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Problem: Define the type of the singular point \( z= \) 0 for the function: \[ z^{3} \sin \frac{3}{z^{2}} \text {. } \]

10.2.2 Singular points and residues

1.27 $

Problem: Find all isolated singular points and determine their type. \[ \frac{1}{z\left(e^{z}-1\right)} \text {. } \]

10.2.3 Singular points and residues

1.27 $

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

10.2.4 Singular points and residues

0 $

Problem: Using the argument principle, find the number of roots of the given equation that lie in the right halfplane: \[ z^{4}+2 z^{3}+3 z^{2}+z+2=0 . \]

10.6.8 Analytic functions

4.57 $

Problem: Using the argument principle, find the number of roots of the given equation lying in the right halfplane: \( 2 z^{3}-z^{2}-7 z+5=0 \).

10.6.9 Analytic functions

3.81 $

Problem: Using the argument principle, find the number of roots of the given equation, lying in the right halfplane: \( z^{5}+5 z^{4}-5=0 \).

10.6.10 Analytic functions

3.81 $

Problem: Using the argument principle, find the number of roots of the given equation, lying in the right halfplane: \( z^{12}-z+1=0 \).

10.6.11 Analytic functions

3.3 $

Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0 ; R) \). a) \( z^{3}+z+1=0, \quad R=\frac{1}{2} \). b) \( z^{2}-\cos z=0, \quad R=2 \).

10.6.12 Analytic functions

3.05 $

Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0 ; R) \). a) \( z^{5}+z^{2}+1=0, \quad R=2 \). b) \( z^{4}-\sin z, \quad R=\pi \).

10.6.13 Analytic functions

3.05 $

Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0, R) \). a) \( z^{8}+6 z+10=0 \), \( R=1 \). b) \( \cosh z=z^{2}-4 z, \quad R=1 \).

10.6.14 Analytic functions

3.05 $

Problem: Using Rouche's theorem, find the number of roots of the equation in disk \( D(0 ; R) \). a) \( z^{8}-6 z^{6}-z^{3}+2=0, \quad R=1 \). b) \( 2^{z}=4 z, \quad R=1 \).

10.6.15 Analytic functions

3.05 $

Problem: On the side \( A C \) of the triangle \( A B C \) the point \( D \) is chosen. Circles, inscribed in triangles \( A B D \) and \( C B D \), are tangent to the segment \( B D \) at points \( P \) and \( Q \). What is the minimum natural value of \( x \) for which we can determine the relation \( A D: D C \) one-valued, if it is given that \( A B=21, B C=27, A C=30, P Q=x \) ?

12.1.9 Olympic geometry

3.05 $

Problem: The circle \( \Omega \) with radius \( R=1 \) and two lines, intersecting at the angle \( \alpha=30^{\circ} \), were constructed on the plane, moreover, one of these lines passes through the centre of the circle \( \Omega \), the other one is tangent to the circle \( \Omega \). How many different circles are there on the plane, each of which is tangent to the circle \( \Omega \) and both mentioned lines?

12.1.10 Olympic geometry

7.62 $

Problem: Cut a regular pentagon into five equal triangles and one smaller regular pentagon.

12.1.11 Olympic geometry

2.54 $

Problem: Square \( A B C D \) is inscribed in the circle \( \omega \). On the smaller \( \operatorname{arc} C D \) of the circle \( \omega \) an arbitrary point \( M \) is chosen. Inside the square such points as \( K \) and \( L \) are marked inside the square so, that \( K L M D \) is a square. Find \( \angle A K D \).

12.1.12 Olympic geometry

2.54 $

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