Subject 3. Solve the equations:
a) $x=\frac{\{x\}}{[x]}$
b) $[x] \cdot\{x\}=[x]-\{x\}$, where $[x]$ represents the integer part of the real number $x$ and $\{x\}$ the fractional part of $x$.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_06_07_4e5824da207017160be1g-3.jpg?height=66&width=1537&top_left_y=688&top_left_x=305)

$$
\begin{aligned}
& x \geq 1 \Rightarrow x \cdot[x] \geq 1 \Rightarrow \text { the equation has no solutions.......................................................................... } 1 \text{p} \\
& x \leq-1 \Rightarrow x \cdot[x] \geq 1 \Rightarrow \text { the equation has no solutions...................................................................... } 1 \text{p}
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_06_07_4e5824da207017160be1g-3.jpg?height=106&width=1537&top_left_y=866&top_left_x=302)

![](https://cdn.mathpix.com/cropped/2024_06_07_4e5824da207017160be1g-3.jpg?height=71&width=1539&top_left_y=1016&top_left_x=301)
For $k=-1$, the equation has no solutions............................................................................. $1 \text{p}$

For $k \neq-1$, the equation has solutions $x=\frac{k^{2}+2 k}{k+1}, k \in \mathbb{Z}$.................................................. $1 \text{p}$