Grasshoppers (in Hungarian)

The talk is about a fun puzzle. In particular, about the following
question, originally asked at the Moscow Oral Team Mathematics Olympiad
in 2001: Four grasshoppers are sitting at the vertices of a square.
Every minute, one of them jumps over another and lands at the point
symmetric to it. Prove that the grasshoppers can never end up sitting at
the vertices of a larger square.

The original problem is not hard to crack. Here we consider two
extensions: (1) What if the grasshoppers start at at the vertices of a
regular k-gon for k not four? This was asked originally by Florestan
Brunck and then it appeared as one of the Schweitzer problems last year.
(2) From what starting positions can the grasshoppers approximate any
configuration arbitrarily well?

The results are joint with Janos Pach.