1. The rectangle drawn in the diagram is divided into 11 squares with different side lengths. The smallest square has a side length of 9. What are the dimensions of the rectangle?

![](https://cdn.mathpix.com/cropped/2024_06_05_d0c212a3407f50323a50g-03.jpg?height=276&width=295&top_left_y=910&top_left_x=1184)
Solution. The square marked with the number 1 has a side length of 9. Let the side length of square 2 be $x$. Then the side length of square 3 is $x+9$, the side length of square 4 is $x-9$, the side length of square 5 is $x+18$, the side length of square 6 is $2 x-9$, the side length of square 7 is $2 x+27$, and the side length of square 8 is $3 x-18$.

If the side of square 9 is $y$, then $y+(x+18)+9=(3 x-18)+(x-9), \quad$ from which $y=3 x-54$. Thus, the side of square 10 is $6 x-72$, and the side of square 11 is $9 x-126$.

The top side of the given rectangle is the sum of the sides of squares 10 and 11, i.e., it has a length

$$
(6 x-72)+(9 x-126)=15 x-198
$$

and its bottom side is the sum of the sides of squares 7, 3, 2, and 6, i.e., it has a length

![](https://cdn.mathpix.com/cropped/2024_06_05_d0c212a3407f50323a50g-03.jpg?height=486&width=522&top_left_y=1437&top_left_x=960)

$$
(2 x+27)+(x+9)+x+(2 x-9)=6 x+27
$$

Equating the obtained expressions, we get $x=25$. Therefore, the sides of the squares marked with numbers from 1 to 11 are $9,25,34,16,43,41,77,57,21,78,99$, respectively. Thus, the length of the given rectangle is 177, and the width is 176.