How many solutions does the equation $\left[\frac{x}{10}\right]=\left[\frac{x}{11}\right]+1$ have in the set of integers?

---

Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
Solution. The integer $x$ can be uniquely written in the following form:

$$
x=11 k+m
$$

where $k$ is an integer and $0 \leq m \leq 10$ is an integer. Substituting this into the expressions on both sides of the equation,

$$
\left[\frac{x}{10}\right]=\left[\frac{11 k+m}{10}\right]=\left[k+\frac{k+m}{10}\right]=k+\left[\frac{k+m}{10}\right]
$$

and since $m \leq 10$, $\left[\frac{m}{11}\right]=0$, therefore

$$
\left[\frac{x}{11}\right]+1=\left[\frac{11 k+m}{11}\right]+1=k+\left[\frac{m}{11}\right]+1=k+1
$$

Substituting these into the equation, we get

$$
k+\left[\frac{k+m}{10}\right]=k+1, \quad \text { that is } \quad\left[\frac{k+m}{10}\right]=1
$$

which means $10 \leq k+m \leq 19$. Thus, $10-m \leq k \leq 19-m$ and $m=0,1,2, \ldots, 10$. From this, we can see that for each possible $m$, there are exactly 10 corresponding values of $k$. Therefore, there are a total of $10 \cdot 11=110$ solutions.

![](https://cdn.mathpix.com/cropped/2024_05_02_082d9c2afa1ec8eb8c6eg-1.jpg?height=510&width=542&top_left_y=1030&top_left_x=778)

Remark. The corresponding $(m ; k)$ pairs can also be represented in a Cartesian coordinate system as the lattice points in the shaded region.