http://www.artofproblemsolving.com/Forum/viewtopic.php?f=51&t=544356&hilit=complex this link should help you.After this it remains to see that $\pi<6$ hence your claim is done.

Split the plane into three sectors (using rays starting from the origin, and going in the directions of 0, 2pi/3, and 4pi/3 radians.) Treat the $Zi$ as vectors, and put them all in the plane starting at the origin. By pidgeonhole, one of our three sectors has at least 1/3 of the total length of the vectors. Wlog this sector is the one with rays in the 0 radian and 2pi/3 radian directions as its boundaries. Now consider all the vectors in this sector. The sum of these vectors must have a magnitude of at least 1/6 going in the direction of pi/3 radians, because each vector contributes at least half its magnitude in that direction.