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Problem: Find all the values of the root \( \sqrt[5]{32} \).

10.3.8 Operations with complex numbers

3.08 $

Problem: Solve the equation in complex numbers: \[ \sin z-i \cos z=1 \text {. } \]

10.3.9 Operations with complex numbers

0.77 $

Problem: A polynomial and one of its roots are given: \[ P(z)=a z^{4}+b z^{3}+c z^{2}+d z+e, \quad z_{1}=\frac{3+3 i \sqrt{3}}{2} . \] 1) Find all the other roots of the polynomial \( P(z) \). 2) Expand \( P(z) \) into irreducible factors in \( \mathbb{C}, \mathbb{R} \). \[ a=1, \quad b=-2, \quad c=8, \quad d=3, \quad e=18 . \]

10.3.11 Operations with complex numbers

3.08 $

Problem: Calculate the curvilinear integral along the segment \( L: z_{1} z_{2} \), where \[ \int_{L} z \operatorname{Re} z d z, \quad L: z_{1}=1 \text { to } z_{2}=-1 . \]

10.1.10 Integral of a complex variable

0.51 $

Problem: In Hilbert space \( L_{2}([0 ; 1]) \) orthogonalize the system of three vectors: \[ f_{0}(t)=1, f_{1}(t)=t, f_{2}(t)=e^{t} \]

19.1.3.1 Orthogonal systems

2.57 $

Problem: Is the system \( \left\{\sin \frac{\mathrm{nt}}{2}\right\}_{\mathrm{n}=1}^{\infty} \) orthogonal on the space \( \mathrm{L}_{2}([0, \pi]) \) ?

19.1.3.2 Orthogonal systems

2.57 $

Problem: Investigate the convergence of the sequence of functions in space: \[ f_{n}(x)=\sqrt{n}(1-x)^{n} \text { in } L_{2}[0,1] \text {. } \]

19.1.2.1 Convergence in metric spaces

1.54 $

Problem: Will the function \( d(x, y) \) \( =\int_{0}^{1} \frac{|x(t)-y(t)|}{e^{t}} d t \) set a metric on \( C([0,1]) \) ? If this function defines a metric, will the sequence \( \left\{x_{n}\right\} \), defined by the formula, \( x_{n}(t)=\left\{\begin{array}{ll}n t, & t \in\left[0 ; \frac{1}{n}\right] \\ 1, & t \in\left[\frac{1}{n} ; 1\right]\end{array}\right. \) be 1) Fundamental; 2) Converging relative to the given metric?

19.1.2.2 Convergence in metric spaces

4.62 $

Problem: Does the sequence of the function \( f_{n}(x)= \) \( n \sqrt{x^{n}}(1-\sqrt{x})^{n} \) converge on the space \( C([0,1]) \) ?

19.1.2.3 Convergence in metric spaces

2.57 $

Problem: Does the sequence of elements converge on the space \( l_{1} \) ? \[ x_{n}=\left(\frac{1-n^{2}}{n^{2}}, \frac{2-n^{2}}{n^{2}}, \ldots, \frac{n-n^{2}}{n^{2}}, 0,0, \ldots, 0, \ldots\right) \text {. } \]

19.1.2.4 Convergence in metric spaces

3.08 $

Problem: What is the minimum height \( h \) of a box, having the shape of a rectangular parallelepiped with a length of 29 and a width of 27 that can fit two balls with radii \( R=12 \) and \( r=10 \) ?

12.1.8 Olympic geometry

5.14 $

Problem: What is the minimum height \( h \) of a box, having the form of a cuboid, with a length of 21 and a width of 18, which can fit two balls with radii \( R=9 \) and \( r=6 \) ?

12.1.3 Olympic geometry

5.14 $

30 Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{l} y=x^{3} \\ x+y=2 \\ y=4 \end{array}\right. \]

9.7.4 The area of a region

1.03 $

Problem: Calculate the area of the domain \( D \), bounded by the given lines: \( D:\left\{\begin{array}{l}y=x^{3} \\ y=4, \quad y=x\end{array}\right. \)

9.7.5 The area of a region

1.28 $

(1) Problem: Calculate the area of the domain \( D \), bounded by the given lines: \[ D:\left\{\begin{array}{l} y^{2}-6 y+x^{2}=0 \\ y=\sqrt{3} x, \quad x=0 \end{array}\right. \]

9.7.7 The area of a region

1.54 $

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