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

6.2.15 Binary relations

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Problem: Calculate the integral using residues: \[ \int_{0}^{2 \pi} \frac{d t}{7+\sqrt{13} \sin t} \text {. } \]

10.7.17 Calculating integrals of a real variable using residues

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Problem: Calculate the integral of the complex variable: \[ \oint_{|z-3|=1} \frac{1-\cos z}{z^{2}-\pi^{2}} d z . \]

10.1.27 Integral of a complex variable

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Problem: Prove the statement using mathematical induction: (8 \( { }^{n}-1 \) ) is a multiple of 7 for all natural \( n \geq 1 \).

6.6.20 Combinatorics

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Problem: Let \( \overline{x y} \) and \( \overline{y x} \) be two two-digit integers. Prove that their sum is a composite number.

12.2.29 Number theory

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Problem: Prove that the sum of the squares of 3 consecutive integers can't be a perfect square.

12.2.40 Number theory

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Problem: From the formula go to the truth table and use the table to construct the perfect CNF. \[ \bar{x} \vee \overline{(y z)} x \text {. } \]

6.3.21 Boolean algebra

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Problem: Convert the formula to Zhegalkin polynomial. \[ (x \overline{y z}) \mid(x \mid(y \mid z)) \text {. } \]

6.3.22 Boolean algebra

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Problem: Prove the statement using mathematical induction: \( \left(n^{3}+11 n\right) \vdots 6 \), for all natural numbers \( n \geq 1 \).

6.6.33 Combinatorics

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Problem: Calculate the indefinite inetrgal: \[ \int \frac{2 x+3}{\sqrt[4]{x^{2}+3 x+1}} d x \]

9.3.1.14 Indefinite integrals

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30 Problem: Calculate the improper integral: \[ \int_{-\infty}^{+\infty} \frac{d x}{x^{2}+2 x+2} \]

9.3.2.6 Improper integrals

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3) Problem: Examine the improper integral of the unbounded function for convergence: \[ \int_{0}^{8} \frac{d x}{2 x^{3}+5 \sqrt{x}} \]

9.3.2.8 Improper integrals

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(3) Problem: Examine the improper integral for convergence: \[ \int_{0}^{+\infty} \frac{2 x d x}{1+x^{2}} \]

9.3.2.9 Improper integrals

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Problem: Examine the improper integral for convergence: \[ \int_{0}^{2} \frac{d x}{x} \]

9.3.2.11 Improper integrals

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Problem: Prove the equality: \[ \int_{0}^{1} x^{m}(1-x)^{n} d x=\int_{0}^{1}(1-x)^{m} x^{n} d x . \]

9.3.3.10 Definite Integrals

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