Putnam 2000, Problem A-5

Three distinct points with integer coordinates lie in the plane on a circle of radius $r > 0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.

Solution:
Let \(a,b,c\) be the distances between the points. Then the area of the triangle with the three points as vertices is \(abc/4r\). On the other hand, the area of a triangle whose vertices have integer coordinates is at least \(1/2\) (for example, by Pick's Theorem). Thus \(abc/4r \geq 1/2\), and so  
\[
\max\{a,b,c\} \geq (abc)^{1/3} \geq (2r)^{1/3} > r^{1/3}.
\]