[Dirichlet's Principle (finite number of points, lines, etc.)]

In a park, there are 10,000 trees planted in a square grid (100 rows of 100 trees each). What is the maximum number of trees that can be cut down so that the following condition is met: if you stand on any stump, you will not see any other stump? (The trees can be considered sufficiently thin.)
Let's divide the trees into 2500 quadruples, as shown in the figure. In each such quadruple, no more than one tree can be cut down. On the other hand, all the trees growing in the upper left corners of the squares formed by our quadruples of trees can be cut down. Therefore, the maximum number of trees that can be cut down is 2500.

![](https://cdn.mathpix.com/cropped/2024_05_06_9e668e47ffca00611739g-28.jpg?height=706&width=992&top_left_y=1259&top_left_x=539)

Send a comment