[ Systems of points and segments. Examples and counterexamples ]

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\text { [ Combinatorial geometry (miscellaneous). ] }
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The grandfather called his grandson to visit him in the village:

- You'll see what an extraordinary garden I've planted! There are four pears growing there, and there are also apple trees, planted in such a way that exactly two pears are 10 meters away from each apple tree.
- So what's so interesting about that, - the grandson replied. - You only have two apple trees.
- And you're wrong, - the grandfather smiled. - I have more apple trees than pears in my garden.

Draw how the apple trees and pears could have grown in the grandfather's garden. Try to place as many apple trees as possible on the drawing without violating the conditions.
Various arrangements of apple and pear trees are possible, for example, as shown in the figure on the left. The maximum number of apple trees can be placed if the pears grow densely enough. For instance, if pears are planted in a row every 5 meters, there will be room for 12 apple trees (figure in the center).
![](https://cdn.mathpix.com/cropped/2024_05_06_39480ed8b0d60d88e837g-32.jpg?height=1368&width=1014&top_left_y=0&top_left_x=364)

Let's prove that more than twelve apple trees are not possible. Consider any two pear trees. At a distance of 10 meters from them, there can be only two apple trees - one on one side of the line of pears, and the other on the opposite side (figure on the right). Therefore, each pair of pears "serves" no more than two apple trees. Since there are six pairs of pears, the maximum number of apple trees is 12.