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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=-2 x+\frac{1}{3} y, \quad y^{\prime}=-2 x+\frac{1}{2} y . \]

8.4.1.12 Stability of the systems of equations

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Problem: Determine the character of the resting point of the following system: \[ x^{\prime}=-y, \quad y^{\prime}=x-2 y \text {. } \]

8.4.1.13 Stability of the systems of equations

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Problem: Reduce the equation of the second-order curve \( f(x, y)=0 \) to canonical form and find its points of intersection with the straight line \( A x+B y+C=0 \). \( 2 x^{2}+4 x+y^{2}-2=0, \quad 2 x+y-2=0 \).

3.1.12 Curves of the 2-nd order

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Problem: Solve the Cauchy problem, using the operational method: \[ y^{\prime \prime}-5 y^{\prime}+6 y=6 t-5, \quad y(0)=1, \quad y^{\prime}(0)=3 \text {. } \]

8.2.1.9 Differential equations

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Problem: Show that the gradient field of the scalar field \( \varphi=6 x^{3} y-2 x y^{4}+z^{4} x^{2} y \) is irrotational.

1.12.3 Vector analysis

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Problem: Are the vectors \( \overrightarrow{c_{1}} \) and \( \overrightarrow{c_{2}} \), formed by the vectors \( \vec{a} \) and \( \vec{b} \) collinear? \[ \begin{array}{ll} \vec{a}=\{-1 ; 2 ;-1\}, & \vec{b}=\{2 ;-7 ; 1\}, \\ \overrightarrow{c_{1}}=6 \vec{a}-2 \vec{b}, & \overrightarrow{c_{2}}=\vec{b}-3 \vec{a} . \end{array} \]

1.1.14 Vector Algebra

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Problem: Find the cosine of the angle between the vectors \( \overrightarrow{A B} \) and \( \overrightarrow{A C} \). \[ A(0 ; 2 ;-4), \quad B(8 ; 2 ; 2), \quad C(6 ; 2 ; 4) \text {. } \]

1.1.15 Vector Algebra

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Problem: Prove the identity: \( \overline{B \backslash A}=\bar{B} \cup A \).

6.8.9 Set theory

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Problem: Prove equalities using the properties of set operations and definitions of operations. \[ C \subseteq D \Rightarrow A \times C \subseteq B \times D \text {. } \]

6.8.11 Set theory

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Problem: Find residues: \[ \operatorname{Res}_{-3} \frac{z+2}{z(z+3)^{2}} . \]

10.2.4 Singular points and residues

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Problem: Calculate the sum: \[ \begin{array}{l} \left(\frac{1+2}{3}+\frac{4+5}{6}+\frac{7+8}{9}+\cdots+\frac{2017+2018}{2019}\right)+ \\ +\left(1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{673}\right) . \end{array} \]

12.5.2 Olympic algebra

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Problem: Simplify the sum: \[ 1-\frac{1}{5}+\frac{1}{4}-\frac{1}{25}+\frac{1}{16}-\frac{1}{125}+\frac{1}{64}-\cdots-\frac{1}{5^{n}}+\frac{1}{4^{n}} . \] Ellipses and summation signs can't be used in the answer.

12.5.3 Olympic algebra

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Problem: Let the set \( A \subset \mathbb{R} \) is open, and \( B \subset \mathbb{R} \) is closed. Prove that the set \( A \backslash B \) is open, and \( B \backslash A \) is closed.

19.7.3 Properties of sets

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Problem: Prove that the set of rational numbers \( \mathbb{Q} \) is countable.

19.7.10 Properties of sets

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Problem: Using the theorem of nested intervals, prove that the segment is uncountable.

19.7.11 Properties of sets

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