2. (7p) Solve in $\mathbf{R}$ the equation:

$$
\{x\}+\{2 x\}+\{3 x\}=1
$$

where $\{a\}$ denotes the fractional part of the real number $a$.

Petru Vlad
2. $x=x+x \Rightarrow x+2 x+3 x=6 x-1$

$6 x-1=k \in Z \Rightarrow x=\frac{k+1}{6}$

The equation becomes $\left[\frac{k+1}{6}\right]+\left[\frac{k+1}{3}\right]+\left[\frac{k+1}{2}\right]=k$

We consider $k$ of the form $6 p, 6 p+1, 6 p+2, \ldots, 6 p+5, p \in \mathbf{Z}$ and we obtain the set of solutions $S=\left\{\frac{6 p+1}{6}, \frac{6 p+2}{6}, \frac{6 p+3}{6}, \left.\frac{6 p+4}{6} \right\rvert\, p \in Z\right\}$