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Problem: Let's prove the relation \( X \times Y=A \times B \Leftrightarrow X=A \), \( Y=B \).

16.9 Topology

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Problem: Find all the values of a, for which the system of inequalities \[ \left\{\begin{array}{l} x^{2}+(5 a+2) x+4 a^{2}+2 a<0 \\ x^{2}+a^{2}=4 \end{array}\right. \] has at least one solution.

17.1 USE problems

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Problem: Solve the inequality for all non-negative values of the parameter a: \[ a^{3} x^{4}+2 a^{2} x^{2}-8 x+a+4 \geq 0, \quad a \geq 0 . \]

17.2 USE problems

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Problem: Solve the inequality, using the majorant method: \[ \cos x-y^{2}-\sqrt{y-x^{2}-1} \geq 0 . \]

17.4 USE problems

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Problem: Solve the inequality, using the majorant method: \[ -|y|+x-\sqrt{x^{2}+y^{2}-1} \geq 1 . \]

17.5 USE problems

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Problem: Solve the problem, using the majorant method: \[ \log _{\frac{1}{2}}(x+1)+\cos ^{-1}\left(x+y^{2}\right) \leq-1 . \]

17.6 USE problems

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Problem: Solve the equation: \[ 2 \log _{6}(\sqrt{x}+\sqrt[4]{x})=\log _{4} x . \]

17.8 USE problems

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Problem: Solve the system of equations: \[ \left\{\begin{array}{l} x-2 y+\frac{3}{x-2 y}=4 \\ \frac{y}{x-2 y-3}=5 \end{array}\right. \]

17.11 USE problems

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Problem: Solve the system of equations: \[ \left\{\begin{array}{l} \sqrt{3 x-5}=\sqrt{3 y+7} \\ y^{2}+x=10 \end{array}\right. \]

17.12 USE problems

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Problem: Solve the irrational equation: \[ \left(x^{2}-4\right) \sqrt{3+5 x-2 x^{2}}=0 \text {. } \]

17.13 USE problems

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Problem: Solve the irrational equation: \[ \sqrt{x-1}+\sqrt{x+2}=\sqrt{x+34}-\sqrt{x+7} . \]

17.14 USE problems

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Problem: Find all values of the parameter a, for which 1) the equation \( a x^{2}+x+17=0 \) has only one root; 2) the equation \( a x^{2}+(5-3 a) x-a=0 \) has two roots of different signs.

17.15 USE problems

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Problem: Find all values of the parameter a, for which 1) the equation \( 4 x^{2}+4 x=a^{2}-1 \) has two different positive roots. 2) the equation \( (a-2) x^{2}+2(a-2) x+ \) \( +2=0 \) has no roots.

17.16 USE problems

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Problem: Determine the number of roots of the equation for each value of the parameter a. \[ a x^{2}+(a+1)^{2} x+a+2=0 . \]

17.17 USE problems

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Problem: Find all values of \( a \) for which the sum of the real roots' squares of the equation \[ x^{2}-a x+a-2=0 \text { is minimal. } \]

17.18 USE problems

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