The 6th problem of the 50th International Mathematical Olympiad (IMO), held
in Germany, 2009, was the following.
Let a1,a2,...,an be distinct positive integers and let M be a set of
n−1 positive integers not containing s=a1+a2+...+an. A grasshopper is to
jump along the real axis, starting at the point 0 and making n jumps to the
right with lengths a1,a2,...,an in some order. Prove that the order can be
chosen in such a way that the grasshopper never lands on any point in M.
The problem was discussed in many on-line forums, as well by communities of
students as by senior mathematicians. Though there have been attempts to solve
the problem using Noga Alon's famous Combinatorial Nullstellensatz, up to now
all known solutions to the IMO problem are elementary and inductive. In this
paper we show that if the condition that the numbers a1,...an are positive
is omitted, it allows us to apply the polynomial method to solve the modified
problem.Comment: Submitted to AMS Monthly on 1st March, 2010; last revision in August,
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