5. Specify all values of the parameter $a$ for which the system of equations

$$
\left\{\begin{array}{l}
\left(\frac{2 y}{3}-\frac{|x-1|}{x-1}-1\right)(y-1)=0, \\
y=a+\frac{|x-1|}{(x-1)(x-y)^{2}}
\end{array}\right. \text { has exactly four distinct solutions. }
$$

Find these solutions for each specified $a$.

(20 points)
Solution: 1) $x>1,\left\{\begin{array}{l}\left(\frac{2 y}{3}-2\right)(y-1)=0, \\ y=a+\frac{1}{(x-y)^{2}}\end{array} \Leftrightarrow\left[\begin{array}{l}y=3,3=a+\frac{1}{(x-3)^{2}} \\ y=1,1=a+\frac{1}{(x-1)^{2}}\end{array} \Leftrightarrow\left[\begin{array}{l}y=3, a=3-\frac{1}{(x-3)^{2}} \\ y=1, a=1-\frac{1}{(x-1)^{2}}\end{array}\right.\right.\right.$

2) $x<1, x=1-\frac{1}{\sqrt{a-1}}$, there is one solution
II. If

1) $y=3, x>1, a=3-\frac{1}{(x-3)^{2}}$,
2) $y=0, x<1, a=\frac{1}{x^{2}}$.

When $a=1$ the system has three different solutions. When $a \neq 1$ we need to determine for which $a$ the system will have three different solutions in case II. For $a \in(0 ; 1)$ we have $y_{1}=0, \quad x_{1}=-\frac{1}{\sqrt{a}}, \quad y_{2 / 3}=3, \quad x_{2 / 3}=3 \pm \frac{1}{\sqrt{3-a}}$.

For $a \in[11 / 4 ; 3) \quad$ we have $\quad y_{1 / 2}=0, \quad x_{1 / 2}= \pm \frac{1}{\sqrt{a}}$,

$$
y_{3}=3, \quad x_{3}=3+\frac{1}{\sqrt{3-a}}
$$

Answer: For $a \in(0 ; 1)$ we have $y_{1}=0, \quad x_{1}=-\frac{1}{\sqrt{a}}$, $y_{2 / 3}=3, \quad x_{2 / 3}=3 \pm \frac{1}{\sqrt{3-a}}, y_{4}=1, \quad x_{4}=1+\frac{1}{\sqrt{1-a}}$.

For $a \in[11 / 4 ; 3) \quad$ we have $\quad y_{1 / 2}=0, \quad x_{1 / 2}= \pm \frac{1}{\sqrt{a}}$,

![](https://cdn.mathpix.com/cropped/2024_05_06_009f2643c33564a3e4a2g-3.jpg?height=754&width=923&top_left_y=774&top_left_x=1092)
$y_{3}=3, \quad x_{3}=3+\frac{1}{\sqrt{3-a}}, \quad y_{4}=1, \quad x_{4}=1-\frac{1}{\sqrt{a-1}}$.