59th Ukrainian National Mathematical Olympiad

Question

Bogdan drawn 2017 vertical and 2018 horizontal lines in the rectangle Q. These lines divide Q into 2018 2019 smaller not necessarily equal rectangles. Two children want to determine the perimeter of Q. Andrew says that he can do it by choosing some 2019 smaller rectangles and getting to know their perimeters. On the contrary, Olesya says that she can do it by choosing 4036 smaller rectangles. Who is right about the number of smaller rectangles, perimeters of which one should know in order to determine the perimeter of Q? (BogdanRublyov)

Accepted Answer

Let us consider two rectangles constructed from 2018 2019 smaller rectangles, of sizes 2 2 and 3 1 respectively. It is clear that all small rectangles in both of them have equal perimeters, namely 8. However, the first one is a rectangle 4036 4038 with the perimeter 16148, not equaling 16146, which is the perimeter of the second, a 6054 2019 rectangle.