Problem 3. On a plane, there are a square and an equilateral triangle such that the area of each of these two figures is numerically equal to the perimeter of the other. Find the side length of the given square.
Answer: $2 \sqrt[3]{2} \sqrt{3}$.
Solution. Let the side of the equilateral triangle be $a$, and the side of the square be $b$. Then $3 a=b^{2}$ and $4 b=\frac{\sqrt{3}}{4} a^{2}$. From this, $b^{4}=9 a^{2}=48 \sqrt{3} b$. It follows that $b^{3}=48 \sqrt{3}$, from which $b=2 \sqrt[3]{2} \sqrt{3}$.

## Criteria

4 6. A complete and justified solution is provided.

In the absence of a correct solution, the highest applicable criterion from those listed below is used:

3 b. Correct equations are set up, but an arithmetic error is made in further calculations.

2 6. Only one of the two equations is set up correctly.

1 6. The correct answer is provided, but there is no justification.

0 6. The problem is not solved or is solved incorrectly.