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Task 3. (15 points) The bases $AB$ and $CD$ of trapezoid $ABCD$ are equal to 101 and 20, respectively, and its diagonals are perpendicular to each other. Find the scalar product of vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$. | Solution.
Fig. 1
Let the point of intersection of the diagonals be $O$ (Fig. 1).
Consider the vectors $\overrightarrow{A O}=\bar{a}$ and $\overrightarrow{B O}=\bar{b}$.
From the similari... | 2020 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,123 |
Task 4. (20 points) In the center of a circular field stands a geologists' cabin. From it, 6 straight roads extend, dividing the field into 6 equal sectors. Two geologists set out on a journey from their cabin at a speed of 4 km/h along a road each arbitrarily chooses. Determine the probability that the distance betwee... | # Solution.
Let's find the distance between the geologists after 1 hour if they are walking on adjacent roads (Fig. 2).
$60^{\circ}$
Fig. 2
Since the triangle is equilateral, $x=4$, whic... | 0.5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,124 |
Task 5. (20 points) Three compressor stations are located not on the same straight line, but are connected by straight roads. The distance from the first station to the third via the second is twice the direct route between them; the distance from the first station to the second via the third is $a$ km longer than the ... | # Solution.
Let $x$ be the distance between the first and second compressor stations, $y$ the distance between the second and third, and $z$ the distance between the first and third (Fig. 4).
 A regular triangular prism $A B C A_{1} B_{1} C_{1}$ is inscribed in a sphere with the base $A B C$ and lateral edges $A A_{1}, B B_{1}, C C_{1}$. The segment $C_{1} D$ is the diameter of this sphere, and point $K$ is the midpoint of edge $C C_{1}$. Find the volume of the prism if $D K=2 \sqrt{6}... | # Solution.
The planes of the bases $ABC$ and $A_1B_1C_1$ of the prism intersect the sphere along the circumcircles of the equilateral triangles $ABC$ and $A_1B_1C_1$, with their centers at points $O$ and $O_1$ respectively.
It is easy to show that the midpoint $M$ of the segment $OO_1$ is the center of the sphere (F... | notfound | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,126 |
8. Finding the minimum force acting on the buoy from the current, knowing the horizontal component of the tension force, is not difficult.
Consider the forces acting on the buoy along the horizontal axis: on one side, there is the force from the current, and on the other side, the horizontal component of the tension f... | Answer: The minimum value of the force acting on the buoy from the current, capable of moving the buoy from its position, is $400 \mathrm{H}$.
## Grading. Maximum 12 points. | 400\mathrm{H} | Other | math-word-problem | Yes | Yes | olympiads | false | 7,128 |
7. If we now take into account that after reflections in the mirrors, the ray EC' has entered the eyepiece, that is, the direction of the ray after two deflections coincides with the direction of DA, it follows that the total deflection of the ray EC' is equal to the angle between the rays EC' and DA
$$
\mathrm{psi}_{... | Answer: There is no need to rotate mirror C relative to point C. The angle of rotation of mirror C relative to the platform is $0^{\circ}$.
## Solution Method 2. | 0 | Geometry | proof | Yes | Yes | olympiads | false | 7,133 |
5. (4 points) An electrical circuit consists of a resistor and a capacitor with capacitance C connected in series. A galvanic cell with electromotive force (emf) $\varepsilon$ and negligible internal resistance is connected to the ends of the circuit. Determine the amount of heat released in the resistor during the cha... | Answer: $Q=\frac{C \varepsilon^{2}}{2}$.
## Grading Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for the charge that has passed through the circuit is written | $\mathbf{1}$ |
| The expressio... | \frac{C\varepsilon^{2}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,136 |
4. (6 points) In an ideal gas, a thermodynamic cycle consisting of two isochoric and two adiabatic processes is carried out. The maximum \( T_{\max} = 900 \, \text{K} \) and minimum \( T_{\min} = 350 \, \text{K} \) absolute temperatures of the gas in the cycle are known, as well as the efficiency of the cycle \( \eta =... | Answer: $k=\frac{T_{\max }}{T_{\text {min }}}(1-\eta) \approx 1.54$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for work in the cycle is written | $\mathbf{1}$ |
| The express... | 1.54 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,137 |
5. (4 points) An electrical circuit consists of a resistor with resistance $R$ and a capacitor with capacitance $C$ connected in series. A galvanic cell with electromotive force (emf) $\mathcal{E}$ and internal resistance $r$ is connected to the ends of the circuit. Determine the amount of heat released in the resistor... | Answer: $Q_{R}=\frac{C \varepsilon^{2} R}{2(R+r)}$.
## Grading Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for the charge that has passed through the circuit is written | $\mathbf{1}$ |
| Th... | Q_{R}=\frac{C\varepsilon^{2}R}{2(R+r)} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,139 |
3. (4 points) In an ideal monatomic gas, a process occurs where the pressure is directly proportional to the volume. Determine the work done by the gas if the change in internal energy $\Delta U$ of the gas in this process is known.
## Possible solution.
According to the first law of thermodynamics, the amount of hea... | Answer: $A=\frac{\Delta U}{3}$.
## Grading Criteria
| Completion | Score |
| :---: | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | 0 |
| The expression for the first law of thermodynamics is written | 1 |
| The expression for the work of the gas is written | 1 |
| T... | \frac{\DeltaU}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,141 |
4. (6 points) In an ideal gas, a thermodynamic cycle consisting of two isochoric and two adiabatic processes is carried out. The ratio of the initial and final absolute temperatures in the isochoric cooling process is \( k = 1.5 \). Determine the efficiency of this cycle, given that the efficiency of the Carnot cycle w... | Answer: $\eta=1-k\left(1-\eta_{C}\right)=0.25=25 \%$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| Participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| Expression for work in the cycle is written | $\mathbf{1}$ |
| Expression for the amount of heat ... | 25 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,142 |
6. (4 points) Behind a thin converging lens with a focal length $F$ and diameter $D$, a flat screen is placed perpendicular to its optical axis at its focus. In front of the lens on the main optical axis, at a distance $d > F$ from the lens, a point light source is placed. Determine the diameter of the light spot on th... | Answer: $D^{\prime}=\frac{F D}{d}$.
## Grading Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | ... | D^{\}=\frac{FD}{} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,144 |
4. (4 points) When moving in the air, a ball is acted upon by a resistance force proportional to the square of its velocity. Immediately before the volleyball player's hit, the ball was flying horizontally at a speed of $V_{1}$. After the hit, the ball flew vertically upwards at a speed of $V_{2}$ with an acceleration ... | Answer: $a_{1}=\left(\frac{V_{1}}{V_{2}}\right)^{2}\left(a_{2}-g\right)$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for acceleration in horizontal flight is written | $\mathb... | a_{1}=(\frac{V_{1}}{V_{2}})^{2}(a_{2}-) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,146 |
3. (6 points) In a horizontal stationary cylindrical vessel closed by a piston of mass $M$, a gas is contained. The gas is heated, and during this process, the piston moves from rest with constant acceleration. The amount of heat supplied to the gas over the time interval $\tau$ is $Q$. Determine the acceleration of th... | Solution. Since the piston moves with uniform acceleration, the process is isobaric. According to the first law of thermodynamics,
$$
Q=p \Delta V+v c \Delta T
$$
Here $p$ - the pressure of the gas, $V$ - its volume, $v$ - the amount of substance of the gas.
According to the Clapeyron-Mendeleev equation,
$$
p \Delt... | \sqrt{\frac{2Q}{M\tau^{2}(1+\frac{}{R})}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,147 |
5. (6 points) Two identical conducting spheres are located at a large distance from each other and have positive charges $Q_{1}$ and $Q_{2}$. A neutral metal ball on a non-conductive suspension is brought close to the first sphere and touches it. Then the ball is brought close to the second sphere and touches it. After... | Solution of this equation:
$$
Q_{2}^{\prime}=\frac{Q_{2}}{2}-q_{2} \pm \sqrt{\frac{Q_{2}^{2}}{4}+Q_{1} q_{2}}
$$
Answer: $Q_{2}^{\prime}=\frac{Q_{2}}{2}-q_{2} \pm \sqrt{\frac{Q_{2}^{2}}{4}+Q_{1} q_{2}}$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or perf... | Q_{2}^{\}=\frac{Q_{2}}{2}-q_{2}\\sqrt{\frac{Q_{2}^{2}}{4}+Q_1q_{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,148 |
3. (6 points) In a horizontal stationary cylindrical vessel closed by a piston of mass $M$, a gas is located. The gas is heated for a time $\tau$, during which the piston moves from rest with a constant acceleration $a$. Find the average power of the heater. The internal energy of one mole of gas is $U=c T$. Neglect th... | Answer: $P=\frac{M a^{2} \tau}{2}\left(1+\frac{c}{R}\right)$.
## Evaluation Criteria
| Participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| :--- | :---: |
| The expression for the first law of thermodynamics is written | $\mathbf{1}$ |
| The Clapeyron-Mendeleev equation... | \frac{M^{2}\tau}{2}(1+\frac{}{R}) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,150 |
5. (6 points) Two identical air capacitors with capacitance $C$ each are charged to a voltage $U$. One of them is submerged in a dielectric liquid with permittivity $\varepsilon$, after which the capacitors are connected in parallel. Determine the amount of heat released upon connecting the capacitors.
Possible soluti... | Answer: $Q=\frac{C U^{2}(\varepsilon-1)^{2}}{2 \varepsilon(\varepsilon+1)}$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| Participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| Expression for the energy of the system before connection is written | $\m... | \frac{CU^{2}(\varepsilon-1)^{2}}{2\varepsilon(\varepsilon+1)} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,151 |
3. (6 points) In a vertical stationary cylindrical vessel closed by a piston of mass $M$, an ideal gas is located. The gas is heated, and the piston moves from rest with a constant acceleration $a$. During the heating time, the piston exits the vessel and then continues to move vertically upward in free space. The aver... | Possible solution. Since the piston moves with uniform acceleration, the process is isobaric. According to the first law of thermodynamics,
$$
Q=p \Delta V+v c_{\mu V} \Delta T
$$
Here $p$ - is the pressure of the gas, $V$ - is its volume, $v$ - is the amount of substance of the gas.
According to the Clapeyron-Mende... | \tau^{\}=\frac{2P}{\operatorname{Mag}(1+\frac{c_{\muV}}{R})} | Other | math-word-problem | Yes | Yes | olympiads | false | 7,152 |
5. (6 points) Two identical air capacitors are charged to a voltage $U$ each. One of them is submerged in a dielectric liquid with permittivity $\varepsilon$ while charged, after which the capacitors are connected in parallel. The amount of heat released upon connecting the capacitors is $Q$. Determine the capacitance ... | Answer: $C=\frac{2 \varepsilon(\varepsilon+1) Q}{U^{2}(\varepsilon-1)^{2}}$
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for the energy of the system before connection is writte... | \frac{2\varepsilon(\varepsilon+1)Q}{U^{2}(\varepsilon-1)^{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,153 |
8. Finally, let's calculate the change in resistance when the switch is closed:
$$
\Delta R=R_{p}-R_{3}=33[\text { Ohms] }-30[\text { Ohms] }=3[\text { Ohms] }
$$ | Answer: 3.0 Ohms
Criteria (maximum 10 points) | 3 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,163 |
Task 1. Calculate the consumption values and fill in the gaps in the table, given that the global fuel consumption across all types of primary energy was 557.10 exajoules in 2020. Calculations - rounded to two decimal places.
Table 1 - Fuel consumption by types of primary energy in individual countries of the world an... | Task 1. Using the conditions of the problem and the data from the table, we find how many exajoules the primary energy consumption in the USA, Germany, and Russia constituted. We subtract from this sum the consumption of all types of fuel except the one we are looking for, and in the remainder, we get the desired value... | 32.54,2.21,14.81 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,166 |
Task 2. By how many units can the city's fleet of natural gas vehicles be increased in 2022, assuming that the capacity of each of the old CNG stations in the city is equal to the capacity of the new station on Narodnaya Street, and that the city's fleet constitutes only $70 \%$ of all vehicles refueling at CNG station... | Task 2. There are a total of 15 stations: 4 new ones and 11 old ones.
The throughput capacity of the old stations is 11 x $200=2200$ vehicles per day, and for the new ones: $200+700=900$ vehicles per day. In total, the stations can refuel: $2200+900=3100$ vehicles per day.
Vehicles from the city fleet account for onl... | 1170 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,167 |
Task 1. (5 points) Find $\frac{a^{12}+4096}{64 a^{6}}$, if $\frac{a}{2}-\frac{2}{a}=5$.
# | # Solution.
$$
\begin{aligned}
& \frac{a^{12}+4096}{64 a^{6}}=\frac{a^{6}}{64}+\frac{64}{a^{6}}=\frac{a^{6}}{64}-2+\frac{64}{a^{6}}+2=\left(\frac{a^{3}}{8}-\frac{8}{a^{3}}\right)^{2}+2= \\
& =\left(\frac{a^{3}}{8}-3 \cdot \frac{a}{2}+3 \cdot \frac{2}{a}-\frac{8}{a^{3}}+3\left(\frac{a}{2}-\frac{2}{a}\right)\right)^{2}+... | 19602 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,169 |
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with side $\sqrt{2+\sqrt{2}}$ cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $45^{\circ}$, and it is known that $\sin 22.5^{\circ}=\frac{\sqr... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. The diagonal of the square $A B C D A C=\sqrt{4+2 \sqrt... | \sqrt{2}+\sqrt{2-\sqrt{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,170 |
Task 3. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}+25 y+19 z=-471 \\
y^{2}+23 x+21 z=-397 \\
z^{2}+21 x+21 y=-545
\end{array}\right.
$$ | # Solution.
Add the first equation to the other two and complete the square for each variable:
$$
\begin{aligned}
& x^{2}+y^{2}+z^{2}+44 x+46 y+40 z=-1413 \\
& x^{2}+44 x+y^{2}+46 y+z^{2}+40 z+1413=0 \\
& x^{2}+44 x+484+y^{2}+46 y+529+z^{2}+40 z+400=0 \\
& (x+22)^{2}+(y+23)^{2}+(z+20)^{2}=0
\end{aligned}
$$
Therefor... | (-22,-23,-20) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,171 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 5 radars around it, each with a coverage radius of $r=13$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | Solution.
To ensure radar coverage of a ring around the platform, the radars should be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
 A point is randomly thrown onto the segment [6; 11] and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-2 k-15\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{7-3 k}{k^{2}-2 k-15} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}-2 k-15} .\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{7-3 k}{k^{2}-2 k-1... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,173 |
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six iden... | # Solution.
When ten identical spheres are arranged in this way, the centers $A, B, C, D$ of four of them are located at the vertices of a regular tetrahedron, and the points of contact are located on the edges of this tetrahedron. Therefore, the edge of the tetrahedron is equal to four radii of these spheres, the rad... | 5(\sqrt{2}+1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,174 |
Task 1. (5 points) Find $\frac{a^{12}+729^{2}}{729 a^{6}}$, if $\frac{a}{3}-\frac{3}{a}=4$.
# | # Solution.
$$
\begin{aligned}
& \frac{a^{12}+729^{2}}{729 a^{6}}=\frac{a^{6}}{729}+\frac{729}{a^{6}}=\frac{a^{6}}{729}-2+\frac{729}{a^{6}}+2=\left(\frac{a^{3}}{27}-\frac{27}{a^{3}}\right)^{2}+2= \\
& =\left(\frac{a^{3}}{27}-3 \cdot \frac{a}{3}+3 \cdot \frac{3}{a}-\frac{27}{a^{3}}+3\left(\frac{a}{3}-\frac{3}{a}\right)... | 5778 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,175 |
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with side length $6+2 \sqrt{5}$ cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $36^{\circ}$, and it is known that $\sin 18^{\circ}=\frac{\sqr... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. The diagonal of the square $A B C D A C=\sqrt{2}(6+2 \s... | 2(2\sqrt{2}+\sqrt{5}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,176 |
Task 3. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}-23 y-25 z=-681 \\
y^{2}-21 x-21 z=-419 \\
z^{2}-19 x-21 y=-313
\end{array}\right.
$$ | # Solution.
Add the first equation to the other two and complete the square for each variable:
$$
\begin{aligned}
& x^{2}+y^{2}+z^{2}-40 x-44 y-46 z=-1413 \\
& x^{2}+y^{2}+z^{2}-40 x-44 y-46 z+1413=0 \\
& x^{2}-40 x+400+y^{2}-44 y+484+z^{2}-46 z+529=0 \\
& (x-20)^{2}+(y-22)^{2}+(z-23)^{2}=0
\end{aligned}
$$
Therefor... | (20;22;23) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,177 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 5 radars around it, each with a coverage radius of $r=25$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, it is necessary to place them at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
 Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the sphere in which the pyramid of balls is inscribed, if the radius of the sphere inscribed in the center of the pyramid of balls, touching six iden... | # Solution.
When ten identical spheres are arranged in this way, the centers $A, B, C, D$ of four of them are located at the vertices of a regular tetrahedron, and the points of contact are located on the edges of this tetrahedron. Therefore, the edge of the tetrahedron is equal to four radii of these spheres, the rad... | \sqrt{6}+1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,180 |
Task 1. (5 points) Find $\frac{a^{12}+729^{2}}{729 a^{6}}$, if $\frac{a}{3}-\frac{3}{a}=2$. | Solution.
$$
\begin{aligned}
& \frac{a^{12}+729^{2}}{729 a^{6}}=\frac{a^{6}}{729}+\frac{729}{a^{6}}=\frac{a^{6}}{729}-2+\frac{729}{a^{6}}+2=\left(\frac{a^{3}}{27}-\frac{27}{a^{3}}\right)^{2}+2= \\
& =\left(\frac{a^{3}}{27}-3 \cdot \frac{a}{3}+3 \cdot \frac{3}{a}-\frac{27}{a^{3}}+3\left(\frac{a}{3}-\frac{3}{a}\right)\r... | 198 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,181 |
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with side $2 \sqrt{3} \mathrm{~cm}$. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $30^{\circ}$, and it is known that $\sin 15^{\circ}=\frac{\sq... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. The diagonal of the square $A B C D A C=2 \sqrt{6}$. If... | 2 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,182 |
Task 3. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}-22 y-69 z+703=0 \\
y^{2}+23 x+23 z-1473=0 \\
z^{2}-63 x+66 y+2183=0
\end{array}\right.
$$ | # Solution.
Add the first equation to the other two and complete the square for each variable:
$$
\begin{aligned}
& x^{2}+y^{2}+z^{2}-40 x+44 y-46 z+1413=0 \\
& x^{2}-40 x+400+y^{2}+44 y+484+z^{2}-46 z+529=0 \\
& (x-20)^{2}+(y+22)^{2}+(z-23)^{2}=0
\end{aligned}
$$
Therefore, \( x=20, y=-22, z=23 \) is the only possi... | (20;-22;23) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,183 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 7 radars around it, each with a coverage radius of $r=41$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, it is necessary to place them at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
};\frac{1440\pi}{\operatorname{tg}(180/7)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,184 |
Problem 5. (20 points) A point is randomly thrown onto the segment $[8 ; 13]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-2 k-35\right) x^{2}+(3 k-9) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$$
\left\{\begin{array}{l}
x_{1}+x_{2}=\frac{9-3 k}{k^{2}-2 k-35} \\
x_{1} \cdot x_{2}=\frac{2}{k^{2}-2 k-35}
\end{array}\right.
$$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{9-3 k}{k^{2}-2... | 0.6 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,185 |
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the ball inscribed in the center of the pyramid, touching six identical balls, if the radius of the sphere in which the pyramid of balls is inscribed... | # Solution.
When ten identical spheres are arranged in this way, the centers $A, B, C, D$ of four of them are located at the vertices of a regular tetrahedron, and the points of contact are located on the edges of this tetrahedron. Therefore, the edge of the tetrahedron is equal to four radii of these spheres, the rad... | (\sqrt{2}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,186 |
Task 1. (5 points) Find $\frac{a^{12}+4096}{64 a^{6}}$, if $\frac{a}{2}-\frac{2}{a}=3$.
# | # Solution.
$$
\begin{aligned}
& \frac{a^{12}+4096}{64 a^{6}}=\frac{a^{6}}{64}+\frac{64}{a^{6}}=\frac{a^{6}}{64}-2+\frac{64}{a^{6}}+2=\left(\frac{a^{3}}{8}-\frac{8}{a^{3}}\right)^{2}+2= \\
& =\left(\frac{a^{3}}{8}-3 \cdot \frac{a}{2}+3 \cdot \frac{2}{a}-\frac{8}{a^{3}}+3\left(\frac{a}{2}-\frac{2}{a}\right)\right)^{2}+... | 1298 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,187 |
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with side $2-\sqrt{5-\sqrt{5}}$ cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $72^{\circ}$, and it is known that $\sin 36^{\circ}=\frac{\sqr... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. The diagonal of the square $A B C D \quad A C=\sqrt{2}(... | \sqrt{5-\sqrt{5}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,188 |
Task 3. (15 points) Solve the system of equations
$$
\left\{\begin{array}{l}
x^{2}-23 y+66 z+612=0 \\
y^{2}+62 x-20 z+296=0 \\
z^{2}-22 x+67 y+505=0
\end{array}\right.
$$ | # Solution.
Add the first equation to the other two and complete the square for each variable:
$$
\begin{aligned}
& x^{2}+y^{2}+z^{2}+40 x+44 y+46 z+1413=0 \\
& x^{2}+40 x+400+y^{2}+44 y+484+z^{2}+46 z+529=0 \\
& (x+20)^{2}+(y+22)^{2}+(z+23)^{2}=0
\end{aligned}
$$
Therefore, \( x=-20, y=-22, z=-23 \) is the only pos... | (-20,-22,-23) | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,189 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 7 radars around it, each with a coverage radius of $r=26$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, they need to be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
Point $O$ is the center of the oil platform, and points $A$ and $B$ are the locations of the radars. The circles repr... | \frac{24}{\sin(180/7)};\frac{960\pi}{\tan(180/7)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,190 |
Task 5. (20 points) A point is randomly thrown onto the segment [3; 8] and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$.
# | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{5-3 k}{k^{2}-2 k-3} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}-2 k-3}\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{5-3 k}{k^{2}-2 k-3} ; ... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,191 |
Problem 6. (30 points) Ten balls of the same radius are arranged in the form of a triangular pyramid such that each ball touches at least three others. Find the radius of the ball inscribed in the center of the pyramid, touching six identical balls, if the radius of the sphere in which the pyramid of balls is inscribed... | # Solution.
When ten identical spheres are arranged in this way, the centers $A, B, C, D$ of four of them are located at the vertices of a regular tetrahedron, and the points of contact are located on the edges of this tetrahedron. Therefore, the edge of the tetrahedron is equal to four radii of these spheres, the rad... | \sqrt{6}-1 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,192 |
Task 1. (5 points) Find $\frac{a^{12}-729}{27 a^{6}}$, if $\frac{a^{2}}{3}-\frac{3}{a^{2}}=4$.
# | # Solution.
$$
\begin{aligned}
& \frac{a^{12}-729}{27 a^{6}}=\left(\frac{a^{2}}{3}-\frac{3}{a^{2}}\right)\left(\frac{a^{4}}{9}+1+\frac{9}{a^{4}}\right)=\left(\frac{a^{2}}{3}-\frac{3}{a^{2}}\right)\left(\frac{a^{4}}{9}-2+\frac{9}{a^{4}}+3\right)= \\
& =\left(\frac{a^{2}}{3}-\frac{3}{a^{2}}\right)\left(\left(\frac{a^{2}... | 76 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,193 |
Task 2. (10 points) A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$, and the point of tangency cuts off a segment of length $6-2\sqrt{5}$ cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $36^{\circ}$, a... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. If radii of the circle are drawn to the points of tange... | (\sqrt{5}-1)(2\sqrt{2}-\sqrt{5}+1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,194 |
Task 3. (15 points) Solve the equation $4 x-x^{2}+\sqrt{\left(9-x^{2}\right)\left(-7+8 x-x^{2}\right)}=7$.
---
The translation maintains the original text's line breaks and format. | Solution.
Rewrite the equation as $\sqrt{\left(9-x^{2}\right)\left(-7+8 x-x^{2}\right)}=x^{2}-4 x+7$.
$$
\sqrt{(x-3)(x+3)(x-1)(x-7)}=x^{2}-4 x+7
$$
Rearrange the factors $(x-3)(x-1)$ and $(x+3)(x-7)$, we get
$$
\sqrt{\left(x^{2}-4 x+3\right)\left(x^{2}-4 x-21\right)}=x^{2}-4 x+7
$$
Let $x^{2}-4 x+3=t$, then $x^{2}... | \frac{4\\sqrt{2}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,195 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 8 radars around it, each with a coverage radius of $r=17$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, they need to be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
 A point is randomly thrown onto the segment $[6 ; 10]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-3 k-10\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{8-3 k}{k^{2}-3 k-10} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}-3 k-10}\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{8-3 k}{k^{2}-3 k-10}... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,197 |
Task 1. (5 points) Find $\frac{a^{12}-4096}{64 a^{6}}$, if $\frac{a^{2}}{4}-\frac{4}{a^{2}}=3$. | # Solution.
$$
\begin{aligned}
& \frac{a^{12}-4096}{64 a^{6}}=\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right)\left(\frac{a^{4}}{16}+1+\frac{16}{a^{4}}\right)=\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right)\left(\frac{a^{4}}{16}-2+\frac{16}{a^{4}}+3\right)= \\
& =\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right)\left(\left(\frac{... | 36 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,199 |
Task 2. (10 points) A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$, and the point of tangency cuts off a segment of length $\sqrt{2}+\sqrt{2-\sqrt{2}}$ cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is ... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. If radii of the circle are drawn to the points of tange... | \sqrt{2+\sqrt{2}} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,200 |
Task 3. (15 points) Solve the equation $5 x-x^{2}+\sqrt{\left(16-x^{2}\right)}\left(-9+10 x-x^{2}\right)=9$.
# | # Solution.
Rewrite the equation as $\sqrt{\left(16-x^{2}\right)}\left(-9+10 x-x^{2}\right)=x^{2}-5 x+9$.
$$
\sqrt{(x-4)(x+4)(x-1)(x-9)}=x^{2}-5 x+9
$$
Rearrange the factors $(x-4)(x-1)$ and $(x+4)(x-9)$, we get
$$
\sqrt{\left(x^{2}-5 x+4\right)\left(x^{2}-4 x-36\right)}=x^{2}-5 x+9
$$
Let $x^{2}-5 x+4=t$, then $x... | \frac{5\\sqrt{7}}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,201 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 8 radars around it, each with a coverage radius of $r=15$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, they need to be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
Point $O$ is the center of the oil platform, and points $A$ and $B$ are the locations of the radars. The circles repr... | \frac{12}{\sin22.5};\frac{432\pi}{\tan22.5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,202 |
Problem 5. (20 points) A point is randomly thrown onto the segment $[5 ; 7]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-3 k-4\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{7-3 k}{k^{2}-3 k-4} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}-3 k-4}\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{7-3 k}{k^{2}-3 k-4} ; ... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,203 |
Task 6. (30 points) When manufacturing a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=8 \\
x y+y z+x z=14
\end{array}\right.
$$
Find the length of the cable. | # Solution.
Transform the second equation of the system: square the first equation and subtract twice the second equation from it, we get:
$\left\{\begin{array}{l}x+y+z=8, \\ (x+y+z)^{2}-2(x y+y z+x z)=8^{2}-2 \cdot 14 .\end{array}\right.$
$\left\{\begin{array}{l}x+y+z=8 \\ x^{2}+y^{2}+z^{2}=36 .\end{array}\right.$
... | 4\pi\sqrt{\frac{11}{3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,204 |
Task 1. (5 points) Find $\frac{a^{12}-729}{27 a^{6}}$, if $\frac{a^{2}}{3}-\frac{3}{a^{2}}=6$. | Solution.
$$
\begin{aligned}
& \frac{a^{12}-729}{27 a^{6}}=\left(\frac{a^{2}}{3}-\frac{3}{a^{2}}\right)\left(\frac{a^{4}}{9}+1+\frac{9}{a^{4}}\right)=\left(\frac{a^{2}}{3}-\frac{3}{a^{2}}\right)\left(\frac{a^{4}}{9}-2+\frac{9}{a^{4}}+3\right)= \\
& =\left(\frac{a^{2}}{3}-\frac{3}{a^{2}}\right)\left(\left(\frac{a^{2}}{... | 234 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,205 |
Task 2. (10 points) A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$, and the point of tangency cuts off a segment of length 2 cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $30^{\circ}$, and it is kno... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. If radii of the circle are drawn to the points of tange... | 2\sqrt{3} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,206 |
Task 3. (15 points) Solve the equation $3 x-x^{2}+\sqrt{\left(9-x^{2}\right)\left(6 x-x^{2}\right)}=0$. | Solution.
Rewrite the equation as $\sqrt{\left(9-x^{2}\right)\left(6 x-x^{2}\right)}=x^{2}-3 x$,
$$
\sqrt{(x-3)(x+3) x(x-6)}=x^{2}-3 x
$$
Rearrange the factors $x(x-3)$ and $(x+3)(x-6)$, we get
$$
\begin{gathered}
\sqrt{x(x-3)\left(x^{2}-3 x-18\right)}=x(x-3) \\
\left\{\begin{array}{c}
x(x-3) \geq 0 \\
x(x-3)\left(... | {0;3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,207 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 9 radars around it, each with a coverage radius of $r=61$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, it is necessary to place them at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2)
 A point is randomly thrown onto the segment $[11 ; 18]$ and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}+2 k-99\right) x^{2}+(3 k-7) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{7-3 k}{k^{2}+2 k-99} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}+2 k-99}\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals.
$\left\{\begin{array}{l}3 x_{2}=\frac{7-3 k}{k^{2}+2 k-99... | \frac{2}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,209 |
Task 6. (30 points) When manufacturing a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$
Find the length of the cable. | # Solution.
Transform the second equation of the system: square the first equation and subtract the second equation from it twice, we get:
$\left\{\begin{array}{l}x+y+z=10, \\ (x+y+z)^{2}-2(x y+y z+x z)=10^{2}-2 \cdot 18 .\end{array}\right.$
$\left\{\begin{array}{l}x+y+z=10 \\ x^{2}+y^{2}+z^{2}=64 .\end{array}\right... | 4\pi\sqrt{\frac{23}{3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,210 |
Task 1. (5 points) Find $\frac{a^{12}-4096}{64 a^{6}}$, if $\frac{a^{2}}{4}-\frac{4}{a^{2}}=5$. | # Solution.
$$
\begin{aligned}
& \frac{a^{12}-4096}{64 a^{6}}=\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right)\left(\frac{a^{4}}{16}+1+\frac{16}{a^{4}}\right)=\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right)\left(\frac{a^{4}}{16}-2+\frac{16}{a^{4}}+3\right)= \\
& =\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right)\left(\left(\frac{... | 140 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,211 |
Task 2. (10 points) A circle touches the extensions of two sides $AB$ and $AD$ of square $ABCD$, and the point of tangency cuts off a segment of length $2+\sqrt{5-\sqrt{5}}$ cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $72^{\c... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. If radii of the circle are drawn to the points of tange... | \frac{\sqrt{\sqrt{5}-1}\cdot\sqrt[4]{125}}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,212 |
Problem 3. (15 points) Solve the equation $6 x-x^{2}+\sqrt{\left(25-x^{2}\right)\left(-11+12 x-x^{2}\right)}=11$.
# | # Solution.
Rewrite the equation as $\sqrt{\left(25-x^{2}\right)\left(-11+12 x-x^{2}\right)}=x^{2}-6 x+11$,
$$
\sqrt{(x-5)(x+5)(x-1)(x-11)}=x^{2}-6 x+11 .
$$
Rearrange the factors $(x-5)(x-1)$ and $(x+5)(x-11)$, we get
$$
\sqrt{\left(x^{2}-6 x+5\right)\left(x^{2}-6 x-55\right)}=x^{2}-6 x+11
$$
Let $x^{2}-6 x+5=t$,... | {\frac{6\\sqrt{14}}{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,213 |
Task 4. (20 points) To protect an oil platform located at sea, it is necessary to distribute 9 radars around it, each with a coverage radius of $r=37$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring around the platform with a width... | # Solution.
To ensure radar coverage of a ring around the platform, they need to be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
Point $O$ is the center of the oil platform, and points $A$ and $B$ are the locations of the radars. The circles repr... | \frac{35}{\sin20};\frac{1680\pi}{\tan20} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,214 |
Problem 5. (20 points) A point is randomly thrown onto the segment [12; 17] and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}+k-90\right) x^{2}+(3 k-8) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{8-3 k}{k^{2}+k-90} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}+k-90}\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{8-3 k}{k^{2}+k-90} ; \\ ... | \frac{2}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,215 |
Task 6. (30 points) When manufacturing a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=8 \\
x y+y z+x z=-18 .
\end{array}\right.
$$
Find the length of the cable. | # Solution.
Transform the second equation of the system: square the first equation and subtract the second equation from it twice, we get:
$$
\begin{aligned}
& \left\{\begin{array}{l}
x+y+z=8, \\
(x+y+z)^{2}-2(x y+y z+x z)=8^{2}-2 \cdot(-18) .
\end{array}\right. \\
& \left\{\begin{array}{l}
x+y+z=8 \\
x^{2}+y^{2}+z^{... | 4\pi\sqrt{\frac{59}{3}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,216 |
4. (6 points) In an ideal gas, a thermodynamic cycle consisting of two isochoric and two adiabatic processes is carried out. The maximum \( T_{\max} = 900 \, \text{K} \) and minimum \( T_{\min} = 350 \, \text{K} \) absolute temperatures of the gas in the cycle are known, as well as the efficiency of the cycle \( \eta =... | Answer: $k=\frac{T_{\max }}{T_{\text {min }}}(1-\eta) \approx 1.54$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| The participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| The expression for work in the cycle is written | $\mathbf{1}$ |
| The express... | 1.54 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,220 |
4. (6 points) In an ideal gas, a thermodynamic cycle consisting of two isochoric and two adiabatic processes is carried out. The ratio of the initial and final absolute temperatures in the isochoric cooling process is \( k = 1.5 \). Determine the efficiency of this cycle, given that the efficiency of the Carnot cycle w... | Answer: $\eta=1-k\left(1-\eta_{C}\right)=0.25=25 \%$.
## Evaluation Criteria
| Performance | Score |
| :--- | :---: |
| Participant did not start the task or performed it incorrectly from the beginning | $\mathbf{0}$ |
| Expression for work in the cycle is written | $\mathbf{1}$ |
| Expression for the amount of heat ... | 25 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,223 |
3. Let's determine the share of the Chinese yuan in the currency structure of the NWF funds as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds allocated to the Chinese yuan as of 01.12.2022:
$CNY_{22}=1388.01-41.89-2.77-478.48-554.91-0.24=309.72$ (billion rubles)... | Answer: increased by 4.5 pp.
## Variant 2
# | 4.5 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,225 |
3. Let's determine the share of the Japanese yen in the currency structure of the NWF funds as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds allocated to Japanese yen as of 01.12.2022:
$J P Y_{22}=1388.01-41.89-2.77-309.72-554.91-0.24=478.48$ (billion rubles)
... | Answer: decreased by 12.6 pp.
## Variant 3
# | decreased\\12.6\pp | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,226 |
3. Let's determine the share of gold in the currency structure of the National Wealth Fund (NWF) funds as of 01.12.2022 using one of the following methods:
First method:
a) Find the total amount of NWF funds allocated to gold as of 01.12.2022:
$G O L D_{22}=1388,01-41,89-2,77-478,48-309,72-0,24=554,91$ (billion rubl... | Answer: increased by 8.2 pp.
## Option 4
# | 8.2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,227 |
3. Let's determine the share of the US dollar in the currency structure of the NWF funds as of 04.01.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds placed in US dollars as of 04.01.2021:
$U S D_{04}=794,26-34,72-8,55-600,3-110,54-0,31=39,84$ (billion rubles)
b) Determin... | Answer: decreased by 44 pp.
##
Variant 5
# | decreased\\44\pp | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,228 |
3. Let's determine the share of the euro in the currency structure of the NWF funds as of 04.01.2021 using one of the following methods: The first method:
a) Find the total amount of NWF funds placed in euros as of 04.01.2021:
$U S D_{04}=794,26-39,84-8,55-600,3-110,54-0,31=34,72$ (billion rubles)
b) Determine the s... | Answer: decreased by 38 p. p.
##
Variant 6
# | decreased\\38\p.p | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,229 |
3. Let's determine the share of British pounds in the currency structure of the NWF funds as of 04.01.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds allocated to British pounds as of 04.01.2021:
$GBR_{04}=794.26-39.84-34.72-600.3-110.54-0.31=8.55$ (billion rubles)
b) Det... | Answer: decreased by 7 pp.
## Variant 7
# | decreased\\7\pp | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,230 |
3. Let's determine the share of the Chinese yuan in the currency structure of the NWF funds as of 01.07.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds allocated to the Chinese yuan as of 01.07.2021:
$C N Y_{22}=1213.76-3.36-38.4-4.25-600.3-340.56-0.29=226.6$ (billion rubl... | Answer: increased by 1.2 pp.
# | 1.2 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,231 |
3. Let's determine the share of the Japanese yen in the currency structure of the NWF funds as of 01.07.2021 using one of the following methods:
First method:
a) Find the total amount of NWF funds allocated to Japanese yen as of 01.07.2021:
$J P Y_{22}=1213.76-3.36-38.4-4.25-226.6-340.56-0.29=600.3$ (billion rubles)... | Answer: decreased by 23.5 pp. | decreased\\23.5\pp | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,232 |
5. Let's determine the mass fraction of potassium sulfate in the initial solution:
$$
\omega\left(K_{2} \mathrm{SO}_{4}\right)=\frac{m\left(K_{2} \mathrm{SO}_{4}\right) \cdot 100 \%}{m_{\mathrm{p}-\mathrm{pa}}\left(K_{2} \mathrm{SO}_{4}\right)}=\frac{2.61 \cdot 100 \%}{160}=1.63 \%
$$
Determining the mass fraction of... | Answer: $1.63 \%$
## TOTAL 10 points
## TASK № 2
Iron was burned in bromine. The resulting salt was added to a solution of potassium carbonate, whereupon a brown precipitate formed. This precipitate was filtered and calcined. The resulting substance was dissolved in hydriodic acid. Write the equations for the four d... | 1.63 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,233 |
1. First, let's determine the amount of gas in each of the vessels after equilibrium is established
$$
\begin{aligned}
& \left\{\begin{array}{l}
v_{1}+v_{2}=4 \\
v_{2}-v_{1}=0.62
\end{array}\right. \\
& v_{2}=v_{1}+0.62 \\
& 2 v_{1}+0.62=4 \\
& v_{1}=1.69 \text { mol, } \\
& v_{2}=4-1.69=2.31 \text { mol. }
\end{align... | Answer: $\mathrm{T}_{2}=272.9 \mathrm{~K}$
## TOTAL 4 points
$$
\begin{gathered}
\frac{v_{1} R T_{1}}{V}=\frac{v_{2} R T_{2}}{V} \\
v_{1} T_{1}=v_{2} T_{2} \\
T_{2}=v_{1} T_{1} / v_{2}=1.69 \cdot 373 / 2.31=272.9 \mathrm{~K}
\end{gathered}
$$
Determination of the temperature of the second thermostat - 2 points
## T... | 272.9\mathrm{~K} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,236 |
18. Determine the amount of substance of sodium carbonate
According to the chemical reaction equation:
$$
\mathrm{n}\left(\mathrm{CO}_{2}\right)=\mathrm{n}\left(\mathrm{Na}_{2} \mathrm{CO}_{3}\right)=0.125 \text { mol }
$$
Determination of the amount of substance - 2 points
19. Calculate the mass of sodium carbonat... | Answer: $10 \%$
## TOTAL 10 points
## TASK № 2
Aluminum oxide was melted with potash. The resulting product was dissolved in hydrochloric acid and treated with excess ammonia water. The precipitate that formed was dissolved in excess sodium hydroxide solution. Write the equations for the four reactions described.
#... | 10 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,240 |
13. $\mathrm{Al}_{2} \mathrm{O}_{3}+\mathrm{K}_{2} \mathrm{CO}_{3}=2 \mathrm{KAlO}_{2}+\mathrm{CO}_{2} \uparrow$
Formulation of the reaction equation and balancing coefficients - 4 points 14. $\mathrm{KAlO}_{2}+\mathrm{HCl}=\mathrm{AlCl}_{3}+2 \mathrm{H}_{2} \mathrm{O}+2 \mathrm{KCl}$
Formulation of the reaction equa... | # Solution:
17. Let's write the general equation for the reaction of alcohols with sodium:
$2 \mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}+1} \mathrm{OH}+2 \mathrm{Na}=2 \mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}+1} \mathrm{ONa}+\mathrm{H}_{2} \uparrow$
Writing the reaction equation and balancing the coeff... | \mathrm{C}_{9}\mathrm{H}_{19}\mathrm{OH} | Other | math-word-problem | Yes | Yes | olympiads | false | 7,241 |
24. Let's determine the amount of substance of the released gas
According to the chemical reaction equation:
$$
\mathrm{n}\left(\mathrm{CO}_{2}\right)=\mathrm{n}\left(\mathrm{CaCO}_{3}\right)=2.4 \text { mol }
$$
Determination of the amount of substance - 2 points
25. Let's calculate the volume of the released gas
... | Answer: 53.76 liters
TOTAL 10 points
## TASK № 2
Potassium chloride was treated with concentrated sulfuric acid, and the resulting salt was added to a solution of potassium hydroxide. Barium nitrate solution was added to the resulting solution. The precipitate formed was filtered, mixed with carbon, and calcined. Wr... | 53.76 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,243 |
18. $\mathrm{KHSO}_{4}+\mathrm{KOH}=\mathrm{K}_{2} \mathrm{SO}_{4}+\mathrm{H}_{2} \mathrm{O}$
Writing the equation of the reaction and balancing the coefficients - 4 points $19 . \mathrm{K}_{2} \mathrm{SO}_{4}+\mathrm{Ba}\left(\mathrm{NO}_{3}\right)_{2}=2 \mathrm{KNO}_{3}+\mathrm{BaSO}_{4} \downarrow$
Writing the equ... | # Solution:
22. Let's derive the general formula for a monobasic carboxylic acid:
$\mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}+2} \mathrm{O}_{2}$
We will set up an equation to calculate the mass fraction of oxygen:
$\omega(O)=16 \cdot 2 /(12 n+2 n+2+16 \cdot 2)=32 /(14 n+34)$
$32 /(14 n+34)=0.4325$
Setting u... | C_3H_6O_2orC_2H_5COOH | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,244 |
28. Let's determine the amount of sulfur substance
According to the chemical reaction equation:
$$
\mathrm{n}(\mathrm{S})=\mathrm{n}\left(\mathrm{SO}_{2}\right)=1 \text { mole }
$$
29. Let's calculate the mass of pure sulfur:
$$
m(S)=n \cdot M=1 \cdot 32=32 \text { g }
$$
Determination of the mass of the substance... | Answer: 6 g.
## TOTAL 10 points
## TASK № 2
Potassium phosphate was calcined with coke in the presence of river sand. The simple substance formed reacted with excess chlorine. The resulting product was added to an excess of potassium hydroxide solution. The formed solution was treated with lime water. Write the equa... | 6 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,246 |
34. Let's determine the amount of calcium carbonate substance. According to the chemical reaction equation:
$n\left(\mathrm{CaCO}_{3}\right)=\frac{1}{2} n(\mathrm{HCl})=\frac{0.822}{2}=0.411$ mol
Determination of the amount of substance - 2 points
35. Let's determine the mass of calcium carbonate $m\left(\mathrm{CaC... | Answer: 42.37 g
## TOTAL 10 points
## TASK № 2
To a solution of copper(II) sulfite, an excess of soda solution was added. The precipitate that formed was calcined, and the solid residue was heated in an atmosphere of excess oxygen. The resulting substance was dissolved in concentrated nitric acid. Write the equation... | 42.37 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,249 |
28. $\mathrm{Cu}+4 \mathrm{HNO}_{3}=\mathrm{Cu}\left(\mathrm{NO}_{3}\right)_{2}+2 \mathrm{NO}_{2} \uparrow+4 \mathrm{H}_{2} \mathrm{O}$
Writing the equation of the reaction and balancing the coefficients - 4 points
## TOTAL 16 points
## TASK № 3
A dibromoalkane contains 85.1% bromine. Determine the formula of the d... | # Solution:
26. Let's derive the general formula for dibromoalkane:
$\mathrm{C}_{\mathrm{n}} \mathrm{H}_{2 \mathrm{n}} \mathrm{Br}_{2}$
We will set up an equation to calculate the mass fraction of bromine:
$\omega(\mathrm{Br})=80 \cdot 2(12 n+2 n+80 \cdot 2)$
$160 /(14 n+160)=0.851$
Setting up the equation for th... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 7,250 |
4. Let's find the total amount of gases in the ampoule after the reactions are completed
$$
v=0.02+0.02=0.04 \text { mol }
$$
If we assume that the composition of the gas phase does not change upon cooling, the pressure in the ampoule will be
$$
p=\frac{v R T}{V}=\frac{0.04 \cdot 8.314 \cdot 298}{0.25}=396.4 \text {... | # Solution:
Tube numbers: № 1 - $\mathrm{CoSO}_{4}$, № 2 - $\mathrm{Co}\left(\mathrm{NO}_{3}\right)_{2}$, № 3 - $\mathrm{Hg}\left(\mathrm{NO}_{3}\right)_{2}$, № 4 $\mathrm{AlCl}_{3}$, № 5 - $\mathrm{BaCl}_{2}$, № 6 - NH4OH, № 7 - $\mathrm{HNO}_{3}$, № 8 - NaOH.
Equations for precipitate formation:
$\mathrm{CoSO}_{4}... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,251 |
39. Let's calculate the volume of carbon dioxide.
According to Avogadro's law, the following rule applies: in equal volumes of different gases taken at the same temperatures and pressures, the same number of molecules is contained.
From the chemical reaction equation, it follows that the ratio of the amount of substa... | Answer: 7 l.
## TOTAL 10 points
## TASK № 2
Chromium (III) oxide was melted with potassium sulfite. The resulting product was added to water. To the precipitate formed, a mixture of bromine and sodium hydroxide was added, resulting in a yellow solution. Upon adding hydrogen sulfide water to the obtained solution, a ... | 7 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,252 |
Problem 5. (20 points) A point is randomly thrown onto the segment [3; 8] and let $k-$ be the obtained value. Find the probability that the roots of the equation $\left(k^{2}-2 k-3\right) x^{2}+(3 k-5) x+2=0$ satisfy the condition $x_{1} \leq 2 x_{2}$. | # Solution.
By Vieta's theorem
$\left\{\begin{array}{l}x_{1}+x_{2}=\frac{5-3 k}{k^{2}-2 k-3} \\ x_{1} \cdot x_{2}=\frac{2}{k^{2}-2 k-3}\end{array}\right.$
Find the value of $k$ under the condition that $x_{1}=2 x_{2}$, and then use the method of intervals. $\left\{\begin{array}{l}3 x_{2}=\frac{5-3 k}{k^{2}-2 k-3} ; ... | \frac{1}{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,258 |
Task 1. (5 points) Find $\frac{a^{8}+256}{16 a^{4}}$, if $\frac{a}{2}+\frac{2}{a}=5$. | Solution.
$$
\begin{aligned}
& \frac{a^{8}+256}{16 a^{4}}=\frac{a^{4}}{16}+\frac{16}{a^{4}}=\frac{a^{4}}{16}+2+\frac{16}{a^{4}}-2=\left(\frac{a^{2}}{4}+\frac{4}{a^{2}}\right)^{2}-2= \\
& =\left(\frac{a^{2}}{4}+2+\frac{4}{a^{2}}-2\right)^{2}-2=\left(\left(\frac{a}{2}+\frac{2}{a}\right)^{2}-2\right)^{2}-2=\left(5^{2}-2\... | 527 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,259 |
Task 2. (10 points) Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$, and the points of tangency cut off segments of length 4 cm from vertex $A$. Two tangents are drawn from point $C$ to this circle. Find the side of the square if the angle between the tangents is $60... | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. If radii of the circle are drawn to the points of tange... | 4(\sqrt{2}-1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,260 |
Task 3. (15 points) Laboratory engineer Sergei received an object for research consisting of about 200 monoliths (a container designed for 200 monoliths, which was almost completely filled). Each monolith has a specific name (sandy loam or clayey loam) and genesis (marine or lake-glacial deposits). The relative frequen... | # Solution.
Let's determine the exact number of monoliths. It is known that the probability of a monolith being loamy sand is $\frac{1}{9}$. The number closest to 200 that is divisible by 9 is 198. Therefore, there are 198 monoliths in total. Monoliths of lacustrine-glacial origin consist of all loamy sands $(198: 9=2... | 77 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,261 |
Task 4. (20 points) To get from the first to the second building of the university, Sasha took a car-sharing vehicle, while Zhenya rented a scooter. Sasha and Zhenya left the first building for the second at the same time, and at the same time, Professor Vladimir Sergeyevich left the second building for the first in a ... | # Solution.
Since the speeds of the scooter and the teacher's car differ by a factor of 4, the teacher's car will travel 4 times the distance in the same amount of time. Therefore, the distance between the buildings is $1+4=5$ km.
Let $x$ be the speed of the scooter, $t$ be the time the car-sharing service spends tra... | 15,40,60 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,262 |
Task 5. (20 points)
Find $x_{0}-y_{0}$, if $x_{0}$ and $y_{0}$ are the solutions to the system of equations:
$$
\left\{\begin{array}{l}
x^{3}-2023 x=y^{3}-2023 y+2020 \\
x^{2}+x y+y^{2}=2022
\end{array}\right.
$$ | # Solution.
Rewrite the system as
$$
\left\{\begin{array}{l}
x^{3}-y^{3}+2023 y-2023 x=2020 \\
x^{2}+x y+y^{2}=2022
\end{array}\right.
$$
Let $x_{0}$ and $y_{0}$ be the solution to the system of equations. Then
$\left\{\begin{array}{l}x_{0}{ }^{3}-y_{0}{ }^{3}+2023 y_{0}-2023 x_{0}=2020, \\ x_{0}{ }^{2}+x_{0} y_{0}... | -2020 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,263 |
Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 5 radars around it, the coverage of each of which is a circle with a radius of $r=13$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring arou... | # Solution.
To ensure radar coverage of a ring around the platform, they need to be placed at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
 Find $\frac{a^{8}-256}{16 a^{4}} \cdot \frac{2 a}{a^{2}+4}$, if $\frac{a}{2}-\frac{2}{a}=3$.
# | # Solution.
$$
\begin{aligned}
& \frac{a^{8}-256}{16 a^{4}} \cdot \frac{2 a}{a^{2}+4}=\left(\frac{a^{4}}{16}-\frac{16}{a^{4}}\right) \cdot \frac{2 a}{a^{2}+4}=\left(\frac{a^{2}}{4}+\frac{4}{a^{2}}\right)\left(\frac{a^{2}}{4}-\frac{4}{a^{2}}\right) \cdot \frac{2 a}{a^{2}+4}= \\
& =\left(\frac{a^{2}}{4}-2+\frac{4}{a^{2}... | 33 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,265 |
Task 2. (10 points) A circle touches the extensions of two sides $A B$ and $A D$ of square $A B C D$ with a side length of 4 cm. Two tangents are drawn from point $C$ to this circle. Find the radius of the circle if the angle between the tangents is $60^{\circ}$. | # Solution.
Fig. 1
The segment cut off from vertex $A$ by the point of tangency of the circle is equal to the radius of this circle. The diagonal of the square $A B C D A C=4 \sqrt{2}$. If... | 4(\sqrt{2}+1) | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,266 |
Task 3. (15 points) Lab engineer Dasha received an object for research consisting of about 100 monoliths (a container designed for 100 monoliths, which was almost completely filled). Each monolith has a specific name (sandy loam or clayey loam) and genesis (marine or lake-glacial deposits). The relative frequency (stat... | # Solution.
Let's determine the exact number of monoliths. It is known that the probability of a monolith being loamy sand is $\frac{1}{7}$. The number closest to 100 that is divisible by 7 is 98. Therefore, there are 98 monoliths in total. Monoliths of lacustrine-glacial origin consist of all loamy sands ($98: 7=14$)... | 35 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,267 |
Task 4. (20 points) To get from the first to the second building of the university, Sasha took a car-sharing vehicle, while Valya rented a scooter. Sasha and Valya set off from the first building to the second at the same time, and at the same time, teacher Sergei Vladimirovich set off from the second building to the f... | # Solution.
Since the speeds of the scooter and the teacher's car differ by 6 times, the teacher's car will travel 6 times more in the same amount of time. Therefore, the distance between the buildings is $1+6=7$ km.
Let $x$ be the speed of the scooter, $t$ be the time the car-sharing service spends traveling 20 km, ... | 10,45,60 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,268 |
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