2014 USAMO Problems/Problem 3
Problem
Prove that there exists an infinite set of points in the plane with the following property: For any three distinct integers
and
, points
,
, and
are collinear if and only if
.
Solution
Consider an elliptic curve with a generator , such that
is not a root of
. By repeatedly adding
to itself under the standard group operation, with can build
as well as
. If we let
then we can observe that collinearity between
,
, and
occurs only if
(by definition of the group operation), which is equivalent to
, or
, or
. We know that all these points
exist because
is never 0 for integer
, so that none of these points need to be point at infinity (the identity element of the group).