  The 6th problem of the 50th International Mathematical Olympiad (IMO), held
in Germany, 2009, is called 'the grasshopper problem'. To this problem Kos
developed theory from unique viewpoints by reference of Noga Alon's
combinatorial Nullstellensatz.
  We have tried to solve this problem by an original method inspired by a
polynomial function that Kos defined, then examined for n=3, 4 and 5. For
almost cases the claim of this problem follows, but there remains imperfection
due to 'singularity'.
