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 |
|---|---|---|---|---|---|---|---|---|---|
17. In how many ways can all param 1 asterisks be replaced with even digits (not necessarily distinct) in the number param 2 so that the resulting number is divisible by 12?
| param1 | param2 | |
| :---: | :---: | :---: |
| 6 | $2017^{*} 1^{*} 5^{* * *} 12^{*}$ | |
| 7 | $2017^{* * * *} 199^{* *} 24^{*}$ | |
| 5 | ... | 17. In how many ways can all param 1 asterisks be replaced with even digits (not necessarily distinct) in the number param 2 so that the resulting number is divisible by 12?
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 6 | $2017^{*} 1^{*} 5^{* * *} 12^{*}$ | 3125 |
| 7 | $2017 * * * * 199^{* *} 24^{*}$ | 1... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,839 |
18. In a music school, there are param1 boys and several girls. On March 8, each of the boys gave gifts to at least two girls. At the same time, for any two boys, there will not be two girls to whom both of them gave gifts. What is the minimum number of girls that could have been in the music school?
| param1 | |
| :... | 18. In a music school, there are param1 boys and several girls. On March 8, each boy gave gifts to at least two girls. At the same time, for any two boys, there will not be two girls to whom both boys gave gifts. What is the minimum number of girls that could have been in the music school?
| param1 | Answer |
| :---: ... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,840 | |
10.2. There are 90 cards - 10 with the digit 1, 10 with the digit $2, \ldots, 10$ with the digit 9. From all these cards, two numbers were formed, one of which is three times the other. Prove that one of these numbers can be factored into four not necessarily distinct natural factors, greater than one. | Solution. Let these numbers be $A$ and $B=3A$. Then the sum of the digits of the number $B$ is divisible by 3. But the sum of the digits on all cards is divisible by 9 (and therefore by 3), so the sum of the digits of the number $A$ is divisible by 3. This means that the number $A$ is divisible by 3. But then the numbe... | 40 | Number Theory | proof | Yes | Yes | olympiads | false | 3,843 |
10.4. Quadrilateral $A B C D$ is inscribed in a circle. The perpendicular to side $B C$, drawn through its midpoint - point $M$, intersects side $A B$ at point $K$. The circle with diameter $K C$ intersects segment $C D$ at point $P(P \neq C)$. Find the angle between the lines $M P$ and $A D$. | Answer: $90^{\circ}$.
Solution. We will prove that the lines $M P$ and $A D$ are perpendicular. Let $\omega$ be the circle constructed with $K C$ as its diameter, then point $M$ lies on $\omega$, since angle $C M K$ is a right angle. Therefore, $\angle C P M = \angle C K M = \alpha$ (they subtend the arc $C M$ of circ... | 90 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,844 |
10.5. In a chess tournament, 10th graders and 9th graders participated. There were 9 times more 10th graders, and they scored 4 times more points than the 9th graders. Which class did the winner of the tournament belong to, and how many points did he score? In chess, 1 point is awarded for a win, 0.5 points for a draw,... | Answer: 9th grader, 9 points.
Solution. Let there be $x$ 9th graders and $9x$ 10th graders in the tournament. Estimate the maximum number of points that 9th graders could score and the minimum number of points that 10th graders could score. This would happen if all 9th graders won their matches against 10th graders. T... | 9thgrader,9points | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,845 |
21. The circle inscribed in isosceles triangle $ABC (AB=BC)$ has a radius of param1 and intersects the height $BH$ of triangle $ABC$ at point $P$. Find the perimeter of triangle $ABC$ if it is known that $BP=$ param2.
| param1 | param2 | |
| :---: | :---: | :---: |
| 3 | 2 | |
| 4 | 1 | |
| 5 | 8 | |
| 12 | 1 | | | 21. The circle inscribed in isosceles triangle $ABC (AB=BC)$ has a radius of param1 and intersects the height $BH$ of triangle $ABC$ at point $P$. Find the perimeter of triangle $ABC$ if it is known that $BP=$ param2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 3 | 2 | 32 |
| 4 | 1 | 54 |
| 5 | 8 | 54 |
|... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,846 | |
22. Given a regular param1. In how many ways can three of its vertices be chosen so that they form the vertices of a triangle with all sides of different lengths? (Two sets of vertices that differ only in the order of the vertices are considered the same.)
| param1 | |
| :---: | :--- |
| 17-gon | |
| 19-gon | |
| 2... | 22. Given a regular param1. In how many ways can three of its vertices be chosen so that they form the vertices of a triangle with all sides of different lengths? (Two sets of vertices that differ only in the order of the vertices are considered the same.)
| param1 | Answer |
| :---: | :---: |
| 17-gon | 544 |
| 19-go... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,847 |
23. It is known that the roots of the quadratic polynomial $f(x)=x^{2}+a x+b$ are each 1 greater than the roots of the quadratic polynomial $g(x)$. Let $M$ be the sum of the coefficients of $f(x)$, and $N$ be the sum of the coefficients of $g(x)$. What is the smallest value that $|M-N|$ can take?
| param1 | |
| :---:... | 23. It is known that the roots of the quadratic polynomial $f(x)=x^{2}+a x+b$ are each 1 greater than the roots of the quadratic polynomial $g(x)$. Let $M$ be the sum of the coefficients of $f(x)$, and $N$ be the sum of the coefficients of $g(x)$. What is the smallest value that $|M-N|$ can take?
| param1 | Answer |
|... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,848 | |
26. It is known that the inequality param 1 is not satisfied for exactly param 2 integer values of $x$. Find the number of integers $m$ for which this is possible.
| param1 | param 2 | |
| :---: | :---: | :---: |
| $x^{2}+10 x+m \geq 0$ | 25 | |
| $x^{2}-14 x+m \geq 0$ | 21 | |
| $x^{2}-30 x+m \geq 0$ | 29 | |
| $... | 26. It is known that the inequality param1 is not satisfied for exactly param 2 integer values of $x$. Find the number of integers $m$ for which this is possible.
| param1 | param | Answer |
| :---: | :---: | :---: |
| $x^{2}+10 x+m \geq 0$ | 25 | 25 |
| $x^{2}-14 x+m \geq 0$ | 21 | 21 |
| $x^{2}-30 x+m \geq 0$ | 29 |... | notfound | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 3,851 |
27. In how many ways can all the param1 asterisks be replaced with digits (not necessarily distinct) in the number param2 so that the resulting number is divisible by $12?$
| param1 | param2 | |
| :---: | :---: | :---: |
| 6 | $2017^{**} 13^{***52} 2^{*}$ | |
| 7 | $2017^{**} 1^{**} 1^{**} 34^{*}$ | |
| 5 | $2017^{... | 27. In how many ways can all param1 asterisks be replaced with digits (not necessarily distinct) in the number ragat 2 so that the resulting number is divisible by $12?$
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 6 | $2017^{**} 13^{***} 52^{*}$ | 100000 |
| 7 | $2017^{**} 1^{**} 1^{**} 3^{*}$ | 1000000 |... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,852 |
28. In a single-round football tournament (each team played against each other exactly once), ragat1 teams participated. For a win in a match, 3 points are awarded, for a draw - 1 point, for a loss - 0 points. It turned out that the teams that took the first three places scored different numbers of points. What is the ... | 28. In a single-round football tournament (each team played against each other exactly once), ragat1 teams participated. For a win in a match, 3 points are awarded, for a draw - 1 point, for a loss - 0 points. It turned out that the teams that took the first three places scored different numbers of points. What is the ... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,853 |
29. The weather forecast from September 1 to $k$ was made by param 1 meteorologists. It turned out that each of them indicated at least two rainy days. At the same time, no two meteorologists had two identical rainy days among those they indicated. For what smallest $k$ could this be?
| param1 | Answer |
| :---: | :--... | 29. The weather forecast from September 1 to $k$ was made by param 1 meteorologists. It turned out that each of them indicated at least two rainy days. At the same time, no two meteorologists had two identical rainy days among those they indicated. For what smallest $k$ could this be?
| param1 | Answer |
| :---: | :--... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,854 | |
30. It is known that for pairwise distinct numbers $a, b, c$, the equality param1 holds. What is the smallest value that the expression $a+b+c$ can take?
| param1 | Answer |
| :---: | :---: |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+2\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -2 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b... | 30. It is known that for pairwise distinct numbers $a, b, c$, the equality param1 holds. What is the smallest value that the expression $a+b+c$ can take?
| param1 | Answer |
| :---: | :---: |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b)+2\left(a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)\right)=0$ | -2 |
| $a^{3}(b-c)+b^{3}(c-a)+c^{3}(a-b... | -10 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,855 |
24. Find the number of integer solutions ( $x ; y ; z$ ) of the equation param1, satisfying the condition param2.
| param1 | param2 | |
| :---: | :---: | :---: |
| $27^{a} \cdot 75^{b} \cdot 5^{c}=75$ | $\|a+b+c\| \leq 101$ | |
| $27^{a} \cdot 75^{b} \cdot 5^{c}=375$ | $\|a+b+c\|<98$ | |
| $27^{a} \cdot 75^{b} \cdo... | 24. Find the number of integer solutions ( $x ; y ; z$ ) of the equation param1, satisfying the condition param2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $27^{a} \cdot 75^{b} \cdot 5^{c}=75$ | $\|a+b+c\| \leq 101$ | 51 |
| $27^{a} \cdot 75^{b} \cdot 5^{c}=375$ | $\|a+b+c\|<98$ | 48 |
| $27^{a} \cdot 7... | 51,48,56,69 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,859 |
25. It is known that $\frac{a}{b+c-3 a}=\frac{b}{a+c-3 b}=\frac{c}{a+b-3 c}$. Find all possible different values of the expression param1. In the answer, write the sum of the found values.
| param1 | |
| :---: | :--- |
| $\frac{2 b}{a}+\frac{2 c}{a}+\frac{3 a}{b}+\frac{3 c}{b}$ | |
| $\frac{3 b}{a}+\frac{3 c}{a}+\fr... | 25. It is known that $\frac{a}{b+c-3 a}=\frac{b}{a+c-3 b}=\frac{c}{a+b-3 c}$. Find all possible different values of the expression ragat1. In the answer, write the sum of the found values.
| param1 | Answer |
| :---: | :---: |
| $\frac{2 b}{a}+\frac{2 c}{a}+\frac{3 a}{b}+\frac{3 c}{b}$ | 5 |
| $\frac{3 b}{a}+\frac{3 c... | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,860 | |
26. It is known that for all pairs of numbers ( $x ; y$ ), for which the equality $x+y=$ param 1 and the inequality $x^{2}+y^{2}>\operatorname{param} 2$ hold, the inequality $x^{3}+y^{3}>m$ also holds. What is the greatest value that $m$ can take?
| param1 | param2 | |
| :---: | :---: | :---: |
| 4 | 17 | |
| 5 | 26... | 26. It is known that for all pairs of numbers ( $x ; y$ ), for which the equality $x+y=$ param1 and the inequality $x^{2}+y^{2}>$ param2 hold, the inequality $x^{3}+y^{3}>m$ also holds. What is the greatest value that $m$ can take?
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| 4 | 17 | 70 |
| 5 | 26 | 132.5... | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 3,861 | |
28. In an acute-angled triangle $ABC$, the altitude $BH$ is drawn. Let $M$ be the midpoint of segment $BH$. The point $M$ is reflected symmetrically with respect to sides $AB$ and $CB$, obtaining points $K$ and $L$ respectively. The radius of the circumcircle of triangle $KLM$ is equal to 1. What is the greatest value ... | 28. In an acute-angled triangle $ABC$, the altitude $BH$ is drawn. Let $M$ be the midpoint of segment $BH$. The point $M$ is reflected symmetrically with respect to sides $AB$ and $CB$, obtaining points $K$ and $L$ respectively. The radius of the circumcircle of triangle $KLM$ is param 1. What is the greatest value tha... | Geometry | MCQ | Yes | Yes | olympiads | false | 3,863 | |
29. In a football tournament held in a single round-robin format (each team must play each other exactly once), there are param1 teams. At some point in the tournament, the coach of team $A$ noticed that any two teams, different from $A$, have played a different number of games. What is the maximum number of games that... | 29. In a football tournament held in a single round-robin format (each team must play every other team exactly once), there are param1 teams participating. At some point in the tournament, the coach of team $A$ noticed that any two teams, different from $A$, have played a different number of games. What is the maximum ... | notfound | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,864 |
30. In the kingdom with param 1 cities, some of them are connected by direct flights. It is known that if there is a direct flight between cities $A$ and $B$, and there is a direct flight between cities $B$ and $C$, then there is no direct flight between cities $A$ and $C$. What is the maximum number of direct flights ... | 30. In the kingdom with param 1 cities, some of them are connected by direct flights. It is known that if there is a direct flight between cities $A$ and $B$, and there is a direct flight between cities $B$ and $C$, then there is no direct flight between cities $A$ and $C$. What is the maximum number of direct flights ... | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,865 | |
1. Calculate param1, given that param2 is 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $\log _{x}\left(x^{4}-27 x+3\right)$ | $x^{9}-3 x^{5}+9 x-1=0$ | |
| $\log _{x}\left(x^{4}-64 x+4\right)$ | $x^{9}-4 x^{5}+16 x-1=0$ | |
| $\log _{x}\left(x^{4}-8 x^{2}+2\right)$ | $x^{10}-2 x^{6}+4 x^{2}-1=0$ | |
... | 1. Calculate param1, given that param2 is 2.
| param1 | param2 | Answer |
| :---: | :---: | :---: |
| $\log _{x}\left(x^{4}-27 x+3\right)$ | $x^{9}-3 x^{5}+9 x-1=0$ | 13 |
| $\log _{x}\left(x^{4}-64 x+4\right)$ | $x^{9}-4 x^{5}+16 x-1=0$ | 13 |
| $\log _{x}\left(x^{4}-8 x^{2}+2\right)$ | $x^{10}-2 x^{6}+4 x^{2}-1=0$ |... | notfound | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,866 |
4. Given a parabola P: param1. Tangents to the parabola P, drawn through points $K_{1}$ and $K_{2}$, intersect the axis at points $P_{1}$ and $P_{2}$, respectively. Lines perpendicular to these tangents and passing through points $P_{1}$ and $P_{2}$, respectively, intersect at point $Q$. What is the maximum area that t... | 4. Given a parabola P: param1. Tangents to the parabola P, drawn through points $K_{1}$ and $K_{2}$, intersect the axis at points $P_{1}$ and $P_{2}$, respectively. Lines perpendicular to these tangents and passing through points $P_{1}$ and $P_{2}$, respectively, intersect at point $Q$. What is the maximum area that t... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,869 |
6. Given a rectangular grid of size param1. In how many ways can it be cut into grid rectangles of size $1 \times 3$ and $1 \times 4$?
| param1 | Answer |
| :---: | :---: |
| $1 \times 60$ | |
| $1 \times 55$ | |
| $1 \times 58$ | |
| $1 \times 59$ | |
| $1 \times 61$ | | | 6. Given a rectangular grid of size param1. In how many ways can it be cut into grid rectangles of size $1 \times 3$ and $1 \times 4$?
| param1 | Answer |
| :---: | :---: |
| $1 \times 60$ | 45665 |
| $1 \times 55$ | 16855 |
| $1 \times 58$ | 30640 |
| $1 \times 59$ | 37432 |
| $1 \times 61$ | 55728 | | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,871 | |
7. What is the maximum volume that the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ can have, given that its diagonals $A_{1} C_{1}, C_{1} D, B D_{1}, B_{1} C$ have lengths param1, param2, param3, param4 in some order? Write the square of the volume in the answer.
| param1 | param2 | param3 | param4 | answer |
| :... | 7. What is the maximum volume that the parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$ can have, given that its diagonals $A_{1} C_{1}, C_{1} D, B D_{1}, B_{1} C$ have lengths param1, param2, param3, param4 in some order? Write the square of the volume in the answer.
| param1 | param2 | param3 | param4 | answer |
| :... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,872 |
8. For each natural $n$, which is not a perfect square, all values of the variable $x$ are calculated for which both numbers $x+\sqrt{n}$ and $x^{3}+$ param1 $\sqrt{n}$ are integers. Find the total number of such values of $x$.
| param1 | answer |
| :---: | :---: |
| 1524 | |
| 1372 | |
| 1228 | |
| 1092 | |
| 964... | 8. For each natural $n$, which is not a perfect square, all values of the variable $x$ are calculated for which both numbers $x+\sqrt{n}$ and $x^{3}+$ param1 $\sqrt{n}$ are integers. Find the total number of such values of $x$.
| param1 | answer |
| :---: | :---: |
| 1524 | 39 |
| 1372 | 33 |
| 1228 | 35 |
| 1092 | 27... | 33 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,873 |
9. A regular param 1 -sided polygon is inscribed in a circle, with different natural numbers written at its vertices. A pair of non-adjacent vertices of the polygon $A$ and $B$ is called interesting if, on at least one of the two arcs $A B$, all the numbers written at the vertices of the arc are greater than the number... | 9. A regular param 1-gon is inscribed in a circle, with different natural numbers written at its vertices. A pair of non-adjacent vertices of the polygon $A$ and $B$ is called interesting if, on at least one of the two arcs $A B$, all the numbers written at the vertices of the arc are greater than the numbers written a... | 92 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,874 |
10. Find the minimum of param1 under the condition param 2.
| param1 | param 2 | Answer |
| :---: | :---: | :---: |
| $x^{2}+y^{2}-4 y$ | $\|4 x-3 y\|+5 \sqrt{x^{2}+y^{2}-20 y+100}=30$ | |
| $y^{2}+x^{2}+2 y$ | $\|4 y-3 x\|+5 \sqrt{x^{2}+y^{2}+20 y+100}=40$ | |
| $x^{2}+y^{2}+2 x$ | $\|4 y+3 x\|+5 \sqrt{x^{2}+y^{2}+... | 10. Find the minimum of param1 under the condition param 2.
| param1 | param 2 | Answer |
| :---: | :---: | :---: |
| $x^{2}+y^{2}-4 y$ | $\|4 x-3 y\|+5 \sqrt{x^{2}+y^{2}-20 y+100}=30$ | 36.96 |
| $y^{2}+x^{2}+2 y$ | $\|4 y-3 x\|+5 \sqrt{x^{2}+y^{2}+20 y+100}=40$ | 28.16 |
| $x^{2}+y^{2}+2 x$ | $\|4 y+3 x\|+5 \sqrt{x^... | 4.76 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,875 |
1. (10 points) Petya added several consecutive even numbers. It turned out that the resulting sum is 30 times greater than the largest addend and 90 times greater than the smallest. Find which numbers Petya added. | Answer: $44,46,48, \ldots, 132$.
Solution. Let the first number be $n$, and the last number be $n+2k$. There are a total of $k+1$ numbers. The numbers form an arithmetic progression, the sum of which is $(n+k)(k+1)$. We obtain the system $\left\{\begin{array}{c}(n+k)(k+1)=30(n+2k), \\ (n+k)(k+1)=90n .\end{array}\right... | 44,46,48,\ldots,132 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,876 |
2. (12 points) The sequence of functions is defined by the formulas:
$$
f_{0}(x)=2 \sqrt{x}, f_{n+1}(x)=\frac{4}{2-f_{n}(x)}, n=0,1,2 \ldots, x \in[4 ; 9]
$$
Find $f_{2023}(4)$. | Answer: -2.
Solution. It is easy to calculate that $f_{3}(x)=f_{0}(x)$, therefore
$$
f_{2023}(x)=f_{1}(x)=\frac{2}{1-\sqrt{x}}
$$
Then $f_{2023}(4)=-2$.
Remark. One can immediately compute the values of the functions at the given point. The sequence will be $f_{0}(4)=4, f_{1}(4)=-2, f_{2}(4)=1, f_{3}(4)=4 \ldots$
... | -2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,877 |
3. (15 points) The vertices of the broken line $A B C D E F G$ have coordinates $A(-1 ; -7), B(2 ; 5), C(3 ; -8), D(-3 ; 4), E(5 ; -1), F(-4 ; -2), G(6 ; 4)$.
Find the sum of the angles with vertices at points $B, E, C, F, D$. | Answer: $135^{\circ}$.
Solution. The closed broken line $B C D E F B$ forms a five-pointed "star". The sum of the angles at the rays of this star is $180^{\circ}$.
We will prove that the sum of the angles at the rays of any five-pointed star $B C D E F B$ is $180^{\circ}$. Let $O$ be the point of intersection of the ... | 135 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,878 |
4. (13 points) A triathlon competitor swam 1 km in the first stage. In the second stage, he cycled 25 km, and in the third stage, he ran 4 km. He completed the entire distance in 1 hour 15 minutes. Before the competition, he tried out the course: he swam for $1 / 16$ hours, cycled and ran for $1 / 49$ hours each, cover... | Answer: $5 / 7$ hours; 35 km/h.
Solution. Let $v_{1}, v_{2}, v_{3}$ be the speeds of the athlete on stages $1, 2, 3$ respectively. From the condition, we have: $\frac{1}{v_{1}}+\frac{25}{v_{2}}+\frac{4}{v_{3}}=\frac{5}{4}$ hours; $\frac{1}{16} v_{1}+\frac{1}{49} v_{2}+$ $\frac{1}{49} v_{3}=\frac{5}{4}$ km. Adding thes... | \frac{5}{7} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,879 |
5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid in the form of a circle with a radius of $R=4$ m. The wagon starts from rest, with the jet force having a constant value. What is the maximum speed the wagon will reach after one full circle, if its acceleration over t... | Answer: $\approx 2 \mathrm{m} / \mathrm{s}$.
Solution. The acceleration of the cart: $a_{1}=\frac{v^{2}}{2 S}=\frac{v^{2}}{4 \pi R}$.
In addition, the cart has a centripetal acceleration:
$$
a_{2}=\frac{v^{2}}{R} .
$$
The total acceleration of the cart: $a^{2}=a_{1}^{2}+a_{2}^{2}=\left(\frac{v^{2}}{4 \pi R}\right)^... | 2.8\mathrm{~}/\mathrm{} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 3,880 |
6. (10 points) On a horizontal surface, there are two identical stationary blocks, each with mass $M$. The distance between them is $S$. A bullet of mass $m$ horizontally hits and gets stuck in the left block. What should be the speed of the bullet so that the final distance between the blocks is also equal to $S$? The... | Answer: $v=\frac{2 M}{m} \sqrt{\mu g S}$.
Solution. The law of conservation of momentum for the collision of the bullet with the left block: $m v=(M+m) u=M u$.
In the case of a perfectly elastic collision between bodies of equal mass, there is an "exchange of velocities".
(3 points)
The law of conservation of energ... | \frac{2M}{}\sqrt{\mu} | Other | math-word-problem | Yes | Yes | olympiads | false | 3,881 |
7. (15 points) A uniformly charged sphere of radius $R$ is fixed on a horizontal surface at point $A$. The charge of the sphere is $Q$. At point $C$, which is located at a distance $L$ from the surface of the sphere, a charged sphere of radius $r$ and mass $m$ is floating. Its charge is $q$. It is known that $r<<R$. De... | Answer: $a=\frac{k q Q r^{3}}{m R^{3}(L+2 R-S)^{2}}$.
Solution. In the initial state for the ball: $F_{\text{el}}=m g$.
In the final state for the ball: $F_{\text{el}}-m g-F_{\text{rem}}=-m a$,
where $F_{\text{rem}}=k \frac{q q_{\text{rem}}}{(L+2 R-S)^{2}}$.
The removed charge: $q_{\text{rem}}=Q\left(\frac{r}{R}\ri... | \frac{kr^{3}}{R^{3}(L+2R-S)^{2}} | Other | math-word-problem | Yes | Yes | olympiads | false | 3,882 |
8. (15 points) A thin ray of light falls on a thin converging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\alpha=45^{\circ}$, and the angle between the refracted ray and the plane of the lens $\beta=30^{\circ}$. Determine its focal length. | Answer: $\approx 13.7$ cm.
Solution. Parallel rays intersect at the focus, so $F$ is the focus of the given lens.
In triangle $O A F$: angle $F A O=30^{\circ}$, angle $O F A=15^{\circ}$, a... | 13.7 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,883 |
1. (10 points) Petya added several consecutive odd numbers. It turned out that the resulting sum is 20 times greater than the largest addend and 60 times greater than the smallest. Find which numbers Petya added. | Answer: $29,31,33, \ldots, 87$.
Solution. Let the first number be $n$, and the last number be $n+2k$. There are a total of $k+1$ numbers. The numbers form an arithmetic progression, the sum of which is $(n+k)(k+1)$. We obtain the system $\left\{\begin{array}{c}(n+k)(k+1)=20(n+2k), \\ (n+k)(k+1)=60n .\end{array}\right.... | 29,31,33,\ldots,87 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,884 |
2. (12 points) The sequence of functions is defined by the formulas:
$$
f_{0}(x)=2 \sqrt{x}, f_{n+1}(x)=\frac{4}{2-f_{n}(x)}, n=0,1,2 \ldots, x \in[4 ; 9]
$$
Find $f_{2023}(9)$. | Answer: -1.
Solution. It is easy to calculate that $f_{3}(x)=f_{0}(x)$, therefore,
$$
f_{2023}(x)=f_{1}(x)=\frac{2}{1-\sqrt{x}}
$$
Then $f_{2023}(9)=-1$.
Remark. One can immediately compute the values of the functions at the given point. The sequence will be $f_{0}(9)=6, f_{1}(9)=-1, f_{2}(9)=\frac{4}{3}, f_{3}(9)=... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,885 |
3. (15 points) The vertices of the broken line $A B C D E F G$ have coordinates $A(0 ; -5), B(3 ; 7), C(4 ; -6), D(-2 ; 6), E(6 ; 1), F(-3 ; 0), G(7 ; 6)$.
Find the sum of the angles with vertices at points $B, E, C, F, D$. | Answer: $135^{\circ}$.
Solution. The closed broken line $B C D E F B$ forms a five-pointed "star". The sum of the angles at the rays of this star is $180^{\circ}$.
We will prove that the sum of the angles at the rays of any five-pointed star $B C D E F B$ is $180^{\circ}$. Let $O$ be the point of intersection of the ... | 135 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,886 |
4. (13 points) A triathlon competitor swam 1 km in the first stage. In the second stage, he cycled 25 km, and in the third stage, he ran 4 km. He completed the entire distance in 1 hour 15 minutes. Before the competition, he tried out the course: he swam for $1 / 16$ hours, cycled and ran for $1 / 49$ hours each, cover... | Answer: $2 / 7$ hour; 14 km/h.
Solution. Let $v_{1}, v_{2}, v_{3}$ be the speeds of the athlete on stages $1,2,3$ respectively. From the condition, we have: $\frac{1}{v_{1}}+\frac{25}{v_{2}}+\frac{4}{v_{3}}=\frac{5}{4}$ hour; $\frac{1}{16} v_{1}+\frac{1}{49} v_{2}+$ $\frac{1}{49} v_{3}=\frac{5}{4}$ km. Adding these eq... | \frac{2}{7} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,887 |
5. (10 points) A small railway wagon with a jet engine is standing on the tracks. The tracks are laid in the form of a circle with a radius of $R=5$ m. The wagon starts from rest, with the jet force having a constant value. What is the maximum speed the wagon will reach after one full circle, if its acceleration over t... | Answer: $\approx 2.23 \mathrm{~m} / \mathrm{s}$.
Solution. The acceleration of the trolley: $a_{1}=\frac{v^{2}}{2 S}=\frac{v^{2}}{4 \pi R}$.
In addition, the trolley has a centripetal acceleration:
$a_{2}=\frac{v^{2}}{R}$.
The total acceleration of the trolley: $a^{2}=a_{1}^{2}+a_{2}^{2}=\left(\frac{v^{2}}{4 \pi R}... | 2.23\mathrm{~}/\mathrm{} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 3,888 |
6. (10 points) On a horizontal surface, there are two identical stationary blocks, each with mass $M$. The distance between them is $S$. A bullet of mass $m$ hits and gets stuck in the left block. What should be the speed of the bullet so that the final distance between the blocks is also equal to $S$? The collision be... | Answer: $v=\frac{2 M}{m} \sqrt{\mu g S}$.
Solution. The law of conservation of momentum for the collision of the bullet with the left block: $m v=(M+m) u=M u$.
In the case of a perfectly elastic collision between bodies of equal mass, there is an "exchange of velocities".
( $\mathbf{\text { point) }}$
The law of co... | \frac{2M}{}\sqrt{\mu} | Other | math-word-problem | Yes | Yes | olympiads | false | 3,889 |
7. (15 points) A uniformly charged sphere of radius $R$ is fixed on a horizontal surface at point $A$. The charge of the sphere is $Q$. At point $C$, which is located at a distance $L$ from the surface of the sphere, a charged sphere of radius $r$ and mass $m$ is floating. Its charge is $q$. It is known that $r<<R$. De... | Answer: $a=\frac{k q Q r^{3}}{m R^{3}(L+2 R-S)^{2}}$.
Solution. For the ball in the initial state: $F_{\text{el}}=m g$.
For the ball in the final state: $F_{\text{el}}-m g-F_{\text {rem }}=-m a$,
where $F_{\text {rem }}=k \frac{q q_{\text {rem }}}{(L+2 R-S)^{2}}$.
The removed charge: $q_{\text {rem }}=Q\left(\frac{... | \frac{kr^{3}}{R^{3}(L+2R-S)^{2}} | Other | math-word-problem | Yes | Yes | olympiads | false | 3,890 |
8. (15 points) A thin ray of light falls on a thin diverging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\alpha=30^{\circ}$, and the angle between the refracted ray and the plane of the lens $\beta=45^{\circ}$. Determine its focal length. | Answer: $\approx 13.7 \mathrm{~cm}$.
Solution. The extensions of the parallel rays intersect at the focus, so $F$ is the focus of the given lens.
In triangle $O A F$: angle $A O F=30^{\cir... | 13.7\mathrm{~} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,891 |
1. Nils has a goose farm. Nils calculated that if he sells 75 geese, the feed will last 20 days longer than if he doesn't sell any. If he buys an additional 100 geese, the feed will run out 15 days earlier than if he doesn't make such a purchase. How many geese does Nils have? | Answer: 300.
Solution. Let $A$ be the total amount of feed (in kg), $x$ be the amount of feed per goose per day (in kg), $n$ be the number of geese, and $k$ be the number of days the feed will last. Then
$$
\begin{gathered}
A=k x n=(k+20) x(n-75)=(k-15) x(n+100) \\
k n=(k+20)(n-75)=(k-15)(n+100)
\end{gathered}
$$
So... | 300 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,892 |
3. Solve the equation
$2 x+1+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+1) \cdot \sqrt{x^{2}+2 x+2}=0$. | Answer: $-\frac{1}{2}$.
Solution. Let $f(x)=x+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}$. The original equation can be rewritten as $f(x)+f(x+1)=0$. Note that the function $f(x)$ is odd. It increases on the positive half-axis (as the sum of increasing functions). Due to its oddness, it increases on the entire real l... | -\frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,893 |
4. There are three alloys of nickel, copper, and manganese. In the first, $30\%$ is nickel and $70\%$ is copper, in the second - $10\%$ is copper and $90\%$ is manganese, and in the third - $15\%$ is nickel, $25\%$ is copper, and $60\%$ is manganese. A new alloy of these three metals is needed with $40\%$ manganese. Wh... | Answer: from $40 \%$ to $43 \frac{1}{3} \%$.
Solution. Let the masses of the original alloys, from which the new alloy is obtained, be denoted by $m_{1}, m_{2}, m_{3}$, and the mass of the new alloy by $m$. The equalities $m_{1}+m_{2}+m_{3}=m$ and $0.9 m_{2}+0.6 m_{3}=0.4 m$ hold. The amount of copper in the new alloy... | 40to43\frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,894 |
1. Nils has a goose farm. Nils calculated that if he sells 50 geese, the feed will last 20 days longer than if he doesn't sell any. If he buys an additional 100 geese, the feed will run out 10 days earlier than if he doesn't make such a purchase. How many geese does Nils have? | Answer: 300.
Solution. Let $A$ be the total amount of feed (in kg), $x$ be the amount of feed per goose per day (in kg), $n$ be the number of geese, and $k$ be the number of days the feed will last. Then
$$
\begin{gathered}
A=k x n=(k+20) x(n-50)=(k-10) x(n+100) \\
k n=(k+20)(n-50)=(k-10)(n+100)
\end{gathered}
$$
So... | 300 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,895 |
3. Solve the equation
$$
2 x+2+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}+\operatorname{arctg}(x+2) \cdot \sqrt{x^{2}+4 x+5}=0
$$ | Answer: -1.
Solution. Let $f(x)=x+\operatorname{arctg} x \cdot \sqrt{x^{2}+1}$. The original equation can be rewritten as $f(x)+f(x+2)=0$. Note that the function $f(x)$ is odd. It is increasing on the positive half-axis (as the sum of increasing functions). Due to its oddness, it is increasing on the entire real line.... | -1 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,896 |
Problem No. 5 (15 points)
Two people are moving in the same direction. At the initial moment, the distance between them is $S_{0}=100 \text{ m}$. The speed of the first, faster, pedestrian is $v_{1}=8 \text{ m} / \text{s}$. Determine the speed $v_{2}$ of the second, if it is known that after $t=5$ min the distance bet... | Answer: 7.5 or 7.83
## Solution and Evaluation Criteria:
For $t=5$ min, the first pedestrian walked: $S_{1}=v_{1} t=8 \cdot 5 \cdot 60=2400$ meters
Therefore, there are two possible answers. In this time, the second pedestrian walked:
or $S_{2}=2400-100-50=2250$ meters, in this case his speed:
$v_{2}=\frac{S_{2}}{... | 7.5or7.83 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,898 |
# Problem No. 6 (10 points)
The density of a body is defined as the ratio of its mass to the volume it occupies. A homogeneous cube with a volume of \( V = 8 \, \text{m}^3 \) is given. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change?
Answer: decrease... | # Solution and evaluation criteria:
Volume of the cube: $v=a^{3}$, where $a$ is the length of the edge, therefore:
$a=2 \partial m=200 mm$.
Final edge length: $a_{\kappa}=204$ mm.
Thus, the final volume: $V_{\kappa}=a_{K}^{3}=2.04^{3}=8.489664 \partial \mu^{3} \approx 1.06 V$.
Therefore, the density:
$\rho_{K}=\f... | 6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,899 |
# Problem № 7 (10 points)
A person and his faithful dog started moving along the perimeter of a block from point A at the same time $t_{0}=0$ min. The person moved with a constant speed clockwise, while the dog ran with a constant speed counterclockwise (see fig.). It is known that they met for the first time after $t... | # Answer: in 9 min
## Solution and evaluation criteria:
In $t_{1}=1$ min, the person and the dog together covered a distance equal to the perimeter of the block, with the person moving 100 meters from the starting point of the journey.
That is, during each subsequent meeting, the person will be 100 meters away from ... | 9 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,900 |
# Problem № 8 (15 points)
In some English-speaking countries, temperature is measured in degrees Fahrenheit. An English schoolboy, observing a thermometer in a glass of cooling water, noticed that it cooled by $10^{\circ} \mathrm{F}$. He became curious about how much heat was released. In books, he found the following... | # Answer: 23.3 kJ
## Solution and grading criteria:
The change in temperature in degrees Fahrenheit is related to the change in temperature in degrees Celsius:
That is, in degrees Celsius... | 23.3 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,901 |
# Problem No. 5 (15 points)
Two people are moving in the same direction. At the initial moment, the distance between them is $S_{0}=200 \mathrm{~m}$. The speed of the first, faster, pedestrian is $v_{1}=7 \mathrm{~m} / \mathrm{c}$. Determine the speed $v_{2}$ of the second pedestrian, given that after $t=5$ minutes, t... | # Answer: 6.0 or 6.67
## Solution and Grading Criteria:
For $t=5$ min, the first pedestrian walked: $S_{1}=v_{1} t=7 \cdot 5 \cdot 60=2100$ meters
(5 points)
Therefore, there are two possible answers. In this time, the second pedestrian walked:
either $S_{2}=2100-200-100=1800$ meters, in which case his speed is:
... | 6or6.67 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,902 |
# Problem No. 6 (10 points)
The density of a body is defined as the ratio of its mass to the volume it occupies. A homogeneous cube with a volume of \( V = 27 \partial \mu^{3} \) is given. As a result of heating, each of its edges increased by 9 mm. By what percentage did the density of this cube change? | # Answer: decreased by $8 \%$
## Solution and evaluation criteria:
Volume of a cube: $v=a^{3}$, where $a$ is the length of the edge, therefore:
$a=3 dm=300 mm$.
Final edge length: $a_{K}=309 mm$.
Thus, the final volume: $V_{K}=a_{K}^{3}=3.09^{3}=29.503629 dm^{3} \approx 1.09 V$.
Therefore, the density:
$\rho_{K}... | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,903 |
# Problem № 7 (10 points)
A person and his faithful dog started moving along the perimeter of a block from point $A$ simultaneously at time $t_{0}=0$ min. The person moved with a constant speed clockwise, while the dog ran with a constant speed counterclockwise (see figure). It is known that they met for the first tim... | # Answer: in 14 min
## Solution and evaluation criteria:
In $t_{1}=2$ min, the person and the dog together covered a distance equal to the perimeter of the block, with the person moving 100 meters from the starting point of the journey.
That is, during each subsequent meeting, the person will be 100 meters away from... | 14 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,904 |
# Problem № 8 (15 points)
In some English-speaking countries, temperature is measured in degrees Fahrenheit. An English schoolboy, observing a thermometer in a glass of cooling water, noticed that it cooled by $30^{\circ} \mathrm{F}$. He became curious about how much heat was released. In books, he found the following... | # Answer: 140 kJ
## Solution and grading criteria:
The change in temperature in degrees Fahrenheit is related to the change in temperature in degrees Celsius:
That is, in degrees Celsius,... | 140 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,905 |
1. Compute the surface area
$$
x^{2}+y^{2}+z^{2}=4(x+y+z)
$$ | Answer: $48 \pi$.
Solution. If we rewrite the equation as
$$
(x-2)^{2}+(y-2)^{2}+(z-2)^{2}=12,
$$
it is easy to see that it represents a sphere, the square of whose radius is 12.
Evaluation. 10 points for the correct solution. If the surface is correctly identified as a sphere with radius $2 \sqrt{3}$, but there is... | 48\pi | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,906 |
3. Solve the equation
$$
2 x^{3}=\left(2 x^{2}+x-1\right) \sqrt{x^{2}-x+1} .
$$ | Answer: $1 ; \frac{-1-\sqrt{13}}{6}$.
First solution. Let $t=\sqrt{x^{2}-x+1}$. Then $x-1=x^{2}-t^{2}$, and the original equation can be rewritten as $2 x^{3}=\left(3 x^{2}-t^{2}\right) t$. Clearly, $x=0$ is not a root of the original equation. Divide the last equation by $x^{3}$. For the new variable $y=\frac{t}{x}$,... | 1;\frac{-1-\sqrt{13}}{6} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,908 |
4. 20 balls of the same mass are moving along a chute towards a metal wall with the same speed. Coming towards them at the same speed are 16 balls of the same mass. When two balls collide, they fly apart with the same speed. After colliding with the wall, a ball bounces off it with the same speed. (The balls move only ... | # Answer: 510.
Solution. We will assume that initially, each ball moving towards the wall has a red flag, and the rest of the balls have blue flags. Imagine that when the balls collide, they exchange flags. Then each blue flag moves at a constant speed in one direction (away from the wall), and each red flag reaches t... | 510 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,909 |
# Problem №1 (10 points)
A stone is thrown from the surface of the Earth at an angle of $60^{\circ}$ to the horizon with an initial velocity $v_{0}=10 \mathrm{~m} / \mathrm{s}$. Determine the radius of curvature of the trajectory at the final point of the flight. The acceleration due to gravity $g=10 \mathrm{~m} / \ma... | # Solution:
The final velocity of the stone is equal to its initial velocity:
The centripetal acceleration at the final point of the flight:
$a_{\text{c}}=g \cdot \cos 60^{\circ}=5 \mathrm... | 20\mathrm{~} | Calculus | math-word-problem | Yes | Yes | olympiads | false | 3,910 |
# Problem №2 (10 points)
In a vertical vessel with straight walls closed by a piston, there is water. Its height $h=2$ mm. There is no air in the vessel. To what height must the piston be raised so that all the water evaporates? The density of water $\rho=1000$ kg / $\mathrm{M}^{3}$, the molar mass of water vapor $M=0... | # Solution:
Mass of water in the vessel:
$m=\rho S h$, where $S-$ is the area of the base of the vessel.
(2 points)
Equation of state for an ideal gas for water vapor:
$p V=\frac{m}{M} R T$
where the volume occupied by the vapor $V=S(h+x)$.
(2 points)
As a result, we get:
$p S(h+x)=\frac{\rho S h}{M} R T$
(2 ... | 24.2\mu | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,911 |
# Problem №3 (15 points)
There is a current source with an internal resistance of $r=20$ Ohms. What external resistance should be connected to the source so that the power dissipated in the external resistance differs from the maximum possible by $25 \%$?
# | # Solution:
Power of the current
$P=I^{2} R=\left(\frac{\varepsilon}{R+r}\right)^{2} R$
(2 points)
It will be maximum when $R=r$, i.e.:
$P_{\text {MAX }}=\left(\frac{\varepsilon}{R+r}\right)^{2} R=\left(\frac{\varepsilon}{20+20}\right)^{2} 20=\frac{\varepsilon^{2}}{80}$
(5 points)
The power in question differs b... | R_{1}=6.7\,\Omega,\,R_{2}=60\,\Omega | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,912 |
# Problem №4 (15 points)
A right isosceles triangle is positioned near a converging lens such that the vertex of the right angle coincides with the double focus of the lens, and one of the legs is perpendicular to the principal optical axis. It is known that the area of the triangle is 8 cm$^{2}$, and the area of the ... | # Solution:
Area of the triangle:
$S=\frac{1}{2} A B \cdot A C$, hence $A B=B C=4 \text{cm}$
We can conclude that the original triangle, its image, and the lens are arranged as follows:
P... | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,913 | |
1. Andrei was driving to the airport of a neighboring city. After an hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 20 minutes late. Then he sharply increased his speed, as a result of which he covered the remaining part of the journey at an average speed of 90 km/h ... | Answer: 180 km.
Solution. Let the distance from Andrey's house to the airport be $s$ km, and the time he intended to spend on the road be $1+t$ hours. Then
$$
s=60+60\left(t+\frac{1}{3}\right)=60+90\left(t-\frac{1}{3}\right)
$$
From this,
$$
60 t+20=90 t-30, \quad t=\frac{5}{3}, \quad s=180
$$
Evaluation. Full sco... | 180 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,914 |
3. Can a corner be cut off from a cube with an edge of 15 cm so that the cut has the shape of a triangle with sides of 5, 6, and 8 cm? | Answer: No.
Solution. The problem boils down to the following. Does there exist a triangular pyramid with right angles at the vertex, with a base triangle with sides 5, 6, and $8$?
Let the lengths of the lateral edges be $a, b$, and $c$. Then, by the Pythagorean theorem, we have
$$
a^{2}+b^{2}=25, \quad b^{2}+c^{2}=... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,916 |
4. A natural number $n$ is such that the number $36 n^{2}$ has exactly 51 different natural divisors. How many natural divisors does the number $5 n$ have? | Answer: 16.
Solution. Let $m=p_{1}^{k_{1}} \cdot p_{2}^{k_{2}} \cdot \ldots \cdot p_{l}^{k_{l}}$, where $p_{1}, p_{2}, \ldots, p_{l}$ are pairwise distinct prime numbers. Then the number of natural divisors of the number $m$ is
$$
\tau(m)=\left(k_{1}+1\right)\left(k_{2}+1\right) \ldots\left(k_{l}+1\right)
$$
Indeed,... | 16 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,917 |
6. (10 points) A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed of $v$ towards a wall. Upon impact, $k=0.02\%$ of the snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $2 v$. The specific latent heat of fusion of snow $\lam... | Answer: $n=0.08\%$
Solution. The law of conservation of energy in the first case: $\frac{m v^{2}}{2}=\lambda \frac{\mathrm{km}}{100}$. (3 points)
That is, the speed of the snowball: $v=\sqrt{\frac{\lambda k}{50}}=\sqrt{132} \mathrm{M} / \mathrm{c}$.
In the second case: $\frac{m(2 v)^{2}}{2}=\lambda \frac{n m}{100}$.... | 0.08 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,919 |
1. Viktor was driving to the airport of a neighboring city. After half an hour of driving at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. Then he increased his speed, as a result of which he covered the remaining part of the journey at an average speed of 80 km/h and... | Answer: 150 km.
Solution. Let the distance from Viktor's house to the airport be $s$ km, and the time he intended to spend on the road $\frac{1}{2}+t$ hours. Then
$$
s=30+60\left(t+\frac{1}{4}\right)=30+80\left(t-\frac{1}{4}\right)
$$
From this,
$$
60 t+15=80 t-20, \quad t=\frac{7}{4}, \quad s=150
$$
Evaluation. F... | 150 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,922 |
3. Can a corner be cut off from a cube with an edge of 20 cm so that the cut has the shape of a triangle with sides 7, 8, and 11 cm? | Answer: No.
Solution. The problem boils down to the following. Does there exist a triangular pyramid with right angles at the vertex, with a base triangle with sides 7, 8, and $11$?
Let the lengths of the lateral edges be $a, b$, and $c$. Then, by the Pythagorean theorem, we have
$$
a^{2}+b^{2}=49, \quad b^{2}+c^{2}... | proof | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,924 |
4. A natural number $n$ is such that the number $100 n^{2}$ has exactly 55 different natural divisors. How many natural divisors does the number 10n have? | Answer: 18.
Solution. Let $m=p_{1}^{k_{1}} \cdot p_{2}^{k_{2}} \cdot \ldots \cdot p_{l}^{k_{l}}$, where $p_{1}, p_{2}, \ldots, p_{l}$ are pairwise distinct prime numbers. Then the number of natural divisors of the number $m$ is
$$
\tau(m)=\left(k_{1}+1\right)\left(k_{2}+1\right) \ldots\left(k_{l}+1\right)
$$
Indeed,... | 18 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,925 |
5. (10 points) The resistance of the wire loop shown in the figure between points $A$ and $B$ is $R=15$ Mm. It is known that the distance between these points is $2 M$. Find the resistance $R_{0}$ of one meter of the wire used to make the loop.
(3 points)
Its resistance: $\frac{1}{R}=\frac{1}{4 R_{0... | R_{0}=15 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,926 |
6. (10 points) A snowball with a temperature of $0^{\circ} \mathrm{C}$ is launched at a speed of $v$ towards a wall. Upon impact, $k=0.02\%$ of the snowball melts. Determine what percentage of the snowball will melt if it is launched towards the wall at a speed of $\frac{v}{2}$? The specific latent heat of fusion of sn... | Answer: $n=0.005\%$
Solution. The law of conservation of energy in the first case: $\frac{m v^{2}}{2}=\lambda \frac{k m}{100}$. (3 points)
That is, the speed of the snowball: $v=\sqrt{\frac{\lambda k}{50}}=\sqrt{132} \mathrm{M} / \mathrm{c}$.
In the second case: $\frac{m\left(\frac{v}{2}\right)^{2}}{2}=\lambda \frac... | 0.005 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,927 |
8. (15 points) A rigid rod is moving on a horizontal table. At a certain moment, the speed of one end of the rod is \( v_{1}=10 \mathrm{M} / \mathrm{c} \), and the speed of the other end is \( v_{2}=6 \mathrm{M} / \mathrm{c} \) and it is directed along the axis of the rod (see figure). Determine the speed of the midpoi... | Answer: $\approx 7.2 \mu / c$
Solution. All points on the rod have a velocity component directed to the right, equal to $v_{2}$.
(4 points)
Therefore, the velocity component directed upwards for point A: $v_{\text {vert } A}=\sqrt{v_{1}^{2}-v_{2}^{2}}=8 \mathrm{M} / \mathrm{c}$.
The velocity component of point $B$,... | 7.2\mathrm{M}/\mathrm{} | Other | math-word-problem | Yes | Yes | olympiads | false | 3,929 |
1. (16 points) Fresh mushrooms contain $90 \%$ water by mass, while dried mushrooms contain $12 \%$ water. How many kg of dried mushrooms can be obtained from 22 kg of fresh mushrooms? | # Answer: 2.5 kg
Solution. The dry matter in fresh mushrooms is 2.2 kg, which constitutes $88\%$ of the weight in dried mushrooms. By setting up the corresponding proportion, we will find the weight of the dried mushrooms. | 2.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,930 |
2. (17 points) The medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular to each other. Find the area of the square with side $AB$, if $BC=28, AC=44$.
# | # Answer: 544
Solution. Let $D$ be the midpoint of $B C$, $E$ be the midpoint of $A C$, and $M$ be the point of intersection of the medians. Let $M D=a$, $M E=b$. Then $A M=2 a$, $B M=2 b$. From the right triangles $B M D$ and $A M E$, we have $a^{2}+4 b^{2}=B D^{2}=14^{2}$ and $4 a^{2}+b^{2}=A E^{2}=22^{2}$. Adding t... | 544 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,931 |
3. (17 points) In a $4 \times 5$ grid, 5 crosses need to be placed such that each row and each column contains at least one cross. How many ways can this be done? | # Answer: 240
Solution. From the condition, it follows that in some row, two cells are marked (while in the others, only one each). This row can be chosen in 4 ways, and the two crosses in it can be chosen in $5 \cdot 4 / 2=10$ ways. The remaining three crosses can be chosen in $3 \cdot 2 \cdot 1=6$ ways. By the rule ... | 240 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,932 |
4. (20 points) A ball was thrown from the surface of the Earth at an angle of $45^{\circ}$ with a speed of $v_{0}=20 \mathrm{M} / \mathrm{s}$. How long will it take for the velocity vector of the ball to turn by an angle of $90^{\circ}$? Neglect air resistance. The acceleration due to gravity is $g=10 \mathrm{M} / \mat... | Answer: $\approx 2.83 s$.
Solution. In fact, the problem requires finding the total flight time of the ball. The $y$-coordinate of the ball changes according to the law:
$$
y=v_{0 y} t-\frac{g t^{2}}{2}=v_{0} \sin 45^{\circ} t-\frac{g t^{2}}{2}=10 \sqrt{2} \cdot t-5 t^{2}=0
$$
From this, we obtain that $t=2 \sqrt{2}... | 2.83 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,933 |
5. (15 points) A massive vertical plate is fixed on a car moving at a speed of $4 \mathrm{M} / \mathrm{c}$. A ball is flying towards it at a speed of $5 \mathrm{M} / \mathrm{c}$ relative to the Earth. Determine the speed of the ball relative to the Earth after a perfectly elastic normal collision. | Answer: $13 \mathrm{M} / \mathrm{c}$
Solution. We will associate the reference frame with the slab. Relative to this reference frame, the ball is flying at a speed of $5 M / c+4 M / c=9 M / c$. After a perfectly elastic collision, the ball's speed relative to the slab is also $9 \mathrm{M} / \mathrm{c}$. Therefore, re... | 13\mathrm{M}/\mathrm{} | Other | math-word-problem | Yes | Yes | olympiads | false | 3,934 |
6. (15 points) Two identical resistors $R_{0}$ are connected in series and connected to a source of constant voltage. An ideal voltmeter is connected in parallel to one of the resistors. Its reading is $U=9 B$. If the voltmeter is replaced by an ideal ammeter, its reading will be $I=2$ A. Determine the value of $R_{0}$... | Answer: 90 m
Solution. The voltage supplied by the source is $U_{\text {total }}=2 U=18$ V. In the second case, the current flows only through one of the resistors. Therefore, $R_{0}=\frac{U_{\text {total }}}{I}=9 \Omega$.
## Qualifying Stage
## Variant 2
## Problems, answers, and evaluation criteria | 9\Omega | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,935 |
1. (16 points) Fresh mushrooms contain $80 \%$ water by mass, while dried mushrooms contain $20 \%$ water. How many kg of dried mushrooms can be obtained from 20 kg of fresh mushrooms? | # Answer: 5 kg
Solution. The dry matter in fresh mushrooms is 4 kg, which constitutes $80 \%$ of the weight in dried mushrooms. By setting up the corresponding proportion, we will find the weight of the dried mushrooms. | 5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,936 |
2. (17 points) The medians drawn from vertices $A$ and $B$ of triangle $ABC$ are perpendicular to each other. Find the area of the square with side $AB$, if $BC=36, AC=48$.
# | # Answer: 720
Solution. Let $D$ be the midpoint of $B C$, $E$ be the midpoint of $A C$, and $M$ be the point of intersection of the medians. Let $M D=a$, $M E=b$. Then $A M=2 a$, $B M=2 b$. From the right triangles $B M D$ and $A M E$, we have $a^{2}+4 b^{2}=B D^{2}=18^{2}$ and $4 a^{2}+b^{2}=A E^{2}=24^{2}$. Adding t... | 720 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,937 |
3. (17 points) In a $3 \times 4$ grid, 4 crosses need to be placed such that each row and each column contains at least one cross. How many ways can this be done? | # Answer: 36
Solution. From the condition, it follows that in some row, two cells are marked (while in the others, only one each). This row can be chosen in 3 ways, and the two crosses in it can be placed in $4 \cdot 3 / 2=6$ ways. The remaining two crosses can be chosen in 2 ways. By the multiplication rule, the tota... | 36 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 3,938 |
4. (20 points) A ball was thrown from the surface of the Earth at an angle of $30^{\circ}$ with a speed of $v_{0}=20 \mathrm{M} / \mathrm{c}$. How long will it take for the velocity vector of the ball to turn by an angle of $60^{\circ}$? Neglect air resistance. The acceleration due to gravity is $g=10 \mathrm{M} / \mat... | Answer: $t=2 s$
Solution. In fact, the problem requires finding the total flight time of the ball. The $y$-coordinate of the ball changes according to the law: $y=v_{0 y} t-\frac{g t^{2}}{2}=v_{0} \sin 30^{\circ} t-\frac{g t^{2}}{2}=10 \cdot t-5 t^{2}=0$
From this, we obtain that: $t=2 s$. | 2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,939 |
5. (15 points) A massive vertical plate is fixed on a car moving at a speed of $5 \mathrm{M} / \mathrm{c}$. A ball is flying towards it at a speed of $6 \mathrm{m} / \mathrm{s}$ relative to the Earth. Determine the speed of the ball relative to the Earth after a perfectly elastic normal collision. | Answer: $16 \mathrm{~m} / \mathrm{c}$
Solution. We will associate the reference frame with the slab. Relative to this reference frame, the ball is flying at a speed of $5 \mathrm{M} / c+6 \mathrm{M} / c=11 \mathrm{M} / c$. After a perfectly elastic collision, the ball's speed relative to the slab is also $11 \mathrm{M... | 16\mathrm{~}/\mathrm{} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,940 |
1. In a row, the numbers $1,2,3, \ldots, 2014,2015$ are written. We will call a number from this row good if, after its removal, the sum of all the remaining 2014 numbers is divisible by 2016. Find all the good numbers. | Answer: The only good number is 1008.
Solution. The remainder of the division of the sum of all 2015 numbers by 2016 is 1008:
$(1+2015)+(2+2014)+\ldots+(1007+1009)+1008=2016 \cdot 1007+1008$.
Therefore, only 1008 can be crossed out.
Scoring. 12 points for a correct solution. If it is shown that 1008 is a good numbe... | 1008 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,942 |
2. In a company of 39 people, each is either a knight (always tells the truth) or a liar (always lies). They took turns making the following statements:
- "The number of knights in our company is a divisor of 1";
- "The number of knights in our company is a divisor of 2";
- "The number of knights in our company is a d... | Answer: 0 or 6.
Solution. Let $k$ be the number of knights in the company. If $k=0$ (everyone is a liar), then everyone lied, as they should. If $k>0$, the correct answers will be the answers with numbers $k, 2k, 3k, \ldots, mk$, where $mk \leqslant 39$. The total number of correct answers will be $m$. If $k=1,2,3,4,5... | 0or6 | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 3,943 |
3. Can 5 lines be drawn on a plane such that no three of them pass through the same point, and there are a) exactly $11;$ b) exactly 9 intersection points? | Answer: a) no; b) yes.
Solution. The maximum number of intersection points will be obtained if there are no two parallel lines among the lines and no three lines pass through the same point. In this case, there will be 4 intersection points on each of the 5 lines, and the total number of intersection points is $\frac{... | )no;b)yes | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,944 |
# Problem №2 (10 points)
There are two cubes. The mass of the second is $25 \%$ less than the mass of the first, and the edge length of the second cube is $25 \%$ greater than that of the first. By what percentage does the density of the second cube differ from the density of the first?
# | # Solution:
Volume of the second cube
$V=a^{3}=(1.25 a)^{3}$
(4 points)
And its density:
$\rho=\frac{m}{V}=\frac{0.75 m}{(1.25 a)^{3}}=0.384 \frac{m}{a^{3}}$.
(4 points)
That is, the density of the second cube is less than that of the first by $61.6 \%$
(2 points)
# | 61.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,946 |
# Problem №3 (15 points)
Raindrops fall vertically at a speed of 2 m/s. The mass of one drop is 5 mg. There are 400 drops in one cubic meter of air. How long will it take to completely fill a cylindrical vessel with a height of 20 cm and a base area of 10 cm $^{2}$ with water? The density of water $\rho_{B}=1000 \kapp... | # Solution:
In one cubic meter of air, there is $400 \cdot 5 \cdot 10^{-6}=0.002$ kg of water.
That is, 0.004 kg of water falls on one square meter of surface in one second (3 points).
Therefore, on 10 cm$^2$, 0.004 kg of water falls in one second, which is $4 \cdot 10^{-6}$ kg of water.
A fully filled container wi... | 5\cdot10^{4} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,947 |
# Problem №4 (15 points)
A train entered a railway bridge 1400 m long at a speed of 54 km/h. A person is walking through the train in the direction opposite to the train's movement at a speed of 3.6 km/h. The length of one carriage is $23 m$. In which carriage will the person be when the train leaves the bridge? At th... | # Solution:
Convert the initial data to the SI system:
$54 \kappa m / h = 15 \mu / s$
$3.6 \kappa m / h = 1 m / s$
The speed of the person relative to the ground:
$v_{\varphi} = 15 - 1 = 14 m / s$
(4 points)
This means the person will be on the bridge for:
$t = \frac{1400}{14} = 100 s$
During this time, he will... | 5or6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,948 |
1. The older brother has $25 \%$ more money than the younger brother. What portion of his money should the older brother give to the younger brother so that they have an equal amount of money? | Answer: $1 / 10$ or $10 \%$.
Solution. Let the younger one have $x$ rubles. Then the older one has $1.25 x$ rubles. To make the amount of money equal, the older one should give the younger one $0.125 x$ rubles, which constitutes $10 \%$ of his money.
Evaluation. 10 points for the correct solution. If the solution sta... | \frac{1}{10}or10 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,949 |
2. Petya wrote a natural number on the board, and Vasya erased the first two digits of it. As a result, the number decreased by 149 times. What could Petya's number be, given that it is odd? | Answer: 745 or 3725.
Solution. Petya's number can be represented as $10^{k} a+b$, where $10 \leqslant a \leqslant 99, b<10^{k}$. According to the condition, $10^{k} a+b=149 b$. From this, $10^{k} a=37 \cdot 4 b$. Since $10^{k}$ and 37 are coprime numbers, the number $a$ must be divisible by 37. Among two-digit numbers... | 745or3725 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,950 |
# Problem №1 (10 points)
Raindrops fall in windless weather at a speed of $v=2 m / s$. It is known that the rear window of a car is inclined at an angle $\alpha=60^{\circ}$ to the horizontal. At what speed $u$ should the car travel along a horizontal flat road so that its rear window remains dry
# | # Solution:
To keep the car's glass dry during movement, the drops relative to the glass must move parallel
We get that:
$$
u=\frac{v}{\operatorname{tg} 60^{\circ}}=\frac{2}{\sqrt{3}} \app... | 15\mathrm{~} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,951 |
1. We have 3 kg of a copper-tin alloy, in which $40\%$ is copper, and 7 kg of another copper-tin alloy, in which $30\%$ is copper. What masses of these alloys should be taken to obtain 8 kg of an alloy containing $p\%$ copper after melting? Find all $p$ for which the problem has a solution. | Answer: $0.8 p-24$ kg; $32-0.8 p$ kg; $31.25 \leqslant p \leqslant 33.75$.
Solution. If the first alloy is taken $x$ kg, then the second $(8-x)$ kg. The conditions of the problem impose restrictions on the possible values of $x$:
$$
\left\{\begin{array}{l}
0 \leqslant x \leqslant 3 ; \\
0 \leqslant 8-x \leqslant 7
\e... | 31.25\leqslantp\leqslant33.75 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,953 |
3. Nils has a goose farm. Nils calculated that if he sells 75 geese, the feed will last 20 days longer than if he doesn't sell any. If he buys an additional 100 geese, the feed will run out 15 days earlier than if he doesn't make such a purchase. How many geese does Nils have? | Answer: 300.
Solution. Let $A$ be the total amount of feed (in kg), $x$ be the amount of feed per goose per day (in kg), $n$ be the number of geese, and $k$ be the number of days the feed will last. Then
$$
\begin{gathered}
A=k x n=(k+20) x(n-75)=(k-15) x(n+100) \\
k n=(k+20)(n-75)=(k-15)(n+100)
\end{gathered}
$$
So... | 300 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,955 |
4. Can the numbers $x^{2}+2 y$ and $y^{2}+2 x$, where $x$ and $y$ are natural numbers, simultaneously be squares of integers? | Answer: No.
Solution. The smallest square greater than $x^{2}$ is $(x+1)^{2}$. Therefore, if the answer to the question is positive, $x^{2}+2 y \geqslant(x+1)^{2}$, from which $2 y \geqslant 2 x+1$. Similarly, we get $2 x \geqslant 2 y+1$. It is clear that the two inequalities contradict each other.
Evaluation. 14 po... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,956 |
1. There are 4 kg of a copper-tin alloy, in which $40\%$ is copper, and 6 kg of another copper-tin alloy, in which $30\%$ is copper. What masses of these alloys need to be taken so that after melting, 8 kg of an alloy containing $p\%$ copper is obtained? Find all $p$ for which the problem has a solution. | Answer: $0.8 p-24$ kg; $32-0.8 p$ kg; $32.5 \leqslant p \leqslant 35$.
Solution. If the first alloy is taken $x$ kg, then the second one is $(8-x)$ kg. The conditions of the problem impose restrictions on the possible values of $x$:
$$
\left\{\begin{array}{l}
0 \leqslant x \leqslant 4 ; \\
0 \leqslant 8-x \leqslant 6... | 32.5\leqslantp\leqslant35 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,957 |
3. Nils has a goose farm. Nils calculated that if he sells 50 geese, the feed will last 20 days longer than if he doesn't sell any. If he buys an additional 100 geese, the feed will run out 10 days earlier than if he doesn't make such a purchase. How many geese does Nils have? | Answer: 300.
Solution. Let $A$ be the total amount of feed (in kg), $x$ be the amount of feed per goose per day (in kg), $n$ be the number of geese, and $k$ be the number of days the feed will last. Then
$$
\begin{gathered}
A=k x n=(k+20) x(n-50)=(k-10) x(n+100) \\
k n=(k+20)(n-50)=(k-10)(n+100)
\end{gathered}
$$
So... | 300 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,959 |
4. Can the numbers $x^{2}+y$ and $y^{2}+x$, where $x$ and $y$ are natural numbers, simultaneously be squares of integers? | Answer: No.
Solution. The smallest square greater than $x^{2}$ is $(x+1)^{2}$. Therefore, if the answer to the question is positive, $x^{2}+y \geqslant(x+1)^{2}$, from which $y \geqslant 2 x+1$. Similarly, we get $x \geqslant 2 y+1$. It is clear that the two obtained inequalities contradict each other.
Evaluation. 14... | proof | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 3,960 |
2. Let
$$
\sqrt{49-a^{2}}-\sqrt{25-a^{2}}=3
$$
Calculate the value of the expression
$$
\sqrt{49-a^{2}}+\sqrt{25-a^{2}} .
$$ | Answer: 8.
Solution. Let
$$
\sqrt{49-a^{2}}+\sqrt{25-a^{2}}=x
$$
Multiplying this equality by the original one, we get $24=3x$.
Evaluation. Full solution: 11 points. | 8 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 3,961 |
3. Let $D$ be the midpoint of the hypotenuse $B C$ of the right triangle $A B C$. On the leg $A C$, a point $M$ is chosen such that $\angle A M B = \angle C M D$. Find the ratio $\frac{A M}{M C}$. | Answer: $1: 2$.
Solution. Let point $E$ be the intersection of rays $B A$ and $D M$.
Then angles $A M E$ and $C M D$ are equal as vertical angles. Therefore, angles $A M B$ and $A M E$ are ... | 1:2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 3,962 |
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