Problem:
For certain ordered pairs of real numbers, the system of equations
has at least one solution, and each solution is an ordered pair of integers. How many such ordered pairs are there?
Solution:
The equation is that of the circle with center and radius , and is an equation of a line. The problem statement is equivalent to requiring that the line and the circle intersect and that each intersection be a lattice point. There are lattice points on the circle: . Any pair of these points determines a line that intersects the circle in those two points. There are such pairs. Also, at each of the twelve points the tangent line intersects the circle at only that point. Thus, there are lines that intersect the circle and do so only at lattice points. Any such line can be uniquely written in the form if and only if the line does not contain the origin. But of the lines do contain the origin. These are the lines determined by diametrically opposite points. It follows that there are ordered pairs of real numbers for which the given system has at least one solution and has only integer solutions.
The problems on this page are the property of the MAA's American Mathematics Competitions