Sharygin I.F.

A rectangle is composed of six squares (see the right figure). Find the side of the largest square if the side of the smallest one is 1.

![](https://cdn.mathpix.com/cropped/2024_05_06_c2dae46d614e2e953886g-48.jpg?height=357&width=417&top_left_y=2457&top_left_x=838)
The side of the largest square is equal to the sum of the sides of two squares: the next one in the clockwise direction and the smallest one.

## Solution

Note that the side of the largest square is equal to the sum of the sides of two squares: the next one in the clockwise direction and the smallest one. Denoting the side of the largest square as $x$, we can sequentially express the sides of the other squares: $x-1, x-2, x-3, x-3$ (see the diagram). Now note that the length of the upper side of the rectangle is $x+(x-1)$, and the length of the lower side is $(x-2)+(x-3)+(x-3)$. But the opposite sides of the rectangle are equal. We get the equation

$$
x+(x-1)=(x-2)+(x-3)+(x-3)
$$

From this, $2x-1=3x-8$ and, therefore, $x=7$.

| $x$ |  | $x-1$ |
| :---: | :---: | :---: |
| $x-3$ | $x-3$ | $x-2$ |

## Answer

7.00