1. The number expressing the area of one rectangle made of matches (assuming the length of a match is 1) is one more than the number expressing the perimeter of this rectangle, and the area of the second rectangle is one less than its perimeter. What are the dimensions of these rectangles?
Solution. Let the lengths of the sides of the first rectangle be denoted by x and y. Then, from the condition of the problem, $xy = 2(x + y) + 1$. From this, we get $y = \frac{2x + 1}{x - 2} = 2 + \frac{5}{x - 2}$. For y to be an integer when x is an integer, we need $x - 2 = 1$ or $x - 2 = 5$. In the first case, we get $x = 3, y = 7$, and in the second case, $x = 7, y = 3$. Similarly, we find the dimensions of the second rectangle.

Answer: The first rectangle is $7 \times 3$, the second is $5 \times 3$.