This geometric problem was given to me by Arkady Berenstein:
There are n points on the plane, but not all of them are on one line. Prove that a line exists that passes through exactly two points of this set.
Arkady gave me a beautiful solution to this problem. First, draw a line through each pair of points. Suppose you calculate all the distances from each point to all the lines that the point doesn’t belong to. Pick the smallest distance. The corresponding line will be the one with two points. To finish the solution you need to fill in the details. That process is usually left to the reader.
I suspect that there might also be a solution using linear algebra. Can you find one?
I would like to reformulate this problem without using geometry. Suppose there is a set of n elements. Let’s call a family of subsets line-like if any two distinct subsets of this family can have as an intersection not more than one element. Then the geometry problem above has a set-theoretical analogue:
You have a set of n elements and a line-like family of subsets of this set such that any two elements of the set belong to a subset from this family, and that the family doesn’t contain the whole set. Is it true that there always exists a subset in this family consisting of two elements?
Usually I give such problems as homework for the reader, but this time I decided to change my habit, so I’m including the picture which contains the solution of this problem by my son Alexey Radul.
Conclusion: geometry is important.