10885  Martin the Gardener
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10885  Martin the Gardener
Hi guys
I don
I don

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No hard papers are required. The idea is very simple. I was very happy when 1 person solved this problem during the contest! He ended up in the 155'th place. :)
The key observation is the following. Suppose that you know how to plant 13 trees at rational coordinates, so that the distance between each pair of trees is a rational number. Then you can simply multiply every coordinate by the common denominator of everything (coordinates and distances), and you will get integer coordinates and integer distances.
Now how do we solve the problem in rational numbers? That's not very hard at all. Here is a hint. We are not allowed to place 3 trees on the same line. If you add the restriction that no 4 trees are allowed on the same circle, then this is a famous unsolved problem. The largest number of trees that anyone knows how to place with the additional no4onacircle constraint is 6.
The key observation is the following. Suppose that you know how to plant 13 trees at rational coordinates, so that the distance between each pair of trees is a rational number. Then you can simply multiply every coordinate by the common denominator of everything (coordinates and distances), and you will get integer coordinates and integer distances.
Now how do we solve the problem in rational numbers? That's not very hard at all. Here is a hint. We are not allowed to place 3 trees on the same line. If you add the restriction that no 4 trees are allowed on the same circle, then this is a famous unsolved problem. The largest number of trees that anyone knows how to place with the additional no4onacircle constraint is 6.
If only I had as much free time as I did in college...

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