3.1. (16 points) A square with a side of 36 cm was cut into three rectangles such that the areas of all three rectangles are equal and any two rectangles share a common boundary segment. What is the total length (in cm) of the cuts made?
Answer: 60.

Solution. One cut has a length of 36 cm and separates one of the rectangles (see fig.). The remaining 2 rectangles have a common side $x$, so they are both $18 \times x$ cm in size. Therefore,

![](https://cdn.mathpix.com/cropped/2024_05_06_b37e9e20178d455ca9cfg-2.jpg?height=417&width=423&top_left_y=91&top_left_x=1479)
$18 \cdot x=36^{2} / 3$, from which $x=24$ cm. Thus, the total length of the cuts is 60 cm.