44th International Mathematical Olympiad IMO2003 problem A1
 

Here is the first problem from the 44th International Mathematical Olympiad (IMO 2003 for short), which took place from July 7-19 2003, in Tokyo. The other five are posted also. I don't know the answers, but I'll be working on them when I get the chance.

[quote][size=14]A1. S is the set {1, 2, 3, ... , 1000000}. Show that for any subset A of S with 101 elements we can find 100 distinct elements x_i of S, such that the sets x_i + A are all pairwise disjoint. [Note that x_i + A is the set {a + x_i | a is in A} ]. [/size][/quote]