IMO 1989 LL USA104

Origin: USA

Problem

For each nonzero complex number z, let arg z be the unique real number t such that −\pi < t \leq\pi and z = |z|(cos t + ı sin t). Given a real number c > 0 and a complex number z ̸= 0 with arg z ̸= \pi, define B(c, z) = {b \inR | |w −z| < b ⇒| arg w −arg z| < c}. Determine necessary and sufficient conditions, in terms of c and z, such that B(c, z) has a maximum element, and determine what this maximum element is in this case.