# Problem 3.

On the coordinate plane, a square $K$ is marked with vertices at points $(0,0)$ and $(10,10)$. Inside this square, draw the set $M$ of points $(x, y)$ whose coordinates satisfy the equation

$$
[x]=[y],
$$

where $[a]$ denotes the integer part of the number $a$ (i.e., the greatest integer not exceeding $a$; for example, $[10]=10,[9.93]=9,[1/9]=0,[-1.7]=-2$). What fraction of the area of square $K$ is the area of set $M$?
Solution. Let $n \leq x < n+1$, where $n$ is an integer from 0 to 9. Then $[x]=n$ and $[y]=n$. The solution to the latter equation is all $y \in [n, n+1)$. Thus, the solution will be the union of unit squares

$$
\{x \in [n, n+1), y \in [n, n+1), n \in \mathbb{Z}\}
$$

Inside the square $K$ specified in the problem, ten such squares will fit. Since $K$ consists of 100 unit squares, the ratio of the areas is $\frac{S_{M}}{S_{K}}=\frac{10}{100}=0.1=10\%$.

The answer is represented by dotted hatching in the figure below (the left and lower boundaries are included in $M$, the right and upper boundaries are not). The area of the set $M$ is $10\%$ of the area of the square $K$.

![](https://cdn.mathpix.com/cropped/2024_05_06_8bc7c2b9a63f91941abeg-2.jpg?height=664&width=680&top_left_y=1446&top_left_x=885)