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Problem: Solve an algebraic equation in finite fields \( \mathbb{Z}_{5} \) and \( \mathbb{Z}_{7} \). \[ x^{3}+x^{2}+3 x+3=0 \text {. } \]

1.10.5 Polynomials

3.05 $

Problem: Decompose the polynomial \( g(x)=1+g_{1} x+ \) \( +g_{2} x^{2}+g_{3} x^{3}+g_{4} x^{4}+x^{5} \in \mathbb{Z}_{2}[x] \) into a product of irreducible factors. \[ g_{1} g_{2} g_{3} g_{4}=1100 \text {. } \]

1.10.6 Polynomials

2.03 $

Problem: Represent the following function as a Fourier integral, extending it evenly to the interval \( (-\infty, 0) \). \[ f(x)=\left\{\begin{array}{ll} 1, & \text { if } 0 \leq x \leq 1 \\ 0, & \text { if } x>1 \end{array} .\right. \]

2.6.1.7 Fourier integral

2.54 $

Problem: Find the equation of the parabola if its focus \( F(2,5) \) and two points \( A(5,1) \) and \( B(2,3) \), that belong to this parabola, are known.

3.1.10 Curves of the 2-nd order

1.27 $

Problem: Given the equations of a side of a triangle and the bisectors of two angles, adjacent to this side. Find the equations of the other two sides of the triangle. The side: \( 4 x+3 y=0 \), bisectors: \( 7 x+4 y+5=0 \), \( y+4=0 \).

3.2.2 Lines on a plane

2.54 $

Problem: On the sides \( A B \) and \( A C \) of the given triangle \( A B C \) points \( M \) and \( N \) are taken respectively, that divide the segments \( A B \) and \( A C \) with respect to \( 3: 1 \) counting from the vertex \( A \), and the point \( Q \) is the midpoint of segment \( B C \). Prove that the straight lines \( A Q, B N, C M \) intersect at one point. Given the coordinates of the vertices of the triangle \( A B C: A(-3,-1), B(1,-5), C(9,3) \).

3.2.3 Lines on a plane

2.54 $

Problem: Find the equation of the straight line at the distance \( \rho \) from the point \( A \) and the component with the axis \( O X \) twice the size of the angle of the straight line \( l \), if: \[ A(-1,2), \quad \rho=\sqrt{34}, \quad l: 2 x-6 y+5=0 . \]

3.2.4 Lines on a plane

2.54 $

Problem: Prove in propositional calculus: \[ (F \supset G) \supset((F \supset \neg G) \supset \neg F) . \]

6.1.1.9 Propositional calculus

2.54 $

Problem: The binary relation \( P \) is given; find its domain and range. Check by the definition whether relation \( P \) is reflexive, symmetric, antisymmetric, transitive. \[ P \subseteq \mathbb{Z}^{2}, P=\{(x, y) \mid x=-y\} . \]

6.2.12 Binary relations

1.52 $

Problem: Two finite sets are given: \( A=\{a, b, c\}, B=\{1,2,3,4\} \); binary relations \( P_{1} \subseteq A \times B, P_{2} \subseteq B^{2} \). Plot \( P_{1}, P_{2} \) graphically. Find \( P=\left(P_{2} \circ P_{1}\right)^{-1} \). Write the domain and range of all three relations: \( P_{1}, P_{2}, P \). Construct the matrix \( \left[P_{2}\right] \), use it to check whether the relation \( P_{2} \) is reflexive, symmetric, antisymmetric, transitive. \[ \begin{array}{l} P_{1}=\{(a, 3),(b, 4),(b, 3),(c, 1),(c, 2),(c, 4)\} ; \\ P_{2}=\{(1,2),(1,3),(1,4),(2,3),(4,3),(4,2)\} . \\ A=\{a, b, c\}, B=\{1,2,3,4\}, P_{1} \subseteq A \times B, P_{2} \subseteq B^{2} . \end{array} \]

6.2.13 Binary relations

3.81 $

Problem: Find the power of the set of Boolean functions of \( n \) variables \( A=T_{o} \cup T_{1} \cup S \), where \( S, T_{0}, T_{1} \) are Post classes.

6.3.9 Boolean algebra

3.05 $

Problem: Find the power of the set of Boolean functions of \( n \) variables \( A=T_{0} \cup T_{1} \cup L \cup S \), where \( L, S, T_{0}, T_{1} \) are Post classes.

6.3.10 Boolean algebra

4.32 $

Problem: Having created the truth tables of the Boolean functions \( f_{1} \) and \( f_{2} \), check them for equivalence and duality. \[ f_{1}=x \&(y \sim z) ; f_{2}=((x \& y) \sim(x \& z)) \sim x . \]

6.3.11 Boolean algebra

1.52 $

Problem: Using the principle of duality, construct a formula that implements the function dual to the function \( f \), and make sure that the resulting formula is equivalent to the formula \( g \). \[ \begin{array}{l} f=(x \vee y \vee \bar{z}) \& \bar{t} \vee y \vee z ; \\ g=(\bar{x} \vee y \vee z) \overline{\&} t \vee x \& y \overline{\&} z . \end{array} \]

6.3.12 Boolean algebra

2.54 $

Problem: List all dummy variables of function \( f \). 1) \( f\left(x^{3}\right)=(10101010) \); 2) \( f\left(x^{4}\right)=(0101111101011111) \).

6.3.13 Boolean algebra

1.27 $

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