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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 $r=25$ 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 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).
 In triangle $ABC$ with sides $AB=8, AC=4, BC=6$, the bisector $AK$ is drawn, and a point $M$ is marked on side $AC$ such that $AM: CM=3: 1$. Point $N$ is the intersection of $AK$ and $BM$. Find $AN$. | # Solution.
Draw a line through point $A$ parallel to line $B C$ and intersecting line $B M$ at point $L$ (Fig. 1).
Fig. 1.
By the property of the angle bisector $\frac{B K}{C K}=\frac{A... | \frac{18\sqrt{6}}{11} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,272 |
Task 3. (15 points) At the research institute, a scientific employee, Ivan Ivanovich, received an object for research containing about 300 oil samples (a container designed for 300 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or hig... | # Solution.
Let's determine the exact number of oil samples. It is known that the relative frequency of a selected sample being a heavy oil sample is $\frac{1}{8}$, and the number closest to 300 that is divisible by $8-296$. Therefore, the total number of samples in the container is 296. The samples of high-sulfur oil... | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,273 |
Task 4. (20 points) Find the smallest natural solution of the inequality $\left(\frac{2023}{2022}\right)^{27+18+12+8+\ldots+27 \cdot\left(\frac{2}{3}\right)^{n}}>\left(\frac{2023}{2022}\right)^{72}$. | Solution.
$\left(\frac{2023}{2022}\right)^{27+18+12+8+\ldots+27 \cdot\left(\frac{2}{3}\right)^{n}}>\left(\frac{2023}{2022}\right)^{72}$
We transition to an equivalent inequality
$27+18+12+8+\ldots+27 \cdot\left(\frac{2}{3}\right)^{n}>72$
$27\left(1+\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\left(\frac{2}{3}\right)^{... | 5 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 7,274 |
Problem 5. (20 points) Solve the inequality $\sqrt[2022]{x^{3}-x-\frac{1}{x}+\frac{1}{x^{3}}+3} \leq 0$.
# | # Solution.
The inequality $\sqrt[2022]{x^{3}-x-\frac{1}{x}+\frac{1}{x^{3}}+3} \leq 0$ has a solution only if $x^{3}-x-\frac{1}{x}+\frac{1}{x^{3}}+3=0$. Let $t=x+\frac{1}{x}$, then $t^{3}-4 t+3=0, t^{3}-t-3 t+3=0,(t-1)\left(t^{2}+t-3\right)=0, t=1$ or $t=\frac{-1 \pm \sqrt{13}}{2}$.
The equations $x+\frac{1}{x}=1$ an... | \frac{-1-\sqrt{13}\\sqrt{2\sqrt{13}-2}}{4} | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 7,275 |
Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 7 radars around it, the coverage of each of which is a circle with a 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 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).
Point $O$ is the center of the oil platform, and points $A$ and $B$ are the locations of the radars. The circles repr... | \frac{40}{\sin(180/7)};\frac{1440\pi}{\operatorname{tg}(180/7)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,276 |
Task 1. (5 points) Calculate
$$
\left(\frac{10001}{20232023}-\frac{10001}{20222022}\right) \cdot 4090506+\sqrt{4092529}
$$ | # Solution.
$$
\begin{aligned}
& \left(\frac{10001}{20232023}-\frac{10001}{20222022}\right) \cdot 4090506+\sqrt{4092529} \\
& =\left(\frac{10001}{2023 \cdot 10001}-\frac{10001}{2022 \cdot 10001}\right) \cdot 4090506+\sqrt{2023^{2}}= \\
& \quad=\left(\frac{1}{2023}-\frac{1}{2022}\right) \cdot 2022 \cdot 2023+2023=\frac... | 2022 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,277 |
Task 2. (10 points) In triangle $ABC$ with sides $AB=9$, $AC=3$, $BC=8$, the bisector $AK$ is drawn, and a point $M$ is marked on side $AC$ such that $AM: CM=3: 1$. Point $N$ is the intersection of $AK$ and $BM$. Find $KN$.
# | # Solution.
Draw a line through point $A$ parallel to line $B C$ and intersecting line $B M$ at point $L$ (Fig. 1).
Fig. 1.
By the property of the angle bisector $\frac{B K}{C K}=\frac{A... | \frac{\sqrt{15}}{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,278 |
Task 3. (15 points) At the research institute, a scientific employee, Tatyana Vasilyevna, received an object for research containing about 150 oil samples (a container designed for 150 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-sulfur or... | # Solution.
Let's determine the exact number of oil samples. It is known that the relative frequency of a sample being a heavy oil sample is $\frac{2}{11}$, and the number closest to 150 that is divisible by $11-143$. Therefore, the total number of samples in the container is 143. Samples of high-sulfur oil consist of... | 66 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,279 |
Task 4. (20 points) Find the smallest natural solution of the inequality $\left(\frac{2023}{2022}\right)^{36+24+16+\ldots+36\left(\frac{2}{3}\right)^{n}}>\left(\frac{2023}{2022}\right)^{96}$. | Solution.
$\left(\frac{2023}{2022}\right)^{36+24+16+\ldots+36\left(\frac{2}{3}\right)^{n}}>\left(\frac{2023}{2022}\right)^{96}$.
$36+24+16+. .+36 \cdot\left(\frac{2}{3}\right)^{n}>96$
$36\left(1+\frac{2}{3}+\left(\frac{2}{3}\right)^{2}+\ldots+\left(\frac{2}{3}\right)^{n}\right)>96$
$\left(1+\frac{2}{3}+\left(\frac{... | 5 | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 7,280 |
Problem 5. (20 points) Solve the inequality $202 \sqrt{x^{3}-2 x-\frac{2}{x}+\frac{1}{x^{3}}+4} \leq 0$.
# | # Solution.
The inequality $202 \sqrt{x^{3}-2 x-\frac{2}{x}+\frac{1}{x^{3}}+4} \leq 0$ has a solution only if $x^{3}-2 x-\frac{2}{x}+\frac{1}{x^{3}}+4=0$.
Let $t=x+\frac{1}{x}$, then $t^{3}-5 t+4=0, t^{3}-t-4 t+4=0,(t-1)\left(t^{2}+t-4\right)=0, t=1$ or $t=\frac{-1 \pm \sqrt{17}}{2}$.
The equations $x+\frac{1}{x}=1$... | \frac{-1-\sqrt{17}\\sqrt{2\sqrt{17}+2}}{4} | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 7,281 |
Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 7 radars around it, the coverage of each of which is a circle with a 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 arou... | 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).
Point $O$ is the center of the oil platform, and points $A$ and $B$ are the locations of the radars. The circles... | \frac{24}{\sin(180/7)};\frac{960\pi}{\operatorname{tg}(180/7)} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,282 |
Task 1. (5 points) Find $\frac{a^{8}+1296}{36 a^{4}}$, if $\frac{a}{\sqrt{6}}+\frac{\sqrt{6}}{a}=5$.
# | # Solution.
$$
\begin{aligned}
& \frac{a^{8}+1296}{36 a^{4}}=\frac{a^{4}}{36}+\frac{36}{a^{4}}=\frac{a^{4}}{36}+2+\frac{36}{a^{4}}-2=\left(\frac{a^{2}}{6}+\frac{6}{a^{2}}\right)^{2}-2= \\
& =\left(\frac{a^{2}}{6}+2+\frac{6}{a^{2}}-2\right)^{2}-2=\left(\left(\frac{a}{\sqrt{6}}+\frac{\sqrt{6}}{a}\right)^{2}-2\right)^{2}... | 527 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,283 |
Task 2. (10 points) A circle with a radius of 15 touches two adjacent sides $AB$ and $AD$ of the square $ABCD$. On the other two sides, the circle intersects at points, cutting off segments of 6 cm and 3 cm from the vertices, respectively. Find the length of the segment that the circle cuts off from vertex $B$ at the p... | # Solution.
Fig. 1
Let $X$ be the desired segment, then $X+15$ is the side of the square. The segment $K L=15+X-6-3=X+6$. Consider $\triangle O N L$. By the Pythagorean theorem, the follow... | 12 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,284 |
Task 3. (15 points) At the quality control department of an oil refinery, Engineer Pavel Pavlovich received a research object consisting of about 100 oil samples (a container designed for 100 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low-su... | # Solution.
Let's determine the exact number of oil samples. It is known that the relative frequency of a sample being a heavy oil sample is $\frac{1}{7}$. The number closest to 100 that is divisible by $7$ is $98$. Therefore, there are 98 samples in total in the container. The samples of high-sulfur oil include all t... | 35 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,285 |
Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+298 x_{n}+x_{n+1}}{300}$ holds. Find $\sqrt{\frac{x_{2023}-x_{2}}{2021} \cdot \frac{2022}{x_{2023}-x_{1}}}-2023$. | # Solution.
From the given relations in the problem, it is easily deduced that for all $n \geq 2$, $x_{n}-x_{n-1}=x_{n+1}-x_{n}$, which implies that the sequence is an arithmetic progression. Indeed,
$$
\begin{gathered}
x_{n}=\frac{x_{n-1}+298 x_{n}+x_{n+1}}{300} \\
2 x_{n}=x_{n-1}+x_{n+1} \\
x_{n}-x_{n-1}=x_{n+1}-x_... | -2022 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,286 |
Problem 5. (20 points) Solve the inequality $202 \sqrt{x^{3}-3 x-\frac{3}{x}+\frac{1}{x^{3}}+5} \leq 0$.
# | # Solution.
The inequality $\sqrt[2022]{x^{3}-3 x-\frac{3}{x}+\frac{1}{x^{3}}+5} \leq 0$ has a solution only if $x^{3}-3 x-\frac{3}{x}+\frac{1}{x^{3}}+5=0$.
Let $t=x+\frac{1}{x}$, then $t^{3}-6 t+5=0, t^{3}-t-5 t+5=0,(t-1)\left(t^{2}+t-5\right)=0, t=1$ or $t=\frac{-1 \pm \sqrt{21}}{2}$.
The equations $x+\frac{1}{x}=... | \frac{-1-\sqrt{21}\\sqrt{2\sqrt{21}+6}}{4} | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 7,287 |
Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 8 radars around it, the coverage of each of which is a circle with a 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 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}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}$, if $\frac{a}{3}-\frac{3}{a}=4$. | Solution.
$\frac{a^{8}-6561}{81 a^{4}} \cdot \frac{3 a}{a^{2}+9}=\left(\frac{a^{4}}{81}-\frac{81}{a^{4}}\right) \cdot \frac{3 a}{a^{2}+9}=\left(\frac{a^{2}}{9}+\frac{9}{a^{2}}\right)\left(\frac{a^{2}}{9}-\frac{9}{a^{2}}\right) \cdot \frac{3 a}{a^{2}+9}=$
$=\left(\frac{a^{2}}{9}-2+\frac{9}{a^{2}}+2\right)\left(\frac{a... | 72 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,289 |
Task 2. (10 points) A circle with a radius of 10 touches two adjacent sides $AB$ and $AD$ of the square $ABCD$. On the other two sides, the circle intersects at points, cutting off segments of 4 cm and 2 cm from the vertices, respectively. Find the length of the segment that the circle cuts off from vertex $B$ at the p... | Solution.
Fig. 1
Let $X$ be the desired segment, then $X+10$ is the side of the square. The segment $K L=10+X-4-2=X+4$. Consider $\triangle O N L$. By the Pythagorean theorem, the followin... | 8 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,290 |
Task 3. (15 points) At the quality control department of an oil refinery, Engineer Valentina Ivanovna received a research object consisting of about 200 oil samples (a container designed for 200 samples, which was almost completely filled). Each sample has certain characteristics in terms of sulfur content - either low... | # Solution.
Let's determine the exact number of oil samples. It is known that the relative frequency of a sample being a heavy oil sample is $\frac{1}{9}$. The number closest to 200 that is divisible by 9 is 198. Therefore, the total number of samples in the container is 198. The samples of high-sulfur oil consist of ... | 77 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,291 |
Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+398 x_{n}+x_{n+1}}{400}$ holds. Find $\sqrt{\frac{x_{2023}-x_{2}}{2021} \cdot \frac{2022}{x_{2023}-x_{1}}}+2021$. | # Solution.
From the given relations in the problem, it is easily deduced that for all $n \geq 2$, $x_{n}-x_{n-1}=x_{n+1}-x_{n}$, which implies that the sequence is an arithmetic progression. Indeed,
$$
\begin{gathered}
x_{n}=\frac{x_{n-1}+398 x_{n}+x_{n+1}}{400} \\
2 x_{n}=x_{n-1}+x_{n+1} \\
x_{n}-x_{n-1}=x_{n+1}-x_... | 2022 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,292 |
Problem 5. (20 points) Solve the inequality $\sqrt[2022]{x^{3}-4 x-\frac{4}{x}+\frac{1}{x^{3}}+6} \leq 0$.
| Solution.
The inequality $\sqrt[2022]{x^{3}-4 x-\frac{4}{x}+\frac{1}{x^{3}}+6} \leq 0$ has a solution only if $x^{3}-4 x-\frac{4}{x}+\frac{1}{x^{3}}+6=0$.
Let $t=x+\frac{1}{x}$, then $t^{3}-7 t+6=0, t^{3}-t-6 t+6=0,(t-1)\left(t^{2}+t-6\right)=0, t=1$, or $t=2$, or $t=-3$.
The equation $x+\frac{1}{x}=1$ has no roots.... | {1;\frac{-3\\sqrt{5}}{2}} | Inequalities | math-word-problem | Yes | Yes | olympiads | false | 7,293 |
Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 8 radars around it, the coverage of each of which is a circle with a 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 arou... | # 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).
 Find $\frac{a^{8}-256}{16 a^{4}} \cdot \frac{2 a}{a^{2}+4}$, if $\frac{a}{2}-\frac{2}{a}=5$. | 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}}{16}-\frac{16}{a^{2}}\right) \cdot \frac{2 a}{a^{2}+4}= \\
& =\left(\frac{a^{2}}{4}-2+\frac{4}{a^{2}... | 81 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,295 |
Task 2. (10 points) A circle touches two adjacent sides $AB$ and $AD$ of square $ABCD$ and cuts off segments of length 4 cm from vertices $B$ and $D$ at the points of tangency. On the other two sides, the circle intersects and cuts off segments of 2 cm and 1 cm from the vertices, respectively. Find the radius of the ci... | # Solution.
Fig. 1
Let $R$ be the radius of the circle, then $R+4$ is the side of the square. The segment $K L=4+R-2-1=R+1$. Consider the triangle $O N L$. By the Pythagorean theorem, the ... | 5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,296 |
Task 3. (15 points) In the educational center "Young Geologist," an object consisting of about 150 monoliths (a container designed for 150 monoliths, which was almost completely filled) was delivered. Each monolith has a specific name (sandy loam or clayey loam) and genesis (marine or lake-glacial deposits). The relati... | # Solution.
Let's determine the exact number of monoliths. It is known that the relative frequency of a monolith being loamy sand is $\frac{2}{11}$. The number closest to 150 that is divisible by 11 is 143. Therefore, there are 143 monoliths in total. Monoliths of lacustrine-glacial origin make up all loamy sands ( $1... | 66 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,297 |
Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+98 x_{n}+x_{n+1}}{100}$ holds. Find $\sqrt{\frac{x_{2023}-x_{1}}{2022} \cdot \frac{2021}{x_{2023}-x_{2}}}+2021$. | # Solution.
From the given relations in the problem, it is easily deduced that for all $n \geq 2$, $x_{n}-x_{n-1}=x_{n+1}-x_{n}$, which implies that the sequence is an arithmetic progression. Indeed,
$$
\begin{gathered}
x_{n}=\frac{x_{n-1}+98 x_{n}+x_{n+1}}{100} \\
2 x_{n}=x_{n-1}+x_{n+1} \\
x_{n}-x_{n-1}=x_{n+1}-x_{... | 2022 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,298 |
Task 5. (20 points) Solve the equation $252 \frac{7}{8}\left(x^{2}+\frac{x^{2}}{(x+1)^{2}}\right)=2023$. | Solution.
$$
\begin{aligned}
& x^{2}+\frac{x^{2}}{(x+1)^{2}}=2023: 252 \frac{7}{8} \\
& x^{2}+\frac{x^{2}}{(x+1)^{2}}=8 \\
& x^{2}-\frac{2 x^{2}}{x+1}+\frac{x^{2}}{(x+1)^{2}}=8-\frac{2 x^{2}}{x+1} \\
& \left(x-\frac{x}{x+1}\right)^{2}=8-\frac{2 x^{2}}{x+1} \\
& \left(\frac{x^{2}}{x+1}\right)^{2}=8-2 \cdot \frac{x^{2}}... | -2;1-\sqrt{3};1+\sqrt{3} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,299 |
Task 6. (30 points) To protect an oil platform located at sea, it is necessary to distribute 9 radars around it, the coverage of each of which is a circle with a radius of $r=61$ km. Determine the maximum distance from the center of the platform at which they should be placed to ensure that the radars cover a ring arou... | # Solution.
To ensure radar coverage of a ring around the platform, it is necessary to place them at the vertices of a regular polygon, the center of which coincides with the center of the platform (Fig. 2).
 Find $\frac{a^{8}+256}{16 a^{4}}$, if $\frac{a}{2}+\frac{2}{a}=3$. | Solution.
$$
\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(3^{2}-2\... | 47 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,301 |
Task 2. (10 points) A circle touches two adjacent sides $AB$ and $AD$ of square $ABCD$ and cuts off segments of length 8 cm from vertices $B$ and $D$ at the points of tangency. On the other two sides, the circle intersects and cuts off segments of 4 cm and 2 cm from the vertices, respectively. Find the radius of the ci... | # Solution.
Fig. 1
Let $R$ be the radius of the circle, then $R+8$ is the side of the square. The segment $K L=8+R-4-2=R+2$. Consider $\triangle O N L$. By the Pythagorean theorem, the fol... | 10 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,302 |
Task 3. (15 points) An educational center "Young Geologist" received an object for research consisting of about 300 monoliths (a container designed for 300 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 rel... | # Solution.
Let's determine the exact number of monoliths. It is known that the relative frequency of a monolith being loamy sand is $\frac{1}{8}$. The number closest to 300 that is divisible by $8-296$. Therefore, there are 296 monoliths in total. Monoliths of lacustrine-glacial origin consist of all loamy sands $(29... | 120 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,303 |
Task 4. (20 points) For the numerical sequence $\left\{x_{n}\right\}$, all terms of which, starting from $n \geq 2$, are distinct, the relation $x_{n}=\frac{x_{n-1}+198 x_{n}+x_{n+1}}{200}$ holds. Find $\sqrt{\frac{x_{2023}-x_{1}}{2022} \cdot \frac{2021}{x_{2023}-x_{2}}}+2022$. | # Solution.
From the given relations in the problem, it is easily deduced that for all $n \geq 2$, $x_{n}-x_{n-1}=x_{n+1}-x_{n}$, which implies that the sequence is an arithmetic progression. Indeed,
$$
\begin{gathered}
x_{n}=\frac{x_{n-1}+198 x_{n}+x_{n+1}}{200} \\
2 x_{n}=x_{n-1}+x_{n+1} \\
x_{n}-x_{n-1}=x_{n+1}-x_... | 2023 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,304 |
Task 5. (20 points) Solve the equation $674 \frac{1}{3}\left(x^{2}+\frac{x^{2}}{(1-x)^{2}}\right)=2023$. | Solution.
$$
\begin{aligned}
& x^{2}+\frac{x^{2}}{(1-x)^{2}}=2023: 674 \frac{1}{3} \\
& x^{2}+\frac{x^{2}}{(1-x)^{2}}=3 \\
& x^{2}-\frac{2 x^{2}}{1-x}+\frac{x^{2}}{(1-x)^{2}}=3-\frac{2 x^{2}}{1-x}
\end{aligned}
$$
$\left(x-\frac{x}{1-x}\right)^{2}=3-\frac{2 x^{2}}{1-x}$
$\left(\frac{x^{2}}{1-x}\right)^{2}=3-2 \cdot ... | -\frac{1+\sqrt{5}}{2};\frac{\sqrt{5}-1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,305 |
Task 6. (30 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, 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).
=\omega \rho V / M=0.062 \cdot 1.055 \text { g/ml } \cdot 22.7 \text { ml / } 56 \text { g/ mol }=0.0265 \text { mol. } \\
v\left(\mathrm{HNO}_{3}\right)=\text { C }_{\m... | Answer: 1.47 kJ of heat will be released.
## Grading system: | 1.47 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,323 |
Task 2. Development of a Method for Optimizing a Mathematical Problem
Problem Statement. At Unit 3 of an oil and gas company, the task is to optimize the operation of the navigation system elements. The task of this unit is to receive encrypted signals from Unit 1 and Unit 2, synthesize the incoming data packets, and ... | Solution to the problem. Solving the problem "head-on" by raising one number to the power of another is bound to exceed the computational power not only of a single computer but even a data center would require a certain amount of time to perform the calculations. Since we only need to send the last digit, let's focus ... | 6 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 7,328 |
# Task 3. Synthesis of Discrete Devices
Text of the task. When comparing drilling data coming to the Central Processing Center with actual production values, a significant discrepancy was identified. A service inspection conducted at the field revealed a malfunction of the sensors, which were decided to be completely ... | To solve the problem, we first need to derive a formula for the function \( y(a, b, c) \) that contains only inversions, conjunctions, and modulo-two sums. We observe that when \( a = 0 \) (the upper half of the table), the column of values matches the column of values for the modulo-two sum \( b \oplus c \), and when ... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,329 |
# Task 4. Processing Numerical Arrays
Task text. An analyzer, through which extracted minerals pass sequentially, shows the carbon mass in each batch, dividing it into layers. The laboratory technician needs to separate those layers in which the sum of the subsequence values of carbon mass per unit area is a multiple ... | Solution to the problem. The task requires calculating the quantity of each separate subsequence, having a different number of nested elements that are multiples of 77. Since the calculation is done in layers, there is no need to define subsequences, for example, consisting of two elements as 3 subsequences, containing... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 7,330 |
# Task 5. Data Encryption
Text of the task. At a polar station, the decryption device has malfunctioned, and now there is no possibility of standard message exchange with other industrial facilities or the mainland. Waiting for specialists from outside would take an unacceptably long time, and there is a spare device ... | Solution to the problem. The text of the message is fully encrypted, so it is logical to assume that not only letter symbols but also numerical ones are encrypted. Let's start with the latter: note that the largest digit in each number is 6. This suggests the hypothesis that the numbers are represented in the septenary... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,331 |
# Task 6. Logic
Text of the problem. Sigmund, Asema, Marina, and Sasha received an invitation to Career Days at their university. Due to conference week, they had little time, so on that day each could have detailed discussions with only one company. Two students went to meet with Eastern Gas, one with Transgas Space ... | Solution to the problem. This problem can be solved by comparing the facts in the table. Let's mark the number of students who attended the meeting with each company. Then exclude for Sizigmund the meeting with Transgas Space Systems. By comparing the combinations, we establish that Asema and Marina attended the meetin... | \begin{pmatrix}\hline | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,332 |
# Task 7. Principles of Encryption and Decryption
Problem statement. Karim has applied to the Spring School of the University for Information Security and needs to solve a test task: decrypt a short phrase. For this, he is provided with an example of the encryption of the word "гипертекст" (hypertext):
## гипертекст ... | To solve the problem, we first need to decipher the encryption algorithm. Let's look at the provided example:
- hypertext - the original word.
- igeptrkets - then there is a swap of adjacent characters.
- kezsftmzfu - a shift to the right by 2 characters in the alphabet.
- ufzmtfszek - the word is reversed.
To decryp... | neftebazyyuzhnogorayona | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,333 |
# Task 9. Data Measurement
Text of the task. Between two working sections of the deposit, it is planned to install an additional channel for exchanging stereo audio signals (messages) for daily reporting communication sessions. Determine the required bandwidth of this channel in kilobits based on the calculation that ... | The solution to the problem. The bandwidth of a channel is a measure of the ratio of the volume of transmitted data to the transmission time, i.e., simply put - how much data can be transmitted over a set time period. It is also worth noting that the indicators provided are for a mono signal, and the calculation needs ... | 2.25 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,335 |
# Task 10. Game Theory
Problem text. An oil and gas company has set the task of revising its sales strategy for natural and liquefied gas during the winter period. In a calm winter, there is a possibility of supplying 2200 cubic meters of natural gas and 3500 cubic meters of liquefied gas, while in the event of severe... | Solution to the problem. Let's model the possible scenarios. Due to unpredictable weather fluctuations, the company can apply two "pure" strategies:
$\mathrm{F}_{1}=(2200,3500) -$ procurement of $2200 \mathrm{m}^{3}$ of natural gas and $3500 \mathrm{m}^{3}$ of liquefied gas
$\mathrm{F}_{2}=(3800,2450) -$ procurement ... | (3032;2954) | Other | math-word-problem | Yes | Yes | olympiads | false | 7,336 |
# Task 1. Bioinformatics
Problem statement. To study the terrain in the area of preparation for a prospective deposit, a group of ecologists was sent with the task of studying the local flora. For this purpose, a set of samples was collected for DNA analysis, which is the main molecule for storing information in biolo... | Solution to the problem. To find the second side of DNA, it is sufficient to input the elements of the first side, replacing the existing elements with "paired" ones. Since the use of conditional operators in the problem is undesirable, as it makes the program unnecessarily bulky, one can resort to a combination of loo... | notfound | Other | math-word-problem | Yes | Yes | olympiads | false | 7,337 |
# Task 3. Boolean Algebra
Problem text. Stepan and Fedor tried to verify one of their hypotheses before tomorrow's presentation at the meeting. In the morning, Stepan found that the notes were often written over previous notes, and much of it was difficult to decipher. Nevertheless, he managed to restore the connectiv... | Solution to the problem. First, let's construct a truth table for the given scheme $\mathrm{Z} \rightarrow(\mathrm{X} \vee \mathrm{Y} \wedge \mathrm{Z}) \equiv \mathrm{X}$
| $\mathbf{X}$ | $\mathbf{Y}$ | $\mathbf{Z}$ | $\mathbf{Y} \wedge \mathbf{Z}$ | $\mathbf{X} \vee \mathbf{Y} \wedge \mathbf{Z}$ | $\mathbf{Z} \right... | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,339 | |
Task 4. Algorithms
Problem statement. The development of a promising gas field in a border zone required additional measures of information security. For this purpose, a team of encryption specialists was involved, but its leader made sure that none of its members knew the entire algorithm, so he divided it into eleme... | # Solution to the problem.
Example of program implementation in Python
def order(sentence):
words = sentence.split()
numbers = range(1, 10)
result = []
for number in numbers:
str_number = str(number)
for word in words:
if str_number in word:
result.a... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,340 |
# Task 5. Logic
Text of the task. On a new site for oil workers, five containers for the accommodation of five shift workers were installed in sequence. Each of them has a favorite movie/series, a favorite dish, and an area of interest that they pursue in their free time. Write down everything you can find out about t... | Solution to the problem. The problem can be solved using tables. First, we will build an empty table with container numbers, and after reading the statements, we will fill in the cells whose information is directly indicated in the statements.
| Number | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{4}$ | $\ma... | Damiir,khinkali,Papa,3Dmodeling | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,341 |
# Task 6. Development of an Optimization Method
Task Description. A geological party conducting mineral exploration lost their bearings and ended up in an area where the navigation equipment malfunctioned. An overall analysis of the direction of movement showed that the general route provided was correct, but it inclu... | # Solution to the problem.
Example of program implementation in Python
```
opposite = {
'north': 'south',
'south': 'north',
'west': 'east',
'east': 'west'
}
def dir_reduc(plan):
new_plan = []
for d in plan:
if new_plan and new_plan[-1] == opposite[d]:
new_plan.pop()
... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,342 |
# Task 7. Number Systems in System Administration
Text of the task. In a workers' settlement of shift workers, a network failure occurred as a result of which one of the devices in the gas extraction complex began to return not the standard display of the device's IP address and subnet mask, but its binary representat... | To convert an IP address and subnet mask from binary to decimal, each of them needs to be divided into octets — combinations of 8 numbers separated by dots. Thus, we get:
IP address: 10110010.10110000.11100110.10101010
Subnet mask: 11111111.11111111.11111111.10000000
Next, each bit of each octet is converted by summ... | 178.176.230.170,255.255.255.128,178.176.230.128 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,343 |
# Task 8. Development of a Complex Sorting Algorithm
Task Description. The management of a mineral processing terminal has set a task for marking incoming samples and has entrusted you with the development of the first part of the marker. The following technical specification has been formulated: the value of the mass... | # Solution to the problem.
Example of program implementation in Python
```
def order_weight(string):
weights = string.split(' ')
order_weights = {}
for weight in weights:
sum = 0
for digit in weight:
sum += int(digit)
order_weights[weight] = sum
weights.sort()
w... | notfound | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 7,344 |
# Task 9. Combinatorics
Problem text. At the end of the financial year, a traditional audit of the activities of several subsidiaries of an oil and gas holding is conducted. This year, 13 oil refineries (NPR) and 15 gas transportation hubs (GTU) are selected. How many options for selecting an audit object exist at the... | # Solution to the problem.
A total of 13 refineries and 15 processing units need to be inspected. After the first week, 11 refineries remain (since 13-2 = 11, i.e., 2 refineries were inspected in the first week, and we exclude them from the total number) and 12 processing units (15-3=12). To solve this, we can use the... | 12,100 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,345 |
5. Determination of the product of the reaction - 2 points:
Total: 10 points.
## TASK 4
## SOLUTION:
To decrease the reaction rate, it is necessary to increase the volume of the system, i.e., to decrease the pressure and, thereby, reduce the concentration of the gaseous component - $\mathrm{NH}_{3}$. The concentrat... | Answer: increase by 1.66 times.
## Scoring system: | 1.66 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,355 |
# Task 1. Automation of Production Activities
Task text. When designing the oil cleaning system, the technical task was to determine the stage of comparing oil mixtures according to specified quality criteria from the Star and Shine extraction sites. Over one time period, ten iterations of comparison are conducted, wi... | Solution to the problem. To solve this, it is necessary to define four counter variables - star, shine, HV_22, and LV_426 to record the required parameters. Next, it is necessary to separate the characters in the original list that can be converted to numerical values from the rest. For this, the appropriate method (sp... | HV-22:15898.83,LV-426:15898.83,Star:10points,Shine:14points | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,365 |
# Task 2. Logic
Text of the task. On a new site for oil workers, five containers for the accommodation of five shift workers were installed in sequence. Each of them has a favorite movie/TV show, a favorite dish, and an area of interest that they pursue in their free time. Write down everything you can find out about ... | Solution to the problem. The problem can be solved using tables. First, we will build an empty table with container numbers, and after reading the statements, we will fill in the cells whose information is directly indicated in the statements.
| Number | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{4}$ | $\ma... | Damiirlivesinthewhitecontainer,heloveskhinkali,oftenre-watchesthemovie"Papa,"isinterestedin3Dmodeling | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,366 |
# Task 4. Data Measurement
Text of the task. Between two working sections of the deposit, it is planned to install an additional channel for exchanging stereo audio signals (messages) for daily reporting communication sessions. Determine the required bandwidth of this channel in kibibytes based on the assumption that ... | The solution to the problem. The bandwidth of a channel is a measure of the ratio of the volume of transmitted data to the transmission time, i.e., simply put - how much data can be transmitted over a set time period. It is also worth noting that the indicators provided are for a mono signal, and the calculation needs ... | 0.28 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,368 |
# Task 5. Boolean Algebra
Problem text. Evdokia and Symbatay tried to develop a solution for verifying their hypotheses that could be quickly used during meetings. However, in the morning, it was discovered that the program was not saved, and from the paper notes, only a part could be deciphered. Nevertheless, they ma... | Solution to the problem. First, let's construct a truth table for the given scheme $\mathrm{Z} \rightarrow(\mathrm{X} \vee \mathrm{Y} \wedge \mathrm{Z}) \equiv \mathrm{X}$
| $\mathbf{X}$ | $\mathbf{Y}$ | $\mathbf{Z}$ | $\mathbf{Y} \wedge \mathbf{Z}$ | $\mathbf{X} \vee \mathbf{Y} \wedge \mathbf{Z}$ | $\mathbf{Z} \right... | notfound | Logic and Puzzles | math-word-problem | Yes | Yes | olympiads | false | 7,369 |
# Task 6. Game Theory
Test problem. Companies Samotlor and BaltikaPromNeft are striving to enter new markets and compete with each other in terms of the amount of oil extracted. At this stage, a forecast of production prospects in millions of barrels for a given period is being calculated. The prepared analytical repo... | Solution to the problem. To find the best strategy for Samotlor company, we will write down the minimum number in each row: $(2,2,3,2)$. The maximum of these is 3 - this number is the lower value of the game and pertains to the strategy СЗ. Next, for BalticPromNeft, we highlight the maximum values of the payoff: $(4,5,... | x_{1}^{*}=2/3,x_{2}^{*}=1/3,v=10/3 | Other | math-word-problem | Yes | Yes | olympiads | false | 7,370 |
# Task 8. Combinatorics
Text of the problem. At the end of the financial year, a traditional audit of the activities of several subsidiaries of an oil and gas holding is conducted. This year, they include 13 oil refineries (NPR) and 15 gas transportation hubs (GTU). How many options for selecting an audit object exist... | Solution to the problem. In total, 13 refineries and 15 processing units need to be inspected. After the first week, 11 refineries remain (since $13-2$ = 11, i.e., 2 refineries were inspected in the first week and were excluded) and 12 processing units ($15-3=12$). To solve this, we can use the combination formula:
$$... | 12100 | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,372 |
Task 1. (5 points) Solve the equation $x^{6}-22 x^{2}+\sqrt{21}=0$. | Solution.
Rewrite the equation as $x^{6}-21 x^{2}-x^{2}+\sqrt{21}=0$.
then
$x^{2}\left(x^{4}-21\right)-\left(x^{2}-\sqrt{21}\right)=0$,
$x^{2}\left(x^{2}-\sqrt{21}\right)\left(x^{2}+\sqrt{21}\right)-\left(x^{2}-\sqrt{21}\right)=0$,
$\left(x^{2}-\sqrt{21}\right)\left(x^{4}+x^{2} \sqrt{21}-1\right)=0$,
$x^{2}-\sqrt... | {\\sqrt[4]{21};\\sqrt{\frac{-\sqrt{21}+5}{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,374 |
Task 2. (10 points) Simplify the expression $2+22+222+\ldots+\underbrace{222 . . .2}_{2021}$. | Solution. Rewrite the sum $2+22+222+\ldots+\underbrace{222 \ldots .2}_{2021}$ as
$2 \cdot 1+2 \cdot 11+2 \cdot 111+\ldots+2 \cdot \underbrace{111 \ldots 1}_{2021}=2 \cdot \frac{10-1}{9}+2 \cdot \frac{10^{2}-1}{9}+2 \cdot \frac{10^{3}-1}{9}+\ldots+2 \cdot \frac{10^{2021}-1}{9}=$ and apply the formula for the geometric ... | \frac{20\cdot10^{2021}-36398}{81} | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 7,375 |
Task 4. (20 points) In a sequence of natural numbers, each subsequent number, starting from the third, is equal to the absolute difference of the two preceding ones. Determine the maximum number of elements such a sequence can contain if the value of each of them does not exceed 2022.
# | # Solution.
To maximize the length of the sequence, the largest elements should be at the beginning of the sequence. Let's consider the options:
1) $n, n-1,1, n-1, n-2, n-3,1, n-4, n-5,1, \ldots, 2,1,1$;
2) $n, 1, n-1, n-2,1, n-3, n-4,1, \ldots, 2,1,1$.
3) $1, n, n-1,1, n-2,1, n-3, n-4,1, \ldots, 2,1,1$.
4) $n-1, n, ... | 3034 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 7,377 |
Task 5. (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 5 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. 4).
$60^{\circ}$
Fig. 4
Since the triangle is equilateral, $x=5$, whic... | 0.5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,378 |
Task 6. (30 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 through the second is four times longer than the direct route between them; the distance from the first station to the second through the third i... | 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. 6).
... | 0<<68;60 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,379 |
Task 1. (5 points) Solve the equation $x^{6}-22 x^{2}-\sqrt{21}=0$. Solution.
Rewrite the equation as $x^{6}-21 x^{2}-x^{2}-\sqrt{21}=0$.
then
$x^{2}\left(x^{4}-21\right)-\left(x^{2}+\sqrt{21}\right)=0$,
$x^{2}\left(x^{2}-\sqrt{21}\right)\left(x^{2}+\sqrt{21}\right)-\left(x^{2}+\sqrt{21}\right)=0$,
$\left(x^{2}+\s... | Answer: $\pm \sqrt{\frac{\sqrt{21}+5}{2}}$. | \\sqrt{\frac{\sqrt{21}+5}{2}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,380 |
Task 2. (10 points) Simplify the expression $3+33+333+\ldots+\underbrace{333 . .3}_{2021}$.
---
The translation maintains the original text's line breaks and format as requested. | Solution. Rewrite the sum $3+33+333+\ldots+\underbrace{333 ...3}_{2021}$ as
$3 \cdot 1+3 \cdot 11+3 \cdot 111+\ldots+3 \cdot \underbrace{111 \ldots 1}_{2021}=3 \cdot \frac{10-1}{9}+3 \cdot \frac{10^{2}-1}{9}+3 \cdot \frac{10^{3}-1}{9}+\ldots+3 \cdot \frac{10^{2021}-1}{9}=$ and apply the formula for the geometric progr... | \frac{10^{2022}-18199}{27} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,381 |
Task 3. (15 points) Point $A$ lies on side $L M$ of triangle $K L M$ with an angle of $60^{\circ}$ at vertex $K$. Circles are inscribed in triangles $A K L$ and $A K M$ with centers $F$ and $O$ respectively. Find the radius of the circumcircle of triangle $F K O$, if $A O=7, A F=4$. | Solution.
Fig. 1
The center of the circle inscribed in an angle lies on the bisector of this angle, so the rays $A F$ and $A O$ are the bisectors of $\angle L A K$ and $\angle M A K$ respec... | \sqrt{65} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,382 |
Task 4. (20 points) In a sequence of natural numbers, each subsequent number, starting from the third, is equal to the absolute difference of the two preceding ones. Determine the maximum number of elements such a sequence can contain if the value of each of them does not exceed 2021.
# | # Solution.
To maximize the length of the sequence, the largest elements should be at the beginning of the sequence. Let's consider the options:
1) $n, n-1,1, n-1, n-2, n-3,1, n-4, n-5,1, \ldots, 2,1,1$;
2) $n, 1, n-1, n-2,1, n-3, n-4,1, \ldots, 2,1,1$.
3) $1, n, n-1,1, n-2,1, n-3, n-4,1, \ldots, 2,1,1$.
4) $n-1, n, ... | 3033 | Number Theory | math-word-problem | Yes | Yes | olympiads | false | 7,383 |
Task 5. (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. 4).
$60^{\circ}$
Fig. 4
Since the triangle is equilateral, $x=4$, whic... | 0.5 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,384 |
Task 6. (30 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 through the second is twice the direct route between them; the distance from the first station to the second through the third is $a$ km longer t... | # 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. 6).
 Solve the equation $x^{6}-20 x^{2}-\sqrt{21}=0$. | Solution.
Rewrite the equation as $x^{6}-21 x^{2}+x^{2}-\sqrt{21}=0$.
Then
$x^{2}\left(x^{4}-21\right)+\left(x^{2}-\sqrt{21}\right)=0$,
$x^{2}\left(x^{2}-\sqrt{21}\right)\left(x^{2}+\sqrt{21}\right)+\left(x^{2}-\sqrt{21}\right)=0$,
$\left(x^{2}-\sqrt{21}\right)\left(x^{4}+x^{2} \sqrt{21}+1\right)=0$
$x^{2}-\sqrt{... | {\\sqrt[4]{21}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,386 |
Problem 3. (15 points) Point $A$ lies on side $L M$ of triangle $K L M$ with an angle of $120^{\circ}$ at vertex $K$. Circles are inscribed in triangles $A K L$ and $A K M$ with centers $F$ and $O$ respectively. Find the radius of the circumcircle of triangle $F K O$, if $A F=3, A O=6$. | # Solution.
$\mathrm{L}$
Fig. 1
The center of the circle inscribed in an angle lies on the bisector of this angle, so rays $A F$ and $A O$ are the bisectors of $\angle L A K$ and $\angle ... | \sqrt{15} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,388 |
Problem 4. (20 points) A finite increasing sequence $a_{1}, a_{2}, \ldots, a_{n}$ ( $n \geq 3$ ) of natural numbers is given, and for all $k \leq n-2$, the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-1$ holds. The sequence must contain a term $a_{k}=2021$. Determine the maximum number of three-digit numbers, divisible by 25, t... | # Solution.
The final sequence can contain all three-digit numbers, as it can consist of a given number of natural numbers starting from the chosen number $a_{i}$.
We will prove that for any term of the arithmetic progression $1,2,3, \ldots$ defined by the formula for the $n$-th term $a_{n}=n$, the equality $a_{k+2}=... | 36 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,389 |
Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 8 straight roads extend, dividing the field into 8 equal sectors. Two geologists set out on a journey from their cabin at a speed of 5 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).
Fig. 2
By the cosine theorem: \( x^{2}=5^{2}+5^{2}-2 \cdot 5 \cdot 5... | 0.375 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,390 |
Task 6. (30 points) At the first deposit, equipment of the highest class was used, and at the second deposit, equipment of the first class was used, with the highest class being less than the first. Initially, $40 \%$ of the equipment from the first deposit was transferred to the second. Then, $20 \%$ of the equipment ... | # Solution.
Let there initially be $x$ units of top-class equipment at the first deposit and $y$ units of first-class equipment at the second deposit $(x1.05 y$, from which $y48 \frac{34}{67} .\end{array}\right.\right.$
This double inequality and the condition “x is divisible by 5” is satisfied by the unique value $x... | 60 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,391 |
Problem 3. (15 points) Point $A$ lies on side $L M$ of triangle $K L M$ with an angle of $60^{0}$ at vertex $K$. Circles are inscribed in triangles $A K L$ and $A K M$ with centers $F$ and $O$ respectively. Find the radius of the circumcircle of triangle $F K O$, if $A O=6, A F=3$. | # Solution.
Fig. 1
The center of the circle inscribed in an angle lies on the bisector of this angle, so the rays $A F$ and $A O$ are the bisectors of $\angle L A K$ and $\angle M A K$ resp... | 3\sqrt{5} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,392 |
Task 4. (20 points) A finite increasing sequence of natural numbers $a_{1}, a_{2}, \ldots, a_{n}(n \geq 3)$ is given, and for all $\kappa \leq n-2$ the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-2$ holds. The sequence must contain $a_{k}=2022$. Determine the maximum number of three-digit numbers, divisible by 4, that this seq... | # Solution.
Since it is necessary to find the largest number of three-digit numbers that are multiples of 4, the deviation between the members should be minimal. Note that the arithmetic
progression with a difference of $d=2$, defined by the formula $a_{k}=2 k$, satisfies the equality $a_{k+2}=3 a_{k+1}-2 a_{k}-2$.
I... | 225 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,393 |
Task 5. (20 points) In the center of a circular field stands a geologists' cabin. From it, 8 straight roads extend, dividing the field into 8 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).
Fig. 2
By the cosine theorem: \( x^{2}=4^{2}+4^{2}-2 \cdot 4 \cdot ... | 0.375 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,394 |
Task 1. (5 points) Solve the equation $x^{9}-22 x^{3}+\sqrt{21}=0$.
# | # Solution.
Rewrite the equation as $x^{9}-21 x^{3}-x^{3}+\sqrt{21}=0$.
$$
\begin{aligned}
& \text { then } \\
& x^{3}\left(x^{6}-21\right)-\left(x^{3}-\sqrt{21}\right)=0 \\
& x^{3}\left(x^{3}-\sqrt{21}\right)\left(x^{3}+\sqrt{21}\right)-\left(x^{3}-\sqrt{21}\right)=0 \\
& \left(x^{3}-\sqrt{21}\right)\left(x^{6}+x^{3... | {\sqrt[6]{21};\sqrt[3]{\frac{-\sqrt{21}\5}{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,396 |
Task 2. (10 points) Calculate $\left(\frac{2}{3}-\frac{4}{9}+\frac{8}{27}-\frac{16}{81}+\ldots\right) \cdot\left(\frac{3}{5}-\frac{9}{25}+\frac{27}{125}-\frac{81}{625}+\ldots\right)$. | # Solution.
Let's regroup the terms in parentheses
$$
\begin{aligned}
& \left(\frac{2}{3}+\frac{8}{27}+\ldots-\frac{4}{9}-\frac{16}{81}-\ldots\right) \cdot\left(\frac{3}{5}+\frac{27}{125}+\ldots-\frac{9}{25}-\frac{81}{625}-\ldots\right)= \\
& =\left(\frac{2}{3}+\frac{8}{27}+\ldots-\left(\frac{4}{9}+\frac{16}{81}+\ldo... | 0.15 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,397 |
Task 4. (20 points) It is known that the function $f(x)$ for each value of $x \in(-\infty ;+\infty)$ satisfies the equation $f(x)+(0.5+x) f(1-x)=1$. Find all such functions $f(x)$. | # Solution.
Substitute the argument $(1-x)$ into the equation and write the system of equations:
$\left\{\begin{array}{l}f(x)+(0.5+x) f(1-x)=1, \\ f(1-x)+(0.5+1-x) f(1-1+x)=1 ;\end{array}\left\{\begin{array}{l}f(x)+(0.5+x) f(1-x)=1, \\ f(1-x)+(1.5-x) f(x)=1 .\end{array}\right.\right.$
Solve the system by the method ... | f(x)={\begin{pmatrix}\frac{1}{0.5-x},x\neq0.5,\\0.5,0.50\end{pmatrix}.} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,398 |
Task 5. (20 points) A meter-long gas pipe has rusted in two places. Determine the probability that all three resulting sections can be used as offsets for gas stoves, if according to regulations, the stove should not be located closer than 25 cm to the main gas pipe.
# | # Solution.
Let the sizes of the parts into which the pipe was cut be denoted as $x, y$, and $(100-x-y)$.
Obviously, the values of $x$ and $y$ can take any values from the interval (0; 100). Then, the set of all possible combinations $(x; y)$ can be represented on the coordinate plane $OXY$ as a right-angled triangle... | \frac{1}{16} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,399 |
Task 6. (30 points) A lot consisting of three packages of shares from oil extraction companies - Razneft, Dvaneft, and Trineft - is up for auction. The total number of shares in the packages of Razneft and Dvaneft matches the number of shares in the Trineft package. The package of Dvaneft shares is 4 times cheaper than... | # Solution.
Let's introduce the following notations:
$x$ - the price of one share of Dvanefte,
$y$ - the price of one share of Raznefte,
$z$ - the price of one share of Trinefte,
$n-$ the number of shares in the Dvanefte package,
$m-$ the number of shares in the Raznefte package.
The other conditions of the prob... | 12.5to15 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,400 |
Task 1. (5 points) Solve the equation $x^{9}-22 x^{3}-\sqrt{21}=0$.
# | # Solution.
Rewrite the equation as $x^{9}-21 x^{3}-x^{3}-\sqrt{21}=0$.
$$
\begin{aligned}
& \text { then } \\
& x^{3}\left(x^{6}-21\right)-\left(x^{3}+\sqrt{21}\right)=0 \\
& x^{3}\left(x^{3}-\sqrt{21}\right)\left(x^{3}+\sqrt{21}\right)-\left(x^{3}+\sqrt{21}\right)=0 \\
& \left(x^{3}+\sqrt{21}\right)\left(x^{6}-x^{3... | {-\sqrt[6]{21};\sqrt[3]{\frac{\sqrt{21}\5}{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,401 |
Task 2. (10 points) Calculate $\left(\frac{3}{4}-\frac{9}{16}+\frac{27}{64}-\frac{81}{256}+\ldots\right) \cdot\left(\frac{2}{7}-\frac{4}{49}+\frac{8}{343}-\frac{16}{2401}+\ldots\right)$. | # Solution.
Rearrange the terms in parentheses $\left(\frac{3}{4}+\frac{27}{64}+\ldots-\frac{9}{16}-\frac{81}{256}-\ldots\right) \cdot\left(\frac{2}{7}+\frac{8}{343}+\ldots-\frac{4}{49}-\frac{16}{2401}-\ldots\right)=$ $=\left(\frac{3}{4}+\frac{27}{64}+\ldots-\left(\frac{9}{16}+\frac{81}{256}+\ldots\right)\right) \cdot... | \frac{2}{21} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,402 |
Task 3. (15 points) The bases $AB$ and $CD$ of trapezoid $ABCD$ are 41 and 24, 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 similar... | 984 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,403 |
Task 4. (20 points) It is known that the function $f(x)$ for each value of $x \in(-\infty ;+\infty)$ satisfies the equation $f(x)-(x-0.5) f(-x-1)=1$. Find all such functions $f(x)$. | Solution.
Substitute the argument $(-x-1)$ into the equation and write the system of equations:
$\left\{\begin{array}{l}f(x)-(x-0.5) f(-x-1)=1, \\ f(-x-1)-(-x-1-0.5) f(x+1-1)=1 ;\end{array}\left\{\begin{array}{l}f(x)-(x-0.5) f(-x-1)=1, \\ f(-x-1)+(x+1.5) f(x)=1 .\end{array}\right.\right.$
Solve the system by the meth... | f(x)={\begin{pmatrix}\frac{1}{0.5+x},x\neq-0.5,\\0.5,-0.50\end{pmatrix}.} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,404 |
Task 5. (20 points) A four-meter gas pipe has rusted in two places. Determine the probability that all three resulting sections can be used as offsets for gas stoves, if according to regulations, the stove should not be located closer than 1 m to the main gas pipe.
# | # Solution.
Let the sizes of the parts into which the pipe was cut be denoted as $x, y$, and $(400-x-y)$.
Obviously, the values of $x$ and $y$ can take any values from the interval (0; 400). Then, the set of all possible combinations $(x; y)$ can be represented on the coordinate plane $OXY$ as a right-angled triangle... | \frac{1}{16} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,405 |
Task 6. (30 points) A lot consisting of three packages of shares of oil companies - Razneft, Dvaneft, and Trineft - is up for auction. The total number of shares in the packages of Razneft and Dvaneft matches the number of shares in the Trineft package. The package of Dvaneft shares is three times cheaper than the pack... | # Solution.
Let's introduce the following notations:
$x$ - the price of one share of Dvanefte,
$y$ - the price of one share of Raznefte,
$z$ - the price of one share of Trinefte,
$n$ - the number of shares in the Dvanefte package,
$m$ - the number of shares in the Raznefte package.
The other conditions of the pr... | 15to25 | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,406 |
Task 1. (5 points) Solve the equation $x^{9}-21 x^{3}-\sqrt{22}=0$. Solution.
Rewrite the equation as $x^{9}-22 x^{3}+x^{3}-\sqrt{22}=0$. Then
$$
\begin{aligned}
& x^{3}\left(x^{6}-22\right)+x^{3}-\sqrt{22}=0, \\
& x^{3}\left(x^{3}-\sqrt{22}\right)\left(x^{3}+\sqrt{22}\right)+\left(x^{3}-\sqrt{22}\right)=0, \\
& \lef... | Answer. $\left\{\sqrt[6]{22} ; \sqrt[3]{\frac{-\sqrt{22} \pm 3 \sqrt{2}}{2}}\right\}$ | {\sqrt[6]{22};\sqrt[3]{\frac{-\sqrt{22}\3\sqrt{2}}{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,407 |
Problem 3. (15 points) In parallelogram $A B C D$, side $A D$ is divided into equal parts by points $A_{1}, A_{2}, \ldots, A_{2020}$. Point $E_{1}$ is the intersection point of lines $B A_{1}$ and $A C$. Determine what fraction of diagonal $A C$ segment $A E_{1}$ constitutes. | Solution.
Fig. 1
The points $A_{1}, A_{2}, \ldots, A_{2020}$ divide the side $A D$ into 2021 equal parts (Fig. 1). Connect points $B$ and $A_{1}$, and then draw lines parallel to $B A_{1}$ ... | \frac{1}{2022} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,408 |
Task 4. (20 points) At a refinery, a tank was filled with crude oil with a sulfur concentration of $2 \%$. Part of this oil was directed to production, and the same amount of oil with a sulfur concentration of $3 \%$ was added to the tank. Then, the same amount of oil as before was directed to production again, and thi... | # Solution.
Let $x$ be the fraction of oil that was sent to production twice ($x>0$). Then the sulfur balance equation in the oil will look as follows:
$$
\frac{2}{100}-\frac{2}{100} x+\frac{3}{100} x-\left(\frac{2}{100}-\frac{2}{100} x+\frac{3}{100} x\right) x+\frac{1.5}{100} x=\frac{2}{100}
$$
where $\frac{2}{100}... | \frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,409 |
Task 5. (20 points) A two-meter gas pipe has rusted in two places. Determine the probability that all three resulting sections can be used as offsets for gas stoves, if according to regulations, the stove should not be located closer than 50 cm to the main gas pipe.
# | # Solution.
Let the sizes of the parts into which the pipe was cut be denoted as $x, y$, and $(200-x-y)$.
Obviously, the values of $x$ and $y$ can take any values from the interval $(0 ; 200)$. Then, the set of all possible combinations $(x, y)$ can be represented on the coordinate plane $O X Y$ as a right-angled tri... | \frac{1}{16} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,410 |
Task 6. (30 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 through the second is three times longer than the direct route between them; the distance from the first station to the second through the third ... | # 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. 3).
 Solve the equation $x^{9}-21 x^{3}+\sqrt{22}=0$.
# | # Solution.
Rewrite the equation as $x^{9}-22 x^{3}+x^{3}+\sqrt{22}=0$.
then
$$
\begin{aligned}
& x^{3}\left(x^{6}-22\right)+x^{3}+\sqrt{22}=0, \\
& x^{3}\left(x^{3}-\sqrt{22}\right)\left(x^{3}+\sqrt{22}\right)+\left(x^{3}+\sqrt{22}\right)=0, \\
& \left(x^{3}+\sqrt{22}\right)\left(x^{6}-x^{3} \sqrt{22}+1\right)=0, \... | {-\sqrt[6]{22};\sqrt[3]{\frac{\sqrt{22}\3\sqrt{2}}{2}}} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,412 |
Problem 3. (15 points) In parallelogram $A B C D$, side $A D$ is divided into equal parts by points $A_{1}, A_{2}, \ldots, A_{2022}$. Point $E_{1}$ is the intersection point of lines $B A_{1}$ and $A C$. Determine what fraction of diagonal $A C$ segment $A E_{1}$ constitutes. | # Solution.
Fig. 1
Points $A_{1}, A_{2}, \ldots, A_{2022}$ divide side $A D$ into 2023 equal parts (Fig. 1). Connect points $B$ and $A_{1}$, and then draw lines parallel to $B A_{1}$ throug... | \frac{1}{2024} | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,413 |
Task 4. (20 points) At a refinery, a tank was filled with crude oil with a sulfur concentration of $1.5 \%$. Part of this oil was directed to production, and the same amount of oil with a sulfur concentration of $0.5 \%$ was added to the tank. Then, the same amount of oil as before was directed to production again, but... | # Solution.
Let $x$ be the fraction of oil that was sent to production twice ($x>0$). Then the balance equation for the amount of sulfur in the oil will look as follows:
$\frac{1.5}{100}-\frac{1.5}{100} x+\frac{0.5}{100} x-\left(\frac{1.5}{100}-\frac{1.5}{100} x+\frac{0.5}{100} x\right) x+\frac{2}{100} x=\frac{1.5}{1... | \frac{1}{2} | Algebra | math-word-problem | Yes | Yes | olympiads | false | 7,414 |
Task 5. (20 points) A three-meter gas pipe has rusted in two places. Determine the probability that all three resulting parts can be used as connections to gas stoves, if according to regulations, the stove should not be located closer than 75 cm to the main gas pipe.
# | # Solution.
Let's denote the sizes of the parts into which the pipe was cut as $x, y$, and $(300-x-y)$.
Obviously, the values of $x$ and $y$ can take any values from the interval $(0; 300)$. Then, the set of all possible combinations $(x; y)$ can be represented on the coordinate plane OXY as a right-angled triangle w... | \frac{1}{16} | Combinatorics | math-word-problem | Yes | Yes | olympiads | false | 7,415 |
Task 6. (30 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 through the second is three times longer than the direct route between them; the distance from the first station to the second through the third ... | # 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. 3).
![](https://cdn.mathpix.com/cropped/2024_05_06_6d5fca905c76f828a3d8g-37.jpg?height=343&width=596&top_left_y=354&top_left_x=730... | 0<<60;33 | Geometry | math-word-problem | Yes | Yes | olympiads | false | 7,416 |
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