# 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 $(x, y)$,
the coordinates of which 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 $n < [y]$. The solution to the latter inequality is all $y \geq n+1$. Thus, the solution will be the union of strips

$$
\{n \leq x < n+1, n+1 \leq y, \quad n \in \mathbb{Z}\}
$$

Inside the square specified in the problem, part of nine such strips will fall. If we divide the square $K$ into 100 unit squares, then the set $M$ will consist of $9+8+7+\ldots+1=45$ such squares. Therefore, the ratio of the areas is $\frac{S_{M}}{S_{K}}=\frac{45}{100}$.

The answer is represented by dotted hatching in the figure below (horizontal segments of the boundary are included in $M$, vertical segments are not). The area of the set $M$ is $45\%$ of the area of the square $K$.

![](https://cdn.mathpix.com/cropped/2024_05_06_b9a0d3de8e16693f5754g-2.jpg?height=664&width=677&top_left_y=1446&top_left_x=968)