problem stringlengths 1 13.6k | solution stringlengths 0 18.5k ⌀ | answer stringlengths 0 575 ⌀ | problem_type stringclasses 8
values | question_type stringclasses 4
values | problem_is_valid stringclasses 1
value | solution_is_valid stringclasses 1
value | source stringclasses 8
values | synthetic bool 1
class | __index_level_0__ int64 0 742k |
|---|---|---|---|---|---|---|---|---|---|
5. Solution. Let the probabilities of heads and tails be $p$ and $q=1-p$ respectively. We form the equation
$$
C_{10}^{5} p^{5} q^{5}=C_{10}^{6} p^{6} q^{4}
$$
from which we find: $252 q=210 p ; \frac{p}{q}=\frac{6}{5} ; p=\frac{6}{11}$. | Answer: $\frac{6}{11}$.
## Grading Criteria
| Solution is correct and well-reasoned | 2 points |
| :--- | :---: |
| Correct equation is set up, but an error is made or the solution is not completed | 1 point |
| Solution is incorrect or missing (including only providing the answer) | 0 points | | \frac{6}{11} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,092 |
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 | 11,094 |
# 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 | 11,095 |
9. Solution. Suppose Olga Pavlovna has \( x \) liters of jam left, and Maria Petrovna has \( y \) liters of jam left. The numbers \( x \) and \( y \) are randomly and independently chosen from the interval from 0 to 1. We will consider that a random point with coordinates \((x; y)\) is selected from the unit square \( ... | Answer: 0.375.
Note. Other solution methods are possible.
## Grading Criteria
| Solution is complete and correct | 3 points |
| :--- | :--- |
| The solution contains correct reasoning for individual cases and an enumeration of these cases, and the formula for total probability is applied to them. However, the answer... | 0.375 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,096 |
4. A city is considered a millionaire city if it has a population of more than one million people. Which event has a higher probability:
$$
A=\{\text { a randomly chosen city resident lives in a millionaire city }\}
$$
$$
B=\{\text { a randomly chosen city is a millionaire city }\} ?
$$ | Justify the answer
Take the statistics on the urban population of Russia from the website http://www.perepis2002.ru/ct/doc/1_TOM_01_05.xls. Check if your previous conclusion is valid for Russia. For this, calculate the probability that a randomly chosen urban resident lives in a city with a population of over a millio... | proof | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,099 |
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 | 11,100 |
8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \%$ of satisfied customers leave a positive review. A certain online store has received 60 angry and 20 positive reviews. Using this s... | Solution. Let $p$ be the probability that a customer is served well, and $q=1-p$ be the probability that they are served poorly. Then the probability that a customer leaves a good review is $0.15 p$, and the probability that there will be a bad review is $0.8(1-p)$. Then
$$
\frac{0.15 p}{0.8(1-p)} \approx \frac{1}{3},... | 0.64 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,101 |
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 | 11,103 |
4. Cities of Anchuria (from 6th grade, 1 point). In Anchuria, there is only one river, Rio-Blanco, which originates somewhere in the mountains and flows into the ocean, and there are only five cities: San-Mateo, Alasan, Coralio, Alforan, and Solitas.
In the map of Anchuria, the city names are not shown, and the cities... | Solution. Alasan is the only city not located on the banks of the Rio Blanco, but gets its water from wells. Therefore, Alasan has the number 2.
The lower a city is along the river, the lower its elevation above sea level. Only the "Elevation above sea level" value is required from all available statistics.
Answer: 1... | 1-Alforan,2-Alasan,3-Solitas,4-San-Mateo,5-Coralio | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,104 |
4. Contest (from 6th grade, 1 point). In one social network, a photo contest was held. Several photos were submitted to the contest, and each participant could rate each photo by giving it either 0 (do not like), 1 (not very like), or 2 (like very much).
Two categories were announced: the most attractive photo, which ... | Solution. We will build a table showing how this can happen with just two photos and three voters.
| | Photo 1 | Photo 2 |
| :--- | :---: | :---: |
| 1st voter | 2 | 1 |
| 2nd voter | 2 | 1 |
| 3rd voter | 0 | 1 |
Answer: yes, this is possible | yes,thisispossible | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,105 |
8. Cards and PIN codes. Once, a pickpocket named Brick ${ }^{1}$ stole a wallet, which contained four credit cards and a note with four PIN codes for these cards. Brick does not know which PIN code corresponds to which card. If the wrong PIN code is entered three times for any card, the card will be blocked.
a) (from ... | Solution. a) Taking the first code and trying it on four cards in turn, Brick will find the card to which this code fits. Taking the second code and trying it on the three remaining cards, he will find the second match. Then - the third. Thus, he will find codes for three cards. The fourth one may turn out to be blocke... | \frac{23}{24} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,106 |
19. Essay on Suitcases (from 10th grade, 6 points). In the book by Szczepan Jelenski ${ }^{7}$, there is a small essay about suitcases.
A department store received 10 suitcases and 10 keys in a separate envelope, with the warning that each key opens only one suitcase and that a suitable key can be found for each suitc... | Solution. We will solve the problem in a general form, assuming that there are $n$ suitcases. Let $Y$ be a random variable equal to the number of attempts to open the first of $n$ suitcases. Obviously, $Y=1$ with probability $1 / n$, $Y=2$ with probability $\frac{n-1}{n} \cdot \frac{1}{n-1}=\frac{1}{n}$, and so on, up ... | 29.62 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,107 |
8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \%$ of satisfied customers leave positive reviews. A certain online store has received 60 angry and 20 positive reviews. Using this st... | Solution. Let $p$ be the probability that a customer is served well, and $q=1-p$ be the probability that they are served poorly. Then the probability that a customer leaves a good review is $0.15 p$, and the probability that there will be a bad review is $0.8(1-p)$. Then
$$
\frac{0.15 p}{0.8(1-p)} \approx \frac{1}{3},... | 0.64 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,108 |
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 | 11,110 |
18. The Paradox of the Last Roll (from 9th grade. 4 points). A fair die is rolled until the sum of the points that fall in sequence reaches the number 2019 (becomes equal to 2019 or more). Prove that the probability of the event “6 points fall on the last roll” is more than ${ }^{5}$ times $1 / 6$.
Comment. It seems p... | Solution. In the conditions of the experiment, before the last throw, the sum of the points that fell in sequence $Y$ became equal to one of the numbers
$$
n-6, n-5, \ldots, n-1
$$[^4]
Let's introduce brief notations for the probabilities of these six events: $p_{n-k}=\mathrm{P}(Y=n-k)$. Obviously, $p_{n-1}+p_{n-2}+\... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 11,111 |
6. Median of the sum of numerical sets. Let there be two numerical sets:
$$
X=\left\{x_{1}, x_{2}, \ldots, x_{n}\right\} \text { and } Y=\left\{y_{1}, y_{2}, \ldots, y_{m}\right\}
$$
The first set has $n$ numbers, and the second set has $m$ numbers. The direct sum or simply the sum $X \oplus Y$ of these sets is the s... | Solution. a) For example, two identical sets: $X=\{0,0,1\}$ and $Y=\{0,0,1\}$. The median of each is 0, and the sum of these sets $\{0,0,0,0,1,1,1,1,2\}$ has a median of 1.
b) We will show that such sets do not exist. Let the first set $X$ consist of numbers $x_{1}, x_{2}, \ldots, x_{n}$, and the second set $Y$ consis... | ){0,0,1} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,113 |
11. The Collector's Two Tasks. A chocolate egg manufacturer with a toy inside announced the release of a new collection called "The Nile Family," featuring ten different charming crocodiles. The crocodiles are evenly and randomly distributed among the chocolate eggs, meaning that in a randomly selected egg, each crocod... | Solution. a) Let $B_{k}$ be the event "at the moment when the last crocodile is acquired for the first collection, the second collection is missing exactly $k$ crocodiles." We need to show that
$$
p_{1}=\mathrm{P}\left(B_{1}\right)=\mathrm{P}\left(B_{2}\right)=p_{2}
$$
Let $A_{j, k}$ be the event "at some point, the ... | proof | Combinatorics | proof | Yes | Yes | olympiads | false | 11,114 |
14. Asymmetric coin. (From 9th grade, 2 points) Billy Bones has two coins - a gold one and a silver one. One of them is symmetric, and the other is not. It is unknown which coin is asymmetric, but it is known that the asymmetric coin lands heads with a probability of \( p = 0.6 \).
Billy Bones tossed the gold coin, an... | Solution. Let's introduce notations for the events:
$$
A=\{\text { the gold coin is biased }\},
$$
$B=\left\{\begin{array}{l}\text { when the gold coin is tossed, heads appear immediately, } \\ \text { and when the silver coin is tossed, heads appear on the second attempt. }\end{array}\right\}$
We need to find the c... | \frac{5}{9} | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,115 |
11. Large Cube. A cube is assembled from 27 playing dice. a) (6 - 11 grades, 1 point) Find the probability that exactly 25 sixes are on the surface of the cube.
b) (7 - 11 grades, 1 point) Find the probability that at least one one is on the surface of the cube.
c) (8 - 11 grades, **1** point) Find the expected numbe... | Solution. a) One of the 27 cubes is in the center and therefore not visible at all. The other 26 cubes are visible. Thus, the required event
$$
A=\{25 \text { sixes }\}
$$
consists of all the cubes showing sixes outward, except for one - let's call it the special cube.
Consider all the cubes. If a cube is in the cen... | )\frac{31}{2^{13}\cdot3^{18}}\approx9.77\cdot10^{-12};b)1-\frac{5^{6}}{2^{2}\cdot3^{18}}\approx0.99998992;)9;)189;e)6-\frac{5^{6} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,117 |
18. (10-11 grades, 5 points) Favorite Pair. On the drying rack, in a random order (as taken out of the washing machine), there are p socks. Among them are two favorite socks of the Absent-Minded Scientist. The socks are hidden by a drying sheet, so the Scientist cannot see them and takes one sock at a time by feel. Fin... | Solution. It is convenient to form a triangular table. The shaded cells correspond to pairs of favorite socks. For example, the pair $(2; 4)$, marked with an "X", corresponds to the case where the first favorite sock was the second one, and the second one was the fourth in the sequence. All pairs are equally likely, an... | \frac{2(n+1)}{3} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,118 |
2. Half-year grade. By the end of the half-year, Vasya Petrov had the following grades in mathematics in his journal: $4,1,2,5,2$. Before assigning the half-year grade, the math teacher told Vasya:
- Vasya, you can choose the method to determine your half-year grade. I offer two options. Method A: the arithmetic mean ... | # Solution.
a) The average of the current grades is 2.8 (rounded to 3), and the median of the grades is 2. It is better to choose Method A.
b) Suppose Method A is the best. Therefore, the teacher should give two fives. However, the average after rounding remains 3, while the median rises to 4. Thus, Method A now give... | 3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,119 |
9. Steel doors (from 9th grade. 2 points). At the factory named after Sailor Zheleznyak, they manufacture rectangles with a length of 2 m and a width of 1 m. The length is measured by worker Ivanov, and the width, independently of Ivanov, is measured by worker Petrov. Both have an average error of zero, but Ivanov has ... | # Solution.
a) Let $X$ be the width and $Y$ be the length of the cut rectangle in meters. According to the problem, $\mathrm{E} X=2$, $\mathrm{E} Y=1$. Since the measurements are independent, $\mathrm{E}(X Y)=\mathrm{E} X \cdot \mathrm{E} Y=2$ (sq.m.).
b) From the condition, it follows that $\mathrm{D} X=0.003^{2}=9 ... | )2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,120 |
15. English Club (from 9th grade. 6 points). Every Friday, ten gentlemen come to the club, and each hands their hat to the doorman. Each hat fits its owner perfectly, but no two hats are the same size. The gentlemen leave one by one in a random order.
When seeing off each gentleman, the doorman tries to put on the fir... | Solution. Let the number of gentlemen be $n$. We will number them in the order of increasing sizes of their hats from 1 to $n$. No hats will be left only if each took his own hat. Let the probability of this be $p_{n}$. If the $k$-th gentleman leaves first (the probability of this is $\frac{1}{n}$), then the probabilit... | 0.000516 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 11,121 |
# 4. Calculator (6-9).
a) (1 pt.) On the calculator keyboard, there are digits from 0 to 9 and symbols for two operations (see figure). Initially, the display shows the number 0. You can press any keys. The calculator performs operations in the sequence of key presses. If the operation symbol is pressed several times ... | Solution. a) Note that at least one addition operation is performed, even if the Scholar entered only one number - thereby adding this number to zero.
Let $A$ be the event "the result is odd". The parity of the result is determined by the last addend. Let's explain this in more detail.
Suppose the penultimate number ... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,122 |
4. Cities of Anchuria (from 6th grade, 1 point). In Anchuria, there is only one river, Rio Blanco, which originates somewhere in the mountains and flows into the ocean, and there are only five cities: San Mateo, Alasan, Coralio, Alforan, and Solitas.
In the map of Anchuria, the city names are missing, and the cities a... | Solution. Alasan is the only city not located on the banks of the Rio Blanco, but gets its water from wells. Therefore, Alasan has the number 2.
The lower a city is along the river, the lower its elevation above sea level. Only the "Elevation above sea level" value is required from all available statistics.
Answer: 1... | 1-Alforan,2-Alasan,3-Solitas,4-San-Mateo,5-Coralio | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,125 |
# 4. Calculator (6-9).
a) (1 pt.) On the calculator keyboard, there are digits from 0 to 9 and symbols for two operations (see figure). Initially, the display shows the number 0. You can press any keys. The calculator performs operations in the sequence of key presses. If the operation symbol is pressed several times ... | Solution. a) Note that at least one addition operation is performed, even if the Scholar entered only one number - thereby adding this number to zero.
Let $A$ be the event "the result is odd". The parity of the result is determined by the last addend. Let's explain this in more detail.
Suppose the penultimate number ... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,126 |
8. Reviews (from 7th grade. 1 point). Angry reviews about the work of an online store are left by $80 \%$ of dissatisfied customers (those who were poorly served in the store). Only $15 \%$ of satisfied customers leave positive reviews. A certain online store has received 60 angry and 20 positive reviews. Using this st... | Solution. Let $p$ be the probability that a customer is served well, and $q=1-p$ be the probability that they are served poorly. Then the probability that a customer leaves a good review is $0.15 p$, and the probability that there will be a bad review is $0.8(1-p)$. Then
$$
\frac{0.15 p}{0.8(1-p)} \approx \frac{1}{3},... | 0.64 | Other | math-word-problem | Yes | Yes | olympiads | false | 11,129 |
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 | 11,131 |
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 | 11,132 |
9. Solution. Suppose Olga Pavlovna has \( x \) liters of jam left, and Maria Petrovna has \( y \) liters of jam left. The numbers \( x \) and \( y \) are randomly and independently chosen from the interval from 0 to 1. We will consider that a random point with coordinates \((x; y)\) is selected from the unit square \( ... | Answer: 0.375.
Note. Other solution methods are possible.
## Grading Criteria
| Solution is complete and correct | 3 points |
| :--- | :--- |
| The solution contains correct reasoning for individual cases and an enumeration of these cases, and the formula for total probability is applied to them. However, the answer... | 0.375 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,133 |
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 | 11,135 |
3. The lid of a vertical well 10 m deep periodically opens and closes instantaneously so that the well is in an open state for one second and in a closed state for one second. A stone is thrown vertically upward from the bottom of the well with an initial velocity $V$ exactly 0.5 seconds before the next opening of the ... | 3. Let $h$ be the depth of the well, $g$ be the acceleration due to gravity, and $\tau$ be the interval during which the lid is open. The data is chosen such that $h /\left(g \tau^{2}\right)=1$.
Note that the ball cannot take too long to rise from the well. The maximum time for the ball to rise is $\sqrt{2 h / g}=\tau... | V\in(\frac{85}{6},\frac{33}{2})\cup(\frac{285}{14},\frac{45}{2}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,136 |
4. In a water-filled and tightly sealed aquarium in the shape of a rectangular parallelepiped measuring 3 m $\times 4$ m $\times 2$ m, there are two small balls: an aluminum one and a wooden one. At the initial moment, the aquarium is at rest, and the distance between the balls is 2 m. What is the greatest distance bet... | 4. The aluminum ball is heavier than water, so in a state of rest, it will occupy the lowest position, while the wooden ball is lighter than water, so it will occupy the highest position. Note that the distance between the balls in a state of rest is equal to the length of the shortest side. Therefore, the largest face... | \sqrt{29} | Other | math-word-problem | Yes | Yes | olympiads | false | 11,137 |
5. An astronomer discovered that the intervals between the appearances of comet $2011 Y$ near planet $12 I V 1961$ are consecutive terms of a decreasing geometric progression. The three most recent intervals (in years) are the roots of the cubic equation $t^{3}-c t^{2}+350 t-1000=0$, where $c-$ is some constant. What w... | 5. Writing the factorization of the polynomial: $t^{3}-c t^{2}+350 t-1000=\left(t-t_{1}\right)\left(t-t_{1}\right)\left(t-t_{1}\right)$, where $t_{1}, t_{2}, t_{3}$ are the roots of the polynomial, we get:
$$
\left\{\begin{array}{c}
t_{1}+t_{2}+t_{3}=c \\
t_{1} t_{2}+t_{2} t_{3}+t_{3} t_{1}=350 \\
t_{1} t_{2} t_{3}=10... | 2.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,138 |
Task 1. A vacuum robot is programmed to move on the floor according to the law:
$\left\{\begin{array}{l}x=t(t-6)^{2} \\ y=0,0 \leq t \leq 7 ; y=(t-7)^{2}, t \geq 7\end{array}\right.$
where the axes are parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and the coordinates ... | Solution. For the first seven minutes, the point moves along the $x$-axis. The velocity of the point for $t \leq 7$ is $\dot{x}=3(t-2)(t-6)$ and equals zero at times $t_{1}=2$ and $t_{2}=6$. The distance traveled will be $L=x(7)+2(x(2)-x(6))=7+2 \cdot 32=71$.
The velocity vector $\bar{V}$ at each moment $t \geq 7$ is ... | 71;\sqrt{445} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 11,139 |
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 | 11,140 |
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 | 11,141 |
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 | 11,142 |
Problem 5. Vertical oscillations of a load of mass $m$ on a spring with stiffness $k$ in a viscous medium are described by the equation $x(t)=a e^{-2 t}+b e^{-t}+m g / k$, where $a, b$ are constants depending on the initial conditions, $t$ is time. At the initial moment, the load was taken out of equilibrium. Given $\m... | Solution. The substitution $y=e^{-t}$ reduces the problem to the following: find the values of the parameter $a$ for which the equation $a y^{2}+(1-2 a) y+1=0$ has two roots in the interval $(0 ; 1)$.
The conditions for this are:
$$
D=4 a^{2}-8 a+1 \geq 0 ; a f(0)=a>0 ; \text { af }(1)=a(a+(1-2 a)+1)>0 ; y_{0}=\frac{... | \in(1+\frac{\sqrt{3}}{2};2) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,143 |
Problem 6. Three beads with masses \( m_{1}=150 \) g, \( m_{3}=30 \) g, \( m_{2}=1 \) g (see figure) can slide along a horizontal rod without friction.
Determine the maximum speeds of the larger beads if at the initial moment of time they were at rest, while the small bead was moving with a speed of \( V=10 \) m/s.
C... | Solution. After each collision, the magnitude of the velocity of the bead with mass $m_{2}$ decreases. After a certain number of collisions, its velocity will be insufficient to catch up with the next bead $m_{1}$ or $m_{3}$. After such a final collision, the velocities of the beads will no longer change. Let $V_{1}$ a... | V_{1}\approx0.28\mathrm{~}/\mathrm{};V_{3}\approx1.72\mathrm{~}/\mathrm{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,144 |
5. Let's define the equation of the process. Since the piston is light, the elastic force of the spring at any moment is equal to the pressure force of the gas:
$$
k x=p(x) S
$$
where $x$ is the length of the cylinder occupied by the gas, which by condition is equal to the compression of the spring, $k-$ is the sprin... | Answer: $2 R / \mu=4155$ J/(kg K). | 2R/\mu=4155 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,146 |
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 | 11,147 |
Problem 6. Three beads with masses \( m_{1}=150 \) g, \( m_{3}=30 \) g, \( m_{2}=1 \) g (see figure) can slide along a horizontal rod without friction.
Determine the maximum speeds of the larger beads if at the initial moment of time they were at rest, while the small bead was moving with a speed of \( V=10 \) m/s.
C... | Solution. After each collision, the magnitude of the velocity of the bead with mass $m_{2}$ decreases. After a certain number of collisions, its velocity will be insufficient to catch up with the next bead $m_{1}$ or $m_{3}$. After such a final collision, the velocities of the beads will no longer change. Let $V_{1}$ a... | V_{1}\approx0.28\mathrm{~}/\mathrm{};V_{3}\approx1.72\mathrm{~}/\mathrm{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,148 |
5. Let's define the equation of the process. Since the piston is light, the spring's elastic force at any moment is equal to the gas pressure force:
$$
k x=p(x) S
$$
where $x$ is the length of the cylinder occupied by the gas, which by condition is equal to the compression of the spring, $k$ is the spring stiffness c... | Answer: $2 R / \mu=4155$ J $/($ kg $K)$. | 2R/\mu=4155J/(K) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,149 |
2. Two schoolchildren left two neighboring schools at the same time and headed towards each other. After ten minutes, the distance between the schoolchildren was $600 \mathrm{m}$, and after another five minutes, they met. Find the distance between the schools (in meters).
$\{=1800\}$
$:: 1.1::$ A weight of 200 g is s... | 1800 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,150 | |
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 | 11,151 |
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 | 11,152 |
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 | 11,153 |
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 | 11,154 |
5. Gavriila took a cast iron skillet in his hand and heated it on the stove until the handle of the skillet felt hot. After that, Gavriila placed the skillet under a stream of cold water from the tap, but he felt that the handle of the skillet became even hotter (and not colder, as Gavriila had expected). Why did this ... | Solution. The water jet exerts a certain force on the pan, so to hold it, Gavrila squeezed the handle of the pan harder. As a result, the contact area increased and the heat flow also increased. The sensation of heat is primarily associated with the rate of heat flow, not the temperature. Equally heated wood and metal ... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,155 |
1. Three athletes start from the same point on a closed running track that is 400 meters long and run in the same direction. The first runs at a speed of 155 m/min, the second at 200 m/min, and the third at 275 m/min. After what least amount of time will they all be at the same point again? How many overtakes will occu... | Answer: $\frac{80}{3}$ minutes $=26$ min. 40 sec.; 13 overtakes.
Solution. The second athlete overtakes the first by 45 m per minute. Therefore, he will catch up with him again after $\frac{400}{45}=\frac{80}{9}$ minutes, meaning the meetings will occur every $\frac{80}{9} n$ minutes ( $n \in N$ ). Similarly, the thir... | \frac{80}{3} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 11,156 |
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 | 11,157 |
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 | 11,158 |
4. In two identical cylinders closed with light pistons, there is the same amount of gases: in one - nitrogen, in the other - water vapor. Both cylinders are maintained at a constant temperature of $100^{\circ} \mathrm{C}$. At the initial moment, the pressure of the gas in both cylinders is 0.5 atm, the volume is 2 lit... | Answer: $2 ;$ Yes.
Solution. Consider the isothermal compression of nitrogen. Since the ratio of the volumes of the gas at the beginning and end of the motion $l /(l-u t)=\alpha=4$, the pressure of the nitrogen will be $p=\alpha p_{0}=2$ atm. Water vapor cannot exist at such a pressure at the given temperature, so con... | 2;Yes | Other | math-word-problem | Yes | Yes | olympiads | false | 11,159 |
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 | 11,160 |
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 | 11,161 |
1. The distance from a point lying in a horizontal plane to the base of a television tower is 100 m. From this point, the tower (from base to top) is visible at an angle of $46^{\circ}$. Without using tables, calculators, or other computational devices, determine which is greater: the height of the tower or 103.3 m? | Answer: The height of the tower is greater. Solution. The height of the tower is $H=100 \cdot \operatorname{tg} 46^{\circ}$
$=100 \cdot \operatorname{tg}\left(45^{\circ}+1^{\circ}\right)=100 \frac{\operatorname{tg} 45^{\circ}+\operatorname{tg} 1^{\circ}}{1-\operatorname{tg} 45^{\circ} \cdot \operatorname{tg} 1^{\circ}... | 103.55 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,162 |
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 | 11,163 |
3. The sledge run consists of a straight slope $AB$ and a horizontal section $BC$. Point $A$ is 5 m away from the nearest point $H$ on the horizontal ground surface. The distance $HC$ is 3 m, and point $B$ lies on the segment $HC$. Find the distance from point $H$ to point $B$ so that the time of the sledge's motion fr... | Answer: $\frac{5 \sqrt{3}}{3}$
m. Solution. Let $S$ be the distance $HC$, $H$ be the distance $AH$, and $x$ be the required distance. Then the time of fall $t_{\text {AH }}=\sqrt{\frac{2 H}{g}}$, the time of descent from point $A$ to point $B$ is $t_{A B}=\frac{t_{A H}}{\sin \angle A B H}$, where $\sin \angle A B H=\f... | \frac{5\sqrt{3}}{3} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 11,164 |
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 | 11,165 |
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 | 11,166 |
6. A reel of film needs to be rewound from one spool to another. The diameters of the empty spools are the same and equal to $a$. Find the time required for rewinding if the length of the film is $L$, the thickness of the film is small and equal to $S$, and the receiving spool rotates at a constant angular velocity $\o... | Answer: $T=\frac{\pi}{S \omega}\left(\sqrt{a^{2}+\frac{4 S L}{\pi}}-a\right)$. Solution. For each revolution of the receiving coil, one turn of the film is wound onto it, meaning the radius increases by $S$. One revolution of the coil takes time $t_{0}=\frac{2 \pi}{\omega}$. Over time $t$, the coil will complete $n=\fr... | \frac{\pi}{S\omega}(\sqrt{^{2}+\frac{4SL}{\pi}}-) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,167 |
5. In the reference frame associated with Gavrila, the balls are thrown upwards from the same point with the same speed at intervals of $\tau$ in time. By introducing the $y$-axis, directed vertically upwards with its origin at the point of throwing, and measuring time from the moment the first ball is thrown, we can w... | Answer: Minimum distance 0, at a distance of $V\left(\frac{\tau}{2}+\frac{u}{g}\right)$ from Glafira. | V(\frac{\tau}{2}+\frac{u}{}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,169 |
9. According to the first law of thermodynamics
$$
Q=\Delta u+A
$$
where $Q$ - the amount of heat, $\Delta u$ - the change in internal energy, $A$ - the work done by the gas. In our case
$$
Q=0, \quad \Delta u=c_{v}\left(T-T_{0}\right), \quad A=\frac{k x^{2}}{2}
$$
where $x$ - the displacement of the piston, $k$ - ... | Answer: $P_{0} \frac{1}{n\left(1+\frac{(n-1) R}{2 n c_{v}}\right)}$. | P_{0}\frac{1}{n(1+\frac{(n-1)R}{2nc_{v}})} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,172 |
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 | 11,177 |
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 | 11,178 |
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 | 11,179 |
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 | 11,180 |
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 | 11,182 |
5.1. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 5.1. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 83.25 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,189 |
5.2. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere was outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its moveme... | 5.2. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere was outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its moveme... | 60.75 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,190 |
5.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 5.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 105.75 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,191 |
5.4. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its m... | 5.4. A smooth sphere with a radius of 1 cm was dipped in blue paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a blue trail. During its m... | 38.25 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,192 |
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 $V$.
A) Can the value of $x$ be found? If so, what is it?
B) Can the value of $V$ be found?... | Answer: a) Yes; 20; b) no. Solutions.
A) The condition of the problem means that the equation is satisfied
$$
\begin{aligned}
& 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} \Lef... | 20,V | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,193 |
2. A cube made of a material with a density $\rho_{1}$, which is 3 times the density of water $\rho_{0}$, is located at the bottom of a container filled with water. With what acceleration and in what direction should the container be moved to make the cube start to float? | Answer: no less than $g$ downward. Solution. If the acceleration of the vessel is greater than or equal to $\mathrm{g}$, the pressure of the column on the upper surface of the cube is absent. Since the reaction force of the vessel is directed upward or is zero, the acceleration of the cube relative to the ground does n... | g | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,194 |
3. Tourists from the USA, when traveling to Europe, often use an approximate formula to convert temperatures in degrees Celsius $C$ to the familiar degrees Fahrenheit $F$: $\mathrm{F}=2 \mathrm{C}+30$. Indicate the range of temperatures (in degrees Celsius) for which the deviation of the temperature in degrees Fahrenhe... | Answer: $1 \frac{11}{29} \leq C \leq 32 \frac{8}{11}$. Solution. Both temperature scales are uniform, so they are related by a linear law: $\mathrm{F}=\mathrm{kC}+\mathrm{b}$. The constants $a$ and $b$ are determined from the condition. The exact formula is obtained: $F=\frac{9}{5} C+32$.
The deviation of temperatures... | 1\frac{11}{29}\leqC\leq32\frac{8}{11} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,195 |
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 | 11,196 |
5. Two samovars - a large one and a small one - were filled with very hot water of the same temperature. Both samovars have the same shape and are made of the same material. Which one will cool down to room temperature first? | Answer: smaller. Solution. If one samovar is $n$ times larger than the other, then its volume is larger by $n^{3}$ times, and the surface area is larger by $n^{2}$ times. Therefore, in the larger samovar, there is $n$ times more volume per unit of surface area. Thus, it will cool down more slowly. | smaller | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,197 |
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 | 11,198 |
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 | 11,200 |
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 first power 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 2 times. Power is the product of force and speed, s... | increased\\4\times | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,201 |
5. Why does ice form on the inside of the windows in trolleybuses in winter, even though it seems warmer inside than outside? | Answer: Inside the trolleybus, due to the breathing of passengers, there is warm and humid air. The glass is warmed by this air from the inside and cooled by the frosty air from the outside. If the heating from the inside is low, then the temperature of the inner surface of the glass will be negative, and water, conden... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 11,202 |
3.1. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at equal speeds and at equal distances from each other. Another part is flying in the same way but along another straight line parallel to the first, at the same speeds but ... | 3.1. A transgalactic ship has encountered an amazing meteor shower. Some of the meteors are flying along a straight line, one after another, at the same speeds and at equal distances from each other. Another part is flying in the same way but along another straight line parallel to the first, at the same speeds but in ... | 9.1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,206 |
3.3. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at equal speeds and at equal distances from each other. Another part is flying in the same way but along another straight line parallel to the first, at the same speeds but ... | 3.3. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at the same speed and at equal distances from each other. Another part is flying in the same way but along another straight line parallel to the first, at the same speeds bu... | 9.1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,208 |
3.4. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at equal speeds and at equal distances from each other. Another part is flying the same way but along another straight line parallel to the first, at the same speeds but in ... | 3.4. A transgalactic ship has encountered an amazing meteorite stream. Some of the meteorites are flying along a straight line, one after another, at the same speeds and at equal distances from each other. Another part is flying in the same way but along another straight line parallel to the first, at the same speeds b... | 9.1 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,209 |
5.2. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 5.2. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 83.25 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,214 |
5.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 5.3. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 83.25 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,215 |
5.4. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 5.4. A smooth sphere with a radius of 1 cm was dipped in red paint and launched between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (this sphere ended up outside the smaller sphere but inside the larger one). Upon touching both spheres, the sphere leaves a red trail. During its mov... | 83.25 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,216 |
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 | 11,217 |
2. A mobile railway platform has a horizontal bottom in the form of a rectangle 10 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 gr... | Answer: 62.5 t. Solution. The calculation shows that the maximum height of the grain pile will be half the width of the platform, i.e., 2.5 m. The pile can be divided into a "horizontally lying along the platform" prism (its height is 5 m, and the base is an isosceles right triangle with legs $\frac{5 \sqrt{2}}{2}$ and... | 62.5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 11,218 |
3. Gavriil saw a squirrel sitting on a tree branch right in front of him at a distance of 3 m 75 cm through the window. He decided to feed the little animal and threw a nut horizontally at a speed of 5 m/s directly towards the squirrel. Will the squirrel be able to catch the nut if it can jump in any direction at a spe... | Answer: Yes. Solution: The coordinates of the squirrel at the initial moment: $x=a ; y=0$. The nut moves according to the law: $x(t)=V_{0} t ; y(t)=\frac{g t^{2}}{2}$.
Therefore, the square of the distance from the sitting squirrel to the flying nut at time $t$ is: $r^{2}=\left(V_{0} t-a\right)^{2}+\left(\frac{g t^{2}... | \frac{5\sqrt{2}}{4}<\frac{9}{5} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,219 |
4. Condition One mole of an ideal gas undergoes a cyclic process, which in terms of $P, T$ is described by the equation
$$
\left(\frac{P}{P_{0}}-a\right)^{2}+\left(\frac{T}{T_{0}}-b\right)^{2}=c^{2}
$$
where $P_{0}, T_{0}, a, b, c-$ are some constants, $c^{2}<a^{2}+b^{2}$. Determine the maximum volume occupied by the... | Answer: $V_{\text {max }}=\frac{R T_{0} a \sqrt{a^{2}+b^{2}-c^{2}}+b c}{b \sqrt{a^{2}+b^{2}-c^{2}}-a c}$
Solution In the axes $P / P_{0}, T / T_{0}$, the process is represented by a circle with center at point $(a, b)$ and radius $c$. For each point, the volume occupied by the gas is $R \frac{T}{P}$, which is proporti... | V_{\max}=\frac{RT_{0}}{P_{0}}\frac{\sqrt{^{2}+b^{2}-^{2}}+}{b\sqrt{^{2}+b^{2}-^{2}}-} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,220 |
4. Condition: Two weightless cylindrical rollers lie on a horizontal surface parallel to each other and have radii $R=1.25$ m and $r=75$ cm. A heavy flat slab of mass $m=100$ kg is placed on them such that it is inclined to the horizontal by $\alpha=\arccos (0.92)$. Find the magnitude and direction of the slab's accele... | Answer: 2 m/s; \arcsin 0.2;
Solution: Let's transition to a coordinate system where the center of the (left) wheel is at rest. In this system, all points on the rim move with the same speed $v$. That is, the surface moves horizontally to the left with this speed. And the point of contact with the slab also has this sp... | \sin(\alpha/2) | Other | math-word-problem | Yes | Yes | olympiads | false | 11,221 |
4. Condition: A certain farmer has created a system of channels $A B C D E F G H$, as shown in the diagram. All channels are the same, and their connections occur at nodes marked by the letters $B, D$, and $G$. Water enters at node $A$ and exits at node $E$. Let's call the flow rate the number of cubic meters of water ... | Answer: $2 q_{0}, \frac{3}{2} q_{0}, \frac{7}{2} q_{0}$
Solution: First, due to symmetry, note that the flow in channel CD is also equal to $q_{0}$. Let the flow in channel $\mathrm{BG}$ and its symmetric counterpart GD be $x$. Then the total flow along the path $\mathrm{BCD}$ is equal to the flow along the path BGD: ... | \frac{7}{2}q_{0} | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 11,222 |
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 | 11,223 |
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 | 11,224 |
3. Gavriil saw a squirrel sitting on a tree branch right in front of him at a distance of 3 m 75 cm through the window. He decided to feed the little animal and threw a nut horizontally at a speed of 2.5 m/s directly towards the squirrel. Will the squirrel be able to catch the nut if it can jump in any direction at a s... | Answer: No. Solution: The coordinates of the squirrel at the initial moment: $x=a ; y=0$. The nut moves according to the law: $x(t)=V_{0} t ; y(t)=\frac{g t^{2}}{2}$.
Therefore, the square of the distance from the sitting squirrel to the flying nut at time $t$ is: $r^{2}=\left(V_{0} t-a\right)^{2}+\left(\frac{g t^{2}}... | \frac{5\sqrt{5}}{4}>\frac{27}{10} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 11,225 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.