One of the questions raised by W.K. Hayman in his “Research problems in function theory” (1967; Zbl 0158.063) was to find if \(\lambda f+(1-\lambda)g\) is starlike univalent in the unit disk when \(f\) and \(g\) are convex. T. H. MacGregor showed that this combination was univalent for \(| z|<1/\sqrt{2}\) but need not be in any larger disk. In this paper the following is proved: Theorem. If \(f\) and \(g\) are starlike of order 1/2 and \(0\leq\lambda\leq 1\), then \(\lambda f+(1- \lambda)g\) is starlike univalent in \(| z|<1/(\sqrt{\lambda} + \sqrt{1-\lambda})\). The result is sharp for each \(\lambda\) and cannot be improved even if \(f\) and \(g\) are restricted to be in the class of convex functions. The authors state that this theorem improves on an earlier result of the first author but has a different (simpler) proof.