This problem is from the 2003 IMO, the third level of the American Mathematics Competition. I do not even know how to approach this.

"Determine all paris (a,b) such that $\displaystyle \frac{a^2}{2ab^2-b^3+1}$ is a positive integer."

The only assertion I can make is that $\displaystyle 2ab^2-b^3+1>0$.