## Task 5

Lidija and Klara are playing with a pile of equal-sized sticks. Each of them has made the perimeter of a rectangle using a certain number of sticks. Each side of Klara's rectangle is made with 5 more sticks than the corresponding side of Lidija's rectangle, and the area of Klara's rectangle is twice the area of Lidija's rectangle. How many sticks could Lidija have used for the perimeter of her rectangle?
## Solution:

By extending the sides for the sticks, we get the following rectangle:

![](https://cdn.mathpix.com/cropped/2024_05_30_6273b59ad7665824c90fg-6.jpg?height=496&width=1405&top_left_y=680&top_left_x=154)

According to the problem, the area of the newly formed rectangle is twice the area of the initial one, which actually means that the areas of the blue and lavender parts in the image are equal.

![](https://cdn.mathpix.com/cropped/2024_05_30_6273b59ad7665824c90fg-6.jpg?height=494&width=1405&top_left_y=1335&top_left_x=154)

The marked areas on the upper rectangle are equal. If we remove these areas from the parts of equal area, the remaining parts will have equal areas.

![](https://cdn.mathpix.com/cropped/2024_05_30_6273b59ad7665824c90fg-6.jpg?height=506&width=1405&top_left_y=1983&top_left_x=151)

Therefore, the remaining parts of the lavender and blue areas are of equal area.

![](https://cdn.mathpix.com/cropped/2024_05_30_6273b59ad7665824c90fg-7.jpg?height=500&width=1408&top_left_y=127&top_left_x=153)

The marked areas on the upper rectangle are equal. If we remove these areas from the parts of equal area, the remaining parts will have equal areas.

![](https://cdn.mathpix.com/cropped/2024_05_30_6273b59ad7665824c90fg-7.jpg?height=500&width=1408&top_left_y=784&top_left_x=153)

Therefore, the remaining parts of the lavender and blue areas are of equal area.

The remaining blue area consists of two squares with a side length of 5 sticks. If we take the length of the stick as the basic unit of measurement, the area of the blue part is \(5 \cdot 5 + 5 \cdot 5 = 50\).

Thus, the product of the lengths of the sides of the remaining lavender rectangle (expressed in the number of sticks) is 50.

The possibilities are:

I. width 1 stick and length 50 sticks

II. width 2 sticks and length 25 sticks

III. width 5 sticks and length 10 sticks.

The lengths of the sides of the initial rectangle are 5 sticks longer.

The possible lengths of the sides of the initial rectangle are:

I. width 6 sticks and length 55 sticks

II. width 7 sticks and length 30 sticks

III. width 10 sticks and length 20 sticks.

Lidija could have used for the perimeter of her rectangle:
I. \(2 \cdot 6 + 2 \cdot 55 = 122\) sticks

II. \(2 \cdot 7 + 2 \cdot 30 = 74\) sticks

III. \(2 \cdot 10 + 2 \cdot 20 = 60\) sticks.