CGMO 2007 Problem 2

Let $ABC$ be an acute triangle. Points $D$ , $E$ , and $F$ lie on segments $BC$ , $CA$ , and $AB$ , respectively, and each of the three segments $AD$ , $BE$ , and $CF$ contains the circumcenter of $ABC$ . Prove that if any two of the ratios $\frac{BD}{DC}$ , $\frac{CE}{EA}$ , $\frac{AF}{FB}$ , $\frac{BF}{FA}$ , $\frac{AE}{EC}$ , $\frac{CD}{DB}$ are integers, then triangle $ABC$ is isosceles.