BALKAN-SL 2007 G3

Let $A_{1}A_{2}A_{3}A_{4}A_{5}$ be a convex pentagon, such that $$[A_{1}A_{2}A_{3}] = [A_{2}A_{3}A_{4}] = [A_{3}A_{4}A_{5}] = [A_{4}A_{5}A_{1}] = [A_{5}A_{1}A_{2}].$$

Prove that there exists a point $M$ in the plane of the pentagon such that $$[A_{1}MA_{2}] = [A_{2}MA_{3}] = [A_{3}MA_{4}] = [A_{4}MA_{5}] = [A_{5}MA_{1}].$$

Here $[XYZ]$ stands for the area of the triangle $\Delta XYZ$.