2. A rectangle with sides $6 \mathrm{~cm}$ and $3 \mathrm{~cm}$ was cut into three rectangles of equal perimeter. What can the perimeter of these rectangles be? Find all possible answers.
Answer: 14 cm, 10 cm, 10.5 cm.

Sketch of the solution.

Let's cut a rectangle with sides $a$ and $b$. Cases 1 and 2. Cut the rectangle into three equal rectangles with two cuts parallel to side $a$.

![](https://cdn.mathpix.com/cropped/2024_05_06_f259ce87565f6e7a499dg-2.jpg?height=208&width=373&top_left_y=1224&top_left_x=1029)
The perimeter of these rectangles is $\mathrm{P}=2(a+b / 3)$. 1) If $a=3$ cm and $b=6$ cm, then $P=10$ cm. 2) If $a=6$ cm and $b=3 \mathrm{~cm}$, then $P=14 \mathrm{cm}$.

Cases 3 and 4. Cut one rectangle parallel to side $a$, and cut the remaining part into two equal rectangles parallel to side $b$ (see figure). If the perimeters of the rectangles are equal, then their semi-perimeters are also equal, so $\mathrm{a}+\mathrm{x}=\mathrm{b}-\mathrm{x}+\mathrm{a} / 2$, from which $2 \mathrm{x}=\mathrm{b}-\mathrm{a} / 2$ under the condition $\mathrm{b}-\mathrm{a} / 2>0$. The perimeter of these rectangles is $\mathrm{P}=2 \mathrm{a}+\mathrm{b}-\mathrm{a} / 2=3 \mathrm{a} / 2+\mathrm{b}$. 1) If $a=3$ cm and $b=6 \mathrm{cm}$, then $P=10.5$ cm. 4) If $a=6$ cm and $b=3$ cm, then $\mathrm{b}-\mathrm{a} / 2=0$ and the condition for the existence of the rectangle is not satisfied.

Criteria. Correct answer, without explanations: 2 points.