Name
Abstract | ||
|---|---|---|
The 6th problem of the 50th International Mathematical Olympiad (IMO), held in Germany, 2009, was the following. Let a(1), a(2), ..., a(n) be distinct positive integers and let M be a set of n - 1 positive integers not containing s = a(1) + a(2) + ... + a(n). A grasshopper is to jump along the real axis, starting at the point 0 and making n jumps to the right with lengths a(1), a(2), ... , a(n) in some order Prove that the order can be chosen in such a way that the grasshopper never lands on any point in M. In this paper we consider a variant of the IMO problem when the numbers a(1), a(2), ..., a(n) can be negative as well. We find the sharp minimum of the cardinality of the set M which blocks the grasshopper, in terms of n. In contrast with the Olympiad problem where the known solutions are purely combinatorial, for the solution of the modified problem we use the polynomial method. |
Year | DOI | Venue |
|---|---|---|
2011 | 10.4169/amer.math.monthly.118.10.877 | AMERICAN MATHEMATICAL MONTHLY |
Field | DocType | Volume |
Integer,Discrete mathematics,Combinatorics,Olympiad,Complex plane,Jump,Mathematics,Polynomial method | Journal | 118 |
Issue | ISSN | Citations |
10 | 0002-9890 | 0 |
PageRank | References | Authors |
0.34 | 0 | 1 |