1998 USAMO Problems/Problem 5
Proof by induction. For n=2, the proof is trivial, since satisfies the condition. Assume now that there is such a set S of n elements,
which satisfy the condition. The key is to note that if
, then if we define
for all
, where k is a positive integer, then
and
, and so
.
Let . Consider the set
. To finish the proof, we simply need to choose a k such that
for all
. Since
, simply choose k so that
.