2001 IMO Shortlist Problems/N6
Problem
Is it possible to find 100 positive integers not exceeding 25,000, such that all pairwise sums of them are different?
Solution
For example, let's say that the integers are, . Now this turns into a problem of solving for the
integers
. This then each ai takes on the form,
. Then we must find the
integers. By doing this process over and over again, we obtain the last
numbers,
. Obviously these 3 integers can have different sums, and the number of different "parts" in every sequence (the number of terms that are different for
, etc.) is
, not exceeding
.