Algebraic

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Problem: Solve the inequality: \[ \frac{\sqrt{x}}{x-2} \leq 3 \sqrt{x} \]

12.3.1.1 Algebraic

0.75 $

Problem: What is the maximum value the expression can take: \[ \left(1-a^{2}\right) \sqrt{1-b^{2}}-\left(1-b^{2}\right) \sqrt{1-a^{2}} . \]

12.3.1.2 Algebraic

2.5 $

Problem: It's known that \( 0

12.3.1.3 Algebraic

3.75 $

Problem: Let \( a_{n} \) be a sequence of positive numbers, for which the following inequality holds. \[ a_{n}^{2}

12.3.1.4 Algebraic

3.75 $

Problem: Find the maximum value of the parameter \( a \), for which the system has a solution: \[ \left\{\begin{array}{l} 4 \sin ^{2} y-a=16 \sin ^{2} \frac{2 x}{7}+9\left(\tan ^{-1}\right) \frac{2 x}{7} \\ \left(\pi^{2} \cos ^{2} 3 x-2 \pi^{2}-72\right) y^{2}=2 \pi^{2}\left(1+y^{2}\right) \sin 3 x \end{array}\right. \]

12.3.1.5 Algebraic

3.75 $

Problem: 4399 optionally different positive numbers \( v_{1}, v_{2}, \ldots, v_{4399} \) are written around the circle. For any 4 numbers \( h, k, z \) and \( p \), in a row in the indicated order clockwise, the inequality holds: \[ 1,6(h+k) \geq \frac{1}{z}+\frac{1}{p} \text {. } \] What is the smallest possible value of the arithmetic mean of these numbers? If the question of the problem allows several answers, then indicate all of them in the form of a set.

12.3.1.6 Algebraic

6.24 $

Problem: Find the sum of natural values of the parameter \( a \), for each of which the system has at least one solution. \[ \left\{\begin{array}{c} \sqrt{(x+a)^{2}+(y+3)^{2}}+\sqrt{(x-4)^{2}+(y-a)^{2}}= \\ =\sqrt{2 a^{2}+14 a+25} \\ y=a^{2}-5 a-7 \end{array}\right. \]

12.3.1.7 Algebraic

3 $

Problem: Find the sum of natural values of the parameter \( a \), for each of which the system has at least one solution \[ \left\{\begin{array}{c} \sqrt{(x-a)^{2}+(y+3)^{2}}+\sqrt{(x+2)^{2}+(y-a)^{2}}= \\ =\sqrt{2 a^{2}+10 a+13} \\ y=a^{2}-4 a-6 \end{array}\right. \]

12.3.1.8 Algebraic

3 $

Problem: Consider the inequality \[ \left|\log _{2} x\right| \cdot \frac{x^{2}-2 x-a}{(10 x-3 a-16)^{2}} \ldots 0 \] Which inequality \( \operatorname{sign}(<,> \), \( \leq \) or \( \geq) \) must be there instead of the ellipses, so that for at least one value of the parameter \( a \) the solution of the inequality with respect to \( x \) there was an interval (a nonempty open connected bounded set on the real line)? Find the sum of all integer values of the parameter \( a \), for which the solution of this inequality with the chosen sign is an interval.

12.3.1.9 Algebraic

2.5 $