Problem 5. Three regular octagons have a common center, their sides are respectively parallel. The sides of two octagons are 7 cm and 42 cm. The third octagon divides the area of the figure enclosed between the first two in the ratio $1: 6$, counting from the smaller octagon. Find the side of the third octagon.
Answer: $7 \sqrt{6}$.

Solution: Let the side of the third octagon be $x$. The areas of similar figures are in the ratio of the squares of corresponding sides. Therefore, the areas enclosed between the octagons are in the ratio $\left(x^{2}-6^{2}\right):\left(42^{2}-x^{2}\right)=1: 6$, from which the answer follows.