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Problem: Let \( x^{0}, x^{1} \in l_{1} \). Find \( \varepsilon- \) neighbourhood of the point \( x^{0} \), where the point \( x^{1} \) lies. \[ l_{1}=\left\{x=\left(x_{1}, \ldots, x_{n}, \ldots, \ldots\right)\left|\sum_{n=1}^{\infty}\right| x_{n}\left|<\infty, \rho_{1}(x, y)=\sum_{n=1}^{\infty}\right| x_{n}-y_{n} \mid\right\}, x^{0}=(0, \ldots, 0, \ldots), x^{1}=\left(1, \frac{1}{4}, \ldots, \frac{1}{n^{2}}, \ldots\right) \text {. } \]

19.1.1.10 Properties of metric spaces

2.5 $

Problem: Prove that the Apollonius identity holds in Euclidean space: \[ 2\|z-x\|^{2}+2\|z-y\|^{2}=\|x-y\|^{2}+4\left\|z-\frac{x+y}{2}\right\|^{2} . \]

19.2.1.5 Properties of normed spaces

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Problem: Find the norm of \( x(t)=\sin ^{3} t \) on \( C_{1}[0 ; 1] \).

19.2.2.8 Convergence in normed spaces

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Problem: Find the limit of the sequence in the space \( C^{1}[0 ; 1] \) : \[ f_{n}(x)=\int_{0}^{1} x\left(t^{n}\right) d t \]

19.2.2.9 Convergence in normed spaces

4.99 $

Problem: Prove that the functional \[ f: C[0,2] \rightarrow \mathbb{R} ; f(x)=\int_{0}^{2}(t-1)^{2} x(t) d t \] is linear, continuous, and find its norm.

19.3.8 Linear operators

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Problem: Find the norm of the function \( y=\tan x \) in the space \( L_{3}\left[\frac{\pi}{4}, \frac{\pi}{3}\right] \).

19.3.9 Linear operators

2.5 $

Problem: Find the solution of the second order linear equation, satisfying the given initial conditions (the Cauchy problem): \[ \begin{array}{l} u_{x x}+2 \sin x \cdot u_{x y}-\cos ^{2} x \cdot u_{y y}+u_{x}+ \\ +(\sin x+\cos x+1) u_{y}=0, \\ \left.u(x, y)\right|_{y=-\cos x}=1+2 \sin x, \\ \left.u_{y}(x, y)\right|_{y=-\cos x}=\sin x . \end{array} \]

11.5.3.8 With variable coefficients

6.24 $

Problem: Given the system of three linear equations with three unknowns. Required: 1) to find its solution using Cramer's formulas; 2) to solve by the Gauss method; 3 ) to write the system in matrix form and solve it using matrix calculus. Check the correctness of the calculation of the inverse matrix using matrix multiplication. \[ \left\{\begin{array}{c} -3 x_{1}+5 x_{2}-6 x_{3}=-5 \\ 2 x_{1}-3 x_{2}+5 x_{3}=8 \\ x_{1}+4 x_{2}-x_{3}=1 \end{array}\right. \]

1.5.13 Systems of algebraic equations

2.5 $

Problem: Given a system of three linear equations with three unknowns. Required: 1) to find its solution using Cramer's formulas; 2) to solve by the Gauss method; 3) to write the system in matrix form and solve it using matrix calculus. Check the correctness of the calculation of the inverse matrix using matrix multiplication. \[ \left\{\begin{array}{c} 2 x_{1}+4 x_{2}-3 x_{3}=-10 \\ -x_{1}+5 x_{2}-2 x_{3}=5 \\ 3 x_{1}-2 x_{2}+4 x_{3}=3 \end{array}\right. \]

1.5.14 Systems of algebraic equations

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Problem: Prove that every ordered field contains an infinite number of elements.

1.6.31 Fields, Groups, Rings

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Problem: Prove that it is impossible to introduce the structure of an ordered field on the field of complex numbers.

1.6.32 Fields, Groups, Rings

2.5 $

Problem: Give an example of a linear operator whose matrix is not diagonal in any basis.

1.7.2 Linear transformations

1.25 $

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 \( \ell: A x+B y+C=0 \). \( 2 x^{2}+y^{2}-12 x+10=0 \), (e) \( x+y-2=0 \).

3.1.11 Curves of the 2-nd order

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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: By the coordinates of the vertices of the pyramid \( A_{1} A_{2} A_{3} A_{4} \) find: 1) the length of the edge \( A_{1} A_{2} ; 2 \) ) the angle between edges \( A_{1} A_{3} \) and \( \left.A_{1} A_{4} ; 3\right) \) the angle between faces \( A_{1} A_{2} A_{3} \) and \( \left.A_{1} A_{2} A_{4} ; 4\right) \) the equation of the straight line passing through the vertices \( A_{4} \) and the center of gravity of the face \( A_{1} A_{2} A_{3} ; 5 \) ) the length and the equation of the height from the vertex \( A_{4} \) to the face \( \left.A_{1} A_{2} A_{3} ; 6\right) \) the distance between crossing edges \( A_{1} A_{2} \) and \( A_{3} A_{4} \). \[ A_{1}(1 ; 1 ; 2), \quad A_{2}(-1 ; 1 ; 3), \quad A_{3}(2 ;-2 ; 4), \quad A_{4}(-1 ; 0 ;-2) \text {. } \]

3.3.3 Line and plane in space

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