## Task A-1.3.

The area of the intersection of the larger and smaller square is two-thirds of the area of the smaller square, and also one-fifth of the area of their union. Determine the ratio of the side lengths of the larger and smaller squares.
## First solution.

Let the side of the larger square be denoted by $a$, and the side of the smaller square by $b$.

Then the area of the intersection is $\frac{2}{3} b^{2}$.

The area of the union is equal to the sum of the areas of the smaller and larger squares minus the area of their intersection.

Therefore, the area of the union is $a^{2}+b^{2}-\frac{2}{3} b^{2}=a^{2}+\frac{1}{3} b^{2}$.

According to the condition of the problem, $\frac{2}{3} b^{2}=\frac{1}{5}\left(a^{2}+\frac{1}{3} b^{2}\right)$.

This implies $\frac{2}{3} b^{2}=\frac{1}{5} a^{2}+\frac{1}{15} b^{2}$, or $\frac{9}{15} b^{2}=\frac{1}{5} a^{2}$. Thus, the ratio of the areas is $a^{2}: b^{2}=9: 3=3: 1$. 1 point From this, it follows that the ratio of the sides is $a: b=\sqrt{3}: 1$.