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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

0 $

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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Problem: The quadratic equation \( x^{2}+p x+q=0 \) has two different roots \( x_{1} \) and \( x_{2} \). Find \( p \) and \( q \), if the numbers of \( x_{1}+1 \) и \( x_{2}+1 \) are the roots of the equation \( x^{2}-p^{2} x+p q=0 \).

17.19 USE problems

2.57 $

Problem: In the cuboid \( A B C D A_{1} B_{1} C_{1} D_{1} \) the lengths of the edges are \( A D=12, A B=5, A A_{1}=8 \). Find the volume of the pyramid \( M B_{1} C_{1} D \), if \( M \) is a point on \( A A_{1} \) moreover \( A M=5 \).

17.21 USE problems

2.57 $

Problem: We have \( A B C \) right triangle, with the right angle \( C \). On the line, containing \( A D \) median, point \( E \) is taken, which is at a distance of 4 from the vertex \( A \). Find the area of the triangle \( B C E \), if \( B C=6 \), \( A C=4 \).

17.23 USE problems

3.08 $

Problem: 1) Appliances are connected to the power outlet, the total resistance of which is \( 60 \mathrm{Ohm} \). An electric heater is supposed to be connected to the outlet in parallel with them. Determine (in ohms) the smallest possible resistance of this electric heater, if it is known that in case of a parallel connection of two conductors with resistances \( R_{1} \) and \( R_{2} \) their total resistance is given by the formla \( R=\frac{R_{1} \cdot R_{2}}{R_{1}+R_{2}} \), and for the normal functioning of the electrical network the total resistance of it must be not less than \( 10 \mathrm{Ohm} \). 2) Find the biggest value of the function: \( y=17 x-5 \sin x+7 \) on the segment \( \left[-\frac{\pi}{2} ; 0\right] \). 3) The first pipe passes 1 liter less water per minute, than the second pipe. How many liters of water per minute passes through the second pipe, if it fills the reservoir with the volume of 624 liters 2 minutes faster, than the first pipe fills the reservoir with the volume of 650 liters?

17.25 USE problems

3.08 $

Problem: Solve for all values of the parameter: \[ a^{2}(x-1)-a(2 x-5)=3 x-6 \text {. } \]

17.26 USE problems

0 $

Problem: Solve for all values of the parameter. \[ 3 x+9

17.27 USE problems

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Problem: Solve for all values of the parameter. \[ \frac{x-5}{x-1}-\frac{2}{k}=\frac{3}{k(x-1)} \text {. } \]

17.28 USE problems

0 $

Problem: Find all values of the parameter for which the equation \( a x^{2}-4 a x+4 a-5=0 \) has negative roots.

17.29 USE problems

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Problem: Solve the irrational equation. \[ \sqrt{3 x^{2}+5 x+8}-\sqrt{3 x^{2}+5 x+5}=1 . \]

17.30 USE problems

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Problem: The children ate sandwiches and sweets at the New Year's table, and each of them ate something, and someone might have eaten both. It is known that the boys who ate sandwiches were no more than \( 5 / 16 \) of the total number of children who ate sandwiches, and the boys who ate sweets were no more than \( 2 / 5 \) of the total number of children who ate sweets. a) Could there be 13 boys at the table if it is additionally known that there were 25 children in total at the table? b) What is the largest number of boys that could be at the table, if it is additionally known that there were 25 children at the table? c) What was the smallest share of girls in the total number of children without taking into account the additional conditions of points \( a \) and \( b \) ?

17.24 USE problems

5.14 $

Problem: Find all values of the parameter \( a \), for each one of which the equation \( \left|\xi^{2}-49\right|-12|\xi-a|- \) \( -12 a=0 \) with respect to the variable \( \xi \) has exactly 4 solutions. If the question of the task allows several variants of answer, show them in the form of a set.

17.31 USE problems

3.85 $

Problem: Find all values of the parameter \( \omega \), for each one of which the equation \( \left|\frac{1}{2 \sigma}-3\right|-\omega \sigma-\frac{3}{2}=0 \) with respect to the positive value of \( \sigma \) has no solution. If the question of the task allows some variants of answer, show them in the form of a set.

17.32 USE problems

5.14 $

Problem: Solve the inequality \[ \frac{2-\sqrt{\left(2+\log _{2} h\right)\left(-1+\log _{2} h\right)}}{\log _{2} h}+1 \geq 0 \] with respect to the variable \( h \). If the question of the task allows several variants of answer, show them in the form of a set.

17.33 USE problems

1.54 $

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