Task 1. A rectangular grid, both sides of which are even numbers, was cut into figures of the form $\square$ and $\square \square$ such that both types of figures are present. What is the smallest area that such a rectangle could have? Provide an example of the corresponding cutting and explain why a smaller area is impossible. The figures can be rotated and flipped. The length of a side of the rectangle is the number of cells adjacent to it. The area of a grid rectangle is the number of cells it contains. (20 points)
Answer: 40.

Solution: Since the area of both figures is 5, and the sum of numbers divisible by 5 is also divisible by 5, the area of the rectangle must be divisible by 5. Since 5 is a prime number, the length of one of the sides must be divisible by 5. Since the lengths of the sides are even numbers, it must be divisible by 10. The smallest even number is 2, but a rectangle of $2 \times 10$ does not work because the second type of figure can only fit horizontally, leaving one uncovered cell inside it, which cannot be covered by other figures. Therefore, the minimum possible value for the length of the other side is 4. A rectangle of $4 \times 10$ can be cut, for example, as shown in the figure. Therefore, the smallest area is 40.

![](https://cdn.mathpix.com/cropped/2024_05_06_3703d6f108624531dcffg-1.jpg?height=235&width=550&top_left_y=1479&top_left_x=297)

Criteria: Only the example of cutting - 10 points.

Only the estimate with incorrect or missing cutting - 10 points.