Míša cut out two identical squares from paper, one rectangle with dimensions $10 \mathrm{~cm} \times 24 \mathrm{~cm}$, and another rectangle. What could be the dimensions of this rectangle if it was possible to form a square from all four shapes without any of the individual pieces overlapping? Several such rectangles can be found; list at least four.

(L. Šimünek)
When searching for possible dimensions of the fourth shape, it is advantageous to proceed as follows: First, to the rectangle with dimensions $10 \mathrm{~cm} \times 24 \mathrm{~cm}$, we suitably choose the placement and size of two identical squares, keeping in mind that the longest side of the figure formed by these three shapes must also be a side of the resulting square. With knowledge of the length of this side, we can easily calculate the dimensions of the fourth shape.
![](https://cdn.mathpix.com/cropped/2024_04_17_df9261f4862336f3ce8cg-3.jpg?height=1570&width=1182&top_left_y=407&top_left_x=470)

The dimensions of the fourth shape in cm could have been $2 \times 24, 19 \times 29, 34 \times 58, 34 \times 44, 14 \times 17$ or $24 \times 38$. The rectangle could not have had any other dimensions.

Note. Work with 4 to 6 options should be rated 1 - excellent, work with 2 to 3 options should be rated 2 - good. If the solver provides only one option, rate such a solution 3 - unsatisfactory.