Putnam 1971 
Topic: Geometry Number Theory

Problem: The three vertices of a triangle of sides $a, b, \text{and } c$ are lattice points and lie on a circle of radius $R$. Find the minimum value of $abc. (Lattice points are points in the Euclidean plane with integral coordinates.)

Solution: For a triangle with sides $a, b, c$, area $= A$ and circumradius $= R$ we have $abc = 4RA$. But if the vertices are lattice points the determinant formula (or Pick’s Theorem or direct calculation) for the area shows that $2A$ is an integer. Hence $2A \geq 1$, so that $abc \geq \boxed{2R}$. To obtain the formula $abc = 4RA$ note that if $\alpha$ is the angle opposite side $a$, then side $a$ subtends an angle $2\alpha$ at the center and $a = 2R \sin \alpha$, $A = \frac{1}{2} bc \sin \alpha$.