1. The teacher drew a rectangle $A B C D$ on the board. Student Petya divided this rectangle into two rectangles with a line parallel to side AB. It turned out that the areas of these parts are in the ratio 1:2, and the perimeters are in the ratio 3:5 (in the same order). Student Vasya divided this rectangle into two parts with a line parallel to side BC. The areas of the new parts also relate as 1:2. What is the ratio of their perimeters?
Answer: 20:19. Solution. Let $A B=a, B C=b$, and the first line divides side $B C$ into segments $x$, $b-x$. Then, according to the condition, $b-x=2 x$ and $5(x+a)=3(b-x+a)$, from which $b=6 a$. Now let the second line divide side $A B$ into segments $y, a-y$. According to the condition, $y=2(a-y)$, that is, $y=2 a / 3$. Then the ratio of the perimeters is $(y+b) /(a-y+b)=20: 19$.