# Problem 3.

A square with a side of 75 mm was cut by two parallel cuts into three rectangles. It turned out that the perimeter of one of these rectangles is half the sum of the perimeters of the other two. What is this perimeter? Give your answer in centimeters.
# Answer: 20

Solution. For three rectangles, one of the sides is the same and equals 75 mm. The sum of the lengths of two such sides of one rectangle is half the sum of the lengths of all such sides of the other two. Therefore, the other side of the first rectangle is also half the sum of the other sides of the two other rectangles. Thus, it equals $75: 3=25$ mm, from which the perimeter of the first rectangle is $2 \cdot(25+75)=200$ mm $=20 \mathrm{cm}$.

## B-2

A square with a side of 105 mm was cut by two parallel cuts into three rectangles. It turned out that the perimeter of one of these rectangles is half the sum of the perimeters of the other two. What is this perimeter? Give your answer in centimeters.

Answer: 28

## B-3

A square with a side of 135 mm was cut by two parallel cuts into three rectangles. It turned out that the perimeter of one of these rectangles is half the sum of the perimeters of the other two. What is this perimeter? Give your answer in centimeters.

Answer: 36

## B-4

A square with a side of 165 mm was cut by two parallel cuts into three rectangles. It turned out that the perimeter of one of these rectangles is half the sum of the perimeters of the other two. What is this perimeter? Give your answer in centimeters.

Answer: 44

## Lomonosov School Olympiad in Mathematics

Preliminary stage $2020 / 21$ academic year for $5-6$ grades

## B-1

#