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Problem: Find the general integral of the non-linear system: \[ \frac{d x}{z^{2}-y^{2}}=\frac{d y}{z}=\frac{d z}{-y} . \]

8.3.18 Systems of ordinary differential equations

2.5 $

Problem: Separate the real and imaginary parts of the function. \[ f(z)=3 z^{3}+3 z^{2}+2 z+1, \quad z=x+i y . \]

10.3.4 Operations with complex numbers

0.5 $

Problem: Separate the real and imaginary parts of the function. \[ f(z)=2 z^{3}-3 z^{2}+2 z+1, \quad z=x+i y . \]

10.3.5 Operations with complex numbers

0.5 $

Problem: Prove that function \( f(z)=u(x, y)+i v(x, y) \) is differentiable on the entire complex plane and find its derivative \( f^{\prime}(z) \). \[ \begin{array}{l} f(z)=\left(3 x^{3}-6 x y^{2}+2 x^{2}-2 y^{2}-3 x+1\right)+ \\ +i\left(6 x^{2} y-2 y^{3}+4 x y-3 y\right) . \end{array} \]

10.6.3 Analytic functions

2 $

Problem: Prove that function \( f(z)=u(x, y)+i v(x, y) \) is differentiable on the entire complex plane and find its derivative \( f^{\prime}(z) \). \[ \begin{array}{l} f(z)=\left(2 x^{3}-6 x y^{2}-3 x^{2}+3 y^{2}+x+2\right)+ \\ +i\left(6 x^{2} y-2 y^{3}-6 x y+y\right) . \end{array} \]

10.6.4 Analytic functions

2 $

Problem: Prove that there is an analytic function with a given real \( (u(x, y)) \) or imaginary \( (v(x, y)) \) part and restore it under the given condition \( f\left(z_{0}\right)=z_{1} \). \[ \begin{array}{l} v(x, y)=3 x^{2} y-y^{3}-6 x y-3 y+2, \\ f(i)=3-2 i . \end{array} \]

10.6.5 Analytic functions

3 $

Problem: Prove that there is an analytic function with a given real \( (u(x, y)) \) or imaginary \( (v(x, y)) \) part and restore it under the given condition \( f\left(z_{0}\right)=z_{1} \). \[ \begin{array}{l} v(x, y)=3 x^{2} y-y^{3}+6 x y+3 y+2, \\ f(-i)=-3 . \end{array} \]

10.6.6 Analytic functions

3 $

Problem: Calculate the value of the derivative of the function at the given point \( z_{0}=x_{0}+i y_{0} \). \[ f(z)=2 z^{3}-3 z^{2}+2 z+1, \quad z_{0}=-1+3 i . \]

10.3.6 Operations with complex numbers

1.25 $

Problem: Calculate the value of the derivative of the function at the given point \( z_{0}=x_{0}+i y_{0} \). \[ f(z)=3 z^{3}+3 z^{2}+2 z+1, \quad z_{0}=-2-2 i . \]

10.6.7 Analytic functions

1.25 $

Problem: Write the following statements in the language of predicate logic: "For any three numbers, if their product is odd, then all three numbers are odd".

6.5.4 Predicate calculus

0.75 $

Problem: Indicate all subformulas, as well as the scope of quantifications, free and bound occurrences of all variables in the following formulas: \[ S(t, w) \vee \exists x \forall w[(Q(x, w) \rightarrow P(x)) \rightarrow R(w)] \text {. } \]

6.5.5 Predicate calculus

1.25 $

Problem: Perform skolem normal form of the following formulas presented in preliminary form: \[ \forall t \exists x \forall w \exists y[(P(t, w) \& Q(x, y) \rightarrow S(y)) \rightarrow R(x)] . \]

6.5.6 Predicate calculus

1.25 $

Problem: Create a truth table, Karnaugh map, perfect disjunctive form (PDNF) and a corresponding combinational circuit for the following options of a logical function (LF), given by a list of unit sets: \[ 1,4,5,7,9,13,14,15 \text {. } \]

6.3.7 Boolean algebra

3.75 $

Problem: Construct the truth table, the Karnaugh map, the perfect disjunctive (PDNF) and a corresponding combinational circuit for the following options of logical functions (LF), given by a list of unit sets: \[ 0,2,5,7,12,13,14,15 \text {. } \]

6.3.8 Boolean algebra

3.75 $

Problem: Represent the Boolean function as a PDNF and draw a circuit, realizing this function. \[ y=\overline{x_{1} \vee x_{2} \vee x_{1} \wedge x_{2}} \vee x_{3} . \]

6.3.6 Boolean algebra

2.5 $

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