Problem 6.2. A square with a side of 1 m is cut into three rectangles with equal perimeters. What can these perimeters be? List all possible options and explain why there are no others.
Answer: $8 / 3$ or $5 / 2$.

Solution. Note that, up to rotation, a square can be cut into three rectangles in only the following two ways:
![](https://cdn.mathpix.com/cropped/2024_05_06_3ea49ea8943c27893726g-1.jpg?height=212&width=640&top_left_y=1890&top_left_x=710)

First cutting. Note that if the rectangles have the same length, the perimeter can only be the same if the width is also the same. Therefore, such rectangles are equal, and the width is $1 / 3$, so the perimeter is $2 + 2 / 3 = 8 / 3$.

Second cutting. Let the left rectangle have dimensions $x \times 1$, and the right rectangles, which have the same width and thus the same length, have dimensions $(1-x) \times 1 / 2$. The equality of the perimeters can be written as the equation

$$
2 + 2x = 2 \cdot (1 - x) + 2 \cdot 1 / 2
$$

From this, $x = \frac{1}{4}$ and the perimeters are $2 + 2 \cdot \frac{1}{4} = \frac{5}{2}$.

## Criteria

## Kurchatov School Olympiad in Mathematics - 2019 Final Stage.

One of the largest applicable criteria is used:

7 6. Any correct solution to the problem.

2 6. The problem is solved for the first case.

4 6. The problem is solved for the second case.

16 . The correct answer is given.

Points are deducted for the following errors in a solution that is otherwise correct:

-1 6. The equation is correctly set up, but an arithmetic error is made. No more than 1 point is deducted for each case.