In the Hupikék Törpikék's $1001 \times 945$ meter forest, there are 1280 pine trees with a 1 meter diameter. The dwarfs would like to mark 7 $20 \times 34$ meter tennis courts in the forest.

Is it possible to do this without cutting down a single pine tree?
Solution. To solve the problem, we will use the pigeonhole principle. In the forest, we mark out areas the size of a tennis court such that any tree can only make one of these areas impossible to realize. We cover the forest with rectangles of size $21 \mathrm{~m} \times 35 \mathrm{~m}$, with a 0.5-meter wide border. Within these, the tennis courts will be located.

![](https://cdn.mathpix.com/cropped/2024_05_02_3cc2f49ac08f18bf2b8bg-1.jpg?height=388&width=599&top_left_y=305&top_left_x=752)

The border between two adjacent such areas is 1 meter wide, so if a tree protrudes into the interior of one, it does not disturb the interior area (the possible tennis court location) of the other.

One of the 1001 m long sides of the forest is divided into a 21 m and a 980 m segment. Draw a line parallel to the other side through the division point. This results in a $21 \mathrm{~m} \times 945 \mathrm{~m}$ strip, which can be exactly divided into 27 pieces of $21 \mathrm{~m} \times 35 \mathrm{~m}$ rectangles; and a $980 \mathrm{~m} \times 945 \mathrm{~m}$ rectangle, on whose $980 \mathrm{~m}$ side the $35 \mathrm{~m}$ can be measured 28 times, and on whose $945 \mathrm{~m}$ side the $21 \mathrm{~m}$ can be measured 45 times. In the latter case, therefore, $45 \cdot 28=1260$ pieces of $21 \mathrm{~m} \times 35 \mathrm{~m}$ rectangles and thus 1260 tennis courts can be marked out. In the entire forest, therefore, we have marked out 1287 tennis courts such that any tree can make at most one of these impossible to realize. Since the number of trees is 1280, at least 7 tennis courts can definitely be realized.