Competition problem — Geometry — Olympiad Maths Prep

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Geometry Difficulty 5.9 AIME, harder Prove it Ukraine

Bogdan drawn $2017$ vertical and $2018$ horizontal lines in the rectangle $Q$ . These lines divide $Q$ into $2018 \times 2019$ 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)

Solution

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

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