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 .
Solution (Group Theory)
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).
Solution 2 (Function Theory)
Consider letting be the point , where . Then if three points are on the same line , they must be the solutions to the equation (i.e. the intersection of and the line). By Vieta's Formulas, the sum of the roots to this equation is 2014. Because each value of x corresponds to a different one of a, b, and c by definition of , . Conversely, if , they must be the solutions to for some real and . Clearly, then, must all lie on the line . Hence, our setting produces a valid infinite set of points.
Note: We could have let , where a, b, and c are arbitrary constants. (a is nonzero.)