 2=4$ such cycles. This means, in these 4 cycles, the younger grandson ate 4 pancakes, the older grandson ate 12 pancakes, and the grandmothe... | 12 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
Problem 3. Experimenters Glafira and Gavriil placed a triangle made of thin wire with sides of 30 mm, 40 mm, and 50 mm on a white flat surface. This wire is covered with millions of mysterious microorganisms. The scientists found that when an electric current is applied to the wire, these microorganisms begin to move c... | Solution. In one minute, the microorganism moves 10 mm. Since in a right triangle with sides $30, 40, 50$, the radius of the inscribed circle is 10, all points inside the triangle are no more than 10 mm away from the sides of the triangle. Therefore, the microorganisms will fill the entire interior of the triangle.
Wh... | 2114 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
Problem 4. All students in the class scored different numbers of points (positive integers) on the test, with no duplicate scores. In total, they scored 119 points. The sum of the three lowest scores is 23 points, and the sum of the three highest scores is 49 points. How many students took the test? How many points did... | Solution. Let's denote all the results in ascending order $a_{1}, a_{2}, \ldots, a_{n}$, where $n$ is the number of students. Since $a_{1}+a_{2}+a_{3}=23$ and $a_{n-2}+a_{n-1}+a_{n}=49$, the sum of the numbers between $a_{3}$ and $a_{n-2}$ is $119-23-49=47$.
Since $a_{1}+a_{2}+a_{3}=23$, then $a_{3} \geq 9$ (otherwise... | 10 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
Problem 5. An electric kettle heats water from room temperature $T_{0}=20^{\circ} \mathrm{C}$ to $T_{m}=100^{\circ} \mathrm{C}$ in $t=10$ minutes. How long will it take $t_{1}$ for all the water to boil away if the kettle is not turned off and the automatic shut-off system is faulty? The specific heat capacity of water... | Solution. The power $P$ of the kettle is fixed and equal to $P=Q / t$. From the heat transfer law $Q=c m\left(T_{m}-T_{0}\right)$ we get $P t=c m\left(T_{m}-T_{0}\right)$.
To evaporate the water, the amount of heat required is $Q_{1}=L m \Rightarrow P t_{1}=L m$.
By comparing these relations, we obtain $\frac{t_{1}}{... | 68 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Upon entering the Earth's atmosphere, the asteroid heated up significantly and exploded near the surface, breaking into a large number of fragments. Scientists collected all the fragments and divided them into groups based on size. It was found that one-fifth of all fragments had a diameter of 1 to 3 meters, another... | 1. Answer: 70. Solution. Let $\mathrm{X}$ be the total number of fragments. The condition of the problem leads to the equation:
$\frac{x}{5}+26+n \cdot \frac{X}{7}=X$, where $n-$ is the unknown number of groups. From the condition of the problem, it follows that the number of fragments is a multiple of 35
$$
X=35 l, ... | 70 | Number Theory | math-word-problem | Yes | Yes | olympiads | false |
2. The mass of the first iron ball is $1462.5 \%$ greater than the mass of the second ball. By what percentage will less paint be needed to paint the second ball compared to the first? The volume of a sphere with radius $R$ is $\frac{4}{3} \pi R^{3}$, and the surface area of a sphere is $4 \pi R^{2}$. | 2. Answer: $84 \%$.
Let's denote the radii of the spheres as $R$ and $r$ respectively. Then the first condition means that
$$
\frac{\frac{4}{3} \pi R^{3}-\frac{4}{3} \pi r^{3}}{\frac{4}{3} \pi r^{3}} \cdot 100=1462.5 \Leftrightarrow \frac{R^{3}-r^{3}}{r^{3}}=14.625 \Leftrightarrow \frac{R^{3}}{r^{3}}=\frac{125}{8} \L... | 84 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. Little Red Riding Hood is walking along a path at a speed of 6 km/h, while the Gray Wolf is running along a clearing perpendicular to the path at a speed of 8 km/h. When Little Red Riding Hood was crossing the clearing, the Wolf had 80 meters left to run to reach the path. But he was already old, his eyesight was fa... | 3. Answer: No.
Solution. The problem can be solved in a moving coordinate system associated with Little Red Riding Hood. Then Little Red Riding Hood is stationary, and the trajectory of the Wolf's movement is a straight line. The shortest distance from a point to a line here is (by similarity considerations):
$80 \cd... | 48 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
5. In the village where Glafira lives, there is a small pond that is filled by springs at the bottom. Curious Glafira found out that a herd of 17 cows completely drank the pond dry in 3 days. After some time, the springs refilled the pond, and then 2 cows drank it dry in 30 days. How many days would it take for one cow... | 5. Answer: In 75 days.
Solution. Let the pond have a volume of a (conditional units), one cow drinks b (conditional units) per day, and the springs add c (conditional units) of water per day. Then the first condition of the problem is equivalent to the equation $a+3c=3 \cdot 17 b$, and the second to the equation $a+30... | 75 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
6. Gavriil was traveling in Africa. On a sunny and windy day, at noon, when the rays from the Sun fell vertically, the boy threw a ball from behind his head at a speed of $5 \sim$ m/s against the wind at an angle to the horizon. After 1 s, the ball hit him in the stomach 1 m below the point of release. Determine the gr... | 6. Answer: 75 cm
Solution. In addition to the force of gravity, a constant horizontal force $F=m \cdot a$ acts on the body, directed opposite. In a coordinate system with the origin at the point of throw, the horizontal axis x, and the vertical axis y, the law of motion has the form:
$$
\begin{aligned}
& x(t)=V \cdot... | 75 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2.1. During the time it took for a slowly moving freight train to cover 1200 m, a schoolboy managed to ride his bicycle along the railway tracks from the end of the moving train to its beginning and back to the end. In doing so, the bicycle's distance meter showed that the cyclist had traveled 1800 m. Find the length o... | Answer: 500. Solution: Let $V$ and $U$ be the speeds of the cyclist and the train, respectively, and $h$ be the length of the train.
Then the conditions of the problem in mathematical terms can be written as follows:
$$
(V-U) t_{1}=h ; \quad(V+U) t_{2}=h ; \quad U\left(t_{1}+t_{2}\right)=l ; \quad V\left(t_{1}+t_{2}\... | 500 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3.1. The friends who came to visit Gavrila occupied all the three-legged stools and four-legged chairs in the room, but there was no place left for Gavrila himself. Gavrila counted that there were 45 legs in the room, including the "legs" of the stools and chairs, the legs of the visiting guests (two for each!), and Ga... | Answer: 9. Solution. If there were $n$ stools and $m$ chairs, then the number of legs in the room is $3 n+4 m+2 \cdot(n+m)+2$, from which we get $5 n+6 m=43$. This equation in integers has the solution $n=5-6 p, m=3+5 p$. The values of $n$ and $m$ are positive only when $p=0$. Therefore, there were 5 stools and 3 chair... | 9 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
4.1. A train of length $L=600$ m, moving by inertia, enters a hill with an angle of inclination $\alpha=30^{\circ}$ and stops when exactly a quarter of the train is on the hill. What was the initial speed of the train $V$ (in km/h)? Provide the nearest whole number to the calculated speed. Neglect friction and assume t... | Answer: 49. Solution. The kinetic energy of the train $\frac{m v^{2}}{2}$ will be equal to the potential energy of the part of the train that has entered the hill $\frac{1}{2} \frac{L}{4} \sin \alpha \frac{m}{4} g$.
Then we get $V^{2}=\frac{L}{4} \frac{1}{8} *(3.6)^{2}=\frac{6000 *(3.6)^{2}}{32}=9 \sqrt{30}$.
Since t... | 49 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5.1. A grenade lying on the ground explodes into a multitude of small identical fragments, which scatter in a radius of $L=90$ m. Determine the time interval (in seconds) between the moments of impact on the ground of the first and the last fragment, if such a grenade explodes in the air at a height of $H=10 \mathrm{m}... | Answer: 6. Solution. From the motion law for a body thrown from ground level at an angle $\alpha$ to the horizontal, the range of flight is determined by the relation $L=\frac{V_{0}^{2}}{g} \sin 2 \alpha$. Therefore, the maximum range of flight is achieved at $\alpha=45^{\circ}$ and is equal to $L=\frac{V_{0}^{2}}{g}$.... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. 2. A car with a load traveled from one city to another at a speed of 60 km/h, and returned empty at a speed of 90 km/h. Find the average speed of the car for the entire route. Give your answer in kilometers per hour, rounding to the nearest whole number if necessary.
$\{72\}$ | Solution. The average speed will be the total distance divided by the total time: $\frac{2 S}{\frac{S}{V_{1}}+\frac{S}{V_{2}}}=\frac{2 V_{1} \cdot V_{2}}{V_{1}+V_{2}}=\frac{2 \cdot 60 \cdot 90}{60+90}=72(\mathrm{km} / \mathrm{h})$. | 72 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Gavriil and Glafira took a glass filled to the brim with water and poured a little water into three ice cube trays, then placed them in the freezer. When the ice froze, they put the three resulting ice cubes back into the glass. Gavriil predicted that some water would spill out of the glass because ice expands in vo... | Solution. Let $V$ be the volume of water in the molds. Then the volume $W$ of ice in the molds can be determined from the law of conservation of mass $V \cdot \rho_{\text {water }}=W \cdot \rho_{\text {ice }}$. When ice of volume $W$ is floating, the submerged part of this volume $U$ can be determined from the conditio... | 1 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1. A tractor is pulling a very long pipe on sled runners. Gavrila walked along the entire pipe in the direction of the tractor's movement and counted 210 steps. When he walked in the opposite direction, the number of steps was 100. What is the length of the pipe if Gavrila's step is 80 cm? Round the answer to the neare... | 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 $y$ (m). Then, if $m$ and $n$ are the number of steps Gavrila takes in each direction, we get two equations: $x=m(a-y), x=n(a+y)$. From this, $\frac{x}{m}+\frac{x}{n}=2 a$, and $x=\frac{2 a... | 108 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
2. What is the greatest whole number of liters of water that can be heated to boiling temperature using the amount of heat obtained from the combustion of solid fuel, if in the first 5 minutes of combustion, 480 kJ is obtained from the fuel, and for each subsequent five-minute period, 25% less than the previous one. Th... | Answer: 5 liters
Solution: The amount of heat required to heat a mass $m$ of water under the conditions of the problem is determined by the relation $Q=4200(100-20) m=336 m$ kJ. On the other hand, if the amount of heat received in the first 5 minutes is $Q_{0}=480$ kJ. Then the total (indeed over an infinite time) amo... | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. At some point on the shore of a wide and turbulent river, 100 m from the bridge, Gavrila and Glafira set up a siren that emits sound signals at equal intervals. Glafira took another identical siren and positioned herself at the beginning of the bridge on the same shore. Gavrila got into a motorboat, which was locate... | Solution. Let's introduce a coordinate system, with the $x$-axis directed along the shore, and the origin at Gavrila's starting point. The siren on the shore has coordinates $(L, 0), L=50$ m, and Glafira is traveling along the line $x=-L$. Since the experimenters are at the same distance from the shore, the equality of... | 41 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
Problem 2. Experimenters Glafira and Gavriil placed a triangle made of thin wire with sides of 30 mm, 40 mm, and 50 mm on a white flat surface. This wire is covered with millions of mysterious microorganisms. The scientists found that when an electric current is applied to the wire, these microorganisms begin to move c... | Solution. In one minute, the microorganism moves 10 mm. Since in a right triangle with sides $30, 40, 50$, the radius of the inscribed circle is 10, all points inside the triangle are at a distance from the sides of the triangle that does not exceed 10 mm. Therefore, the microorganisms will fill the entire interior of ... | 2114 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
3. The villages of Arkadino, Borisovo, and Vadimovo are connected by straight roads. A square field adjoins the road between Arkadino and Borisovo, one side of which completely coincides with this road. A rectangular field adjoins the road between Borisovo and Vadimovo, one side of which completely coincides with this ... | Solution. The condition of the problem can be expressed by the following relation:
$r^{2}+4 p^{2}+45=12 q$
where $p, q, r$ are the lengths of the roads opposite the settlements Arkadino, Borisovo, and Vadimovo, respectively. This condition is in contradiction with the triangle inequality:
$r+p>q \Rightarrow 12 r+12 ... | 135 | Geometry | math-word-problem | Yes | Yes | olympiads | false |
1. Density is the ratio of the mass of a body to the volume it occupies. Since the mass did not change as a result of tamping, and the volume after tamping $V_{2}=$ $0.8 V_{1}$, the density after tamping became $\rho_{2}=\frac{1}{0.8} \rho_{1}=1.25 \rho_{1}$, that is, it increased by $25 \%$. | Answer: increased by $25 \%$. | 25 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.1. Gavriil found out that the front tires of the car last for 20000 km, while the rear tires last for 30000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 1.1. Gavriil found out that the front tires of the car last for 20,000 km, while the rear tires last for 30,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{24000\}$. | 24000 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1.2. Gavriila found out that the front tires of the car last for 24000 km, while the rear tires last for 36000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 1.2. Gavriil found out that the front tires of the car last for 24,000 km, while the rear tires last for 36,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{28800\}$. | 28800 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1.3. Gavriila found out that the front tires of the car last for 42,000 km, while the rear tires last for 56,000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 1.3. Gavriil found out that the front tires of the car last for 42000 km, while the rear tires last for 56000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km).
Answer. $\{48000\}$. | 48000 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. A new model car travels $4 \frac{1}{6}$ kilometers more on one liter of gasoline compared to an old model car. At the same time, its fuel consumption per 100 km is 2 liters less. How many liters of gasoline does the new car consume per 100 km? | Answer: 6 liters.
Instructions. The fuel consumption of the new car is $x$ liters, and the consumption of the old car is $x+2$
liters. Equation: $\frac{100}{x}-\frac{100}{x+2}=\frac{25}{6} \Leftrightarrow \frac{4(x+2-x)}{x(x+2)}=\frac{1}{6} \Leftrightarrow x^{2}+2 x-48=0 \Leftrightarrow x=-8 ; x=6$. Therefore, $x=6$... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
4. Gavrila placed 7 smaller boxes into a large box. After that, Glafira placed 7 small boxes into some of these seven boxes, and left others empty. Then Gavrila placed 7 boxes into some of the empty boxes, and left others empty. Glafira repeated this operation and so on. At some point, there were 34 non-empty boxes. Ho... | Answer: 205.
Instructions. Filling one box increases the number of empty boxes by 7-1=6, and the number of non-empty boxes by 1. Therefore, after filling $n$ boxes (regardless of the stage), the number of boxes will be: empty $-1+6 n$; non-empty $-n$. Thus, $n=34$, and the number of non-empty boxes will be $1+6 \cdot ... | 205 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false |
1. A car was moving at a speed of $V$. Upon entering the city, the driver reduced the speed by $x \%$, and upon leaving the city, increased it by $0.5 x \%$. It turned out that this new speed was $0.6 x \%$ less than the speed $V$. Find the value of $x$. | Answer: 20. Solution. The condition of the problem means that the equation is satisfied
$$
v\left(1-\frac{x}{100}\right)\left(1+\frac{0.5 x}{100}\right)=v\left(1-\frac{0.6 x}{100}\right) \Leftrightarrow\left(1-\frac{x}{100}\right)\left(1+\frac{x}{200}\right)=1-\frac{3 x}{500} \Leftrightarrow \frac{x^{2}}{20000}=\frac{... | 20 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
1. Gavriil decided to weigh a football, but he only had weights of 150 g, a long light ruler with the markings at the ends worn off, a pencil, and many threads at his disposal. He suspended the ball from one end of the ruler and the weight from the other, and balanced the ruler on the pencil. Then he attached a second ... | 1. Let the distances from the pencil to the ball and to the weight be $l_{1}$ and $l_{2}$ respectively at the first equilibrium. Denote the magnitude of the first shift by $x$, and the total shift over two times by $y$. Then the three conditions of lever equilibrium will be:
$$
\begin{gathered}
M l_{1}=m l_{2} \\
M\le... | 600 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
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 |
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 |
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 |
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 |
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
$$
x=\frac{n(n+5)}{n+9} \Rightarrow x=... | 24 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. Among all six-digit natural numbers, the digits of which are arranged in ascending order (from left to right), numbers containing the digit 1 and not containing this digit are considered. Which numbers are more and by how many? | Solution. First, let's calculate how many six-digit natural numbers there are in total, with their digits arranged in ascending order. For this, we will write down all the digits from 1 to 9 in a row. To get six-digit numbers of the considered type, we need to strike out any three digits. Thus, the number of six-digit ... | 28 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false |
8. Solve the equation $\sqrt{15 x^{2}-52 x+45} \cdot(3-\sqrt{5 x-9}-\sqrt{3 x-5})=1$. | Solution. Rewrite our equation in the form
$$
\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
$$
Apply the... | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
8. Solve the equation $\log _{5}(3 x-4) \cdot \log _{5}(7 x-16) \cdot\left(3-\log _{5}\left(21 x^{2}-76 x+64\right)\right)=1$. | Solution. Rewrite our equation in the form
$$
\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
$$
Such a transformation is possible because the solution to the original equation exists only for $x>\frac{16}{7}$. Let $\log _{5}(3 x-4)=a>0, \log _{5}(7 x-16)=b>0$. Note t... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
5. The Ivanovs' income as of the beginning of June:
$$
105000+52200+33345+9350+70000=269895 \text { rubles }
$$ | Answer: 269895 rubles
## Evaluation Criteria:
Maximum score - 20, if everything is solved absolutely correctly, the logic of calculations is observed, and the answer is recorded correctly.
## 20 points, including:
6 points - the final deposit amount is calculated correctly;
4 points - the size of the mother's sala... | 269895 | Algebra | math-word-problem | Yes | Yes | olympiads | false |
3. (15 points) Purchase a meat grinder at "Technomarket" first, as it is more expensive than a, which means the highest bonuses can be earned on it, and then purchase a blender using the accumulated bonuses. In this case, she will spend
$$
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 |
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 |
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 |
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