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Problem: Find the solution of the inhomogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{\bar{x}}=\left(\begin{array}{ll} 0 & 3 \\ 3 & 0 \end{array}\right) \bar{x}+\left(\begin{array}{c} -1 \\ t \end{array}\right) \\ \bar{x}=\left(\begin{array}{l} 0 \\ 1 \end{array}\right) \end{array}\right. \]

8.3.8 Systems of ordinary differential equations

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Problem: Solve the system of homogeneous linear differential equations, using Euler's method: \[ \dot{x}=\left(\begin{array}{ccc} 4 & -5 & 7 \\ 1 & -4 & 9 \\ -4 & 5 & 0 \end{array}\right) \cdot X \text {. } \]

8.3.9 Systems of ordinary differential equations

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Problem: Find the general solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{c} x^{\prime}=7 x+y \\ y^{\prime}=-z \\ z^{\prime}=2 z+y-2 x \end{array}\right. \]

8.3.10 Systems of ordinary differential equations

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Problem: Find the general solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=2 x-y-z \\ \dot{y}=3 x-2 y-3 z \\ \dot{z}=-x+y+2 z \end{array}\right. \]

8.3.11 Systems of ordinary differential equations

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Problem: Find the general solution of the linear homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=8 x-3 y \\ \dot{y}=2 x+y \end{array}\right. \]

8.3.12 Systems of ordinary differential equations

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Problem: Find the general solution of the linear homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=-y \\ \dot{y}=x \end{array} .\right. \]

8.3.13 Systems of ordinary differential equations

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Problem: Find the general solution of the homogeneous system of differential equations: \[ \left\{\begin{array}{c} \dot{x}=x+2 y-z \\ \dot{y}=x+y+z \\ \dot{z}=x-z \end{array}\right. \]

8.3.14 Systems of ordinary differential equations

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Problem: Let's prove that the set of all intervals of \( R \) with rational ends is countable.

16.1 Topology

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Problem: Can the intersection of two perfect sets be imperfect?

16.2 Topology

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Problem: Show that any metric space satisfies the first axiom of countability in any of its points.

16.3 Topology

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Problem: Show that any group with a discrete topology is a topological group.

16.4 Topology

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Problem: How many solutions in natural numbers does the system have: \[ \left\{\begin{array}{l} x+y=1999 \\ u+v=2000 \end{array}\right. \]

6.6.4 Combinatorics

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Problem: How many diagonals are there in a convex 1999-gon?

6.6.5 Combinatorics

1.29 $

Problem: What is the remainder of the division of \( x_{2018}+ \) \( +x_{2019} \) by 9 , if \( x_{1}=x_{2}=1, x_{2 n+1}=2 x_{2 n}+1 \), \( x_{2 n+2}=x_{1}++x_{2}+\cdots+x_{2 n+1} \).

12.2.12 Number theory

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Problem: Find the minimum of \( x^{2}+y^{2}-4 y \) under the condition \[ |4 x-3 y|+5 \sqrt{x^{2}+y^{2}-20 y+100}=30 . \]

12.3.2.1 Geometric

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