2014 USAMO Problems/Problem 3
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 .
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).