CGMO 2011 Problem 5

A real number $\alpha \geq 0$ is given. Find the smallest $\lambda = \lambda (\alpha ) > 0$ , such that for any complex numbers ${z_1},{z_2}$ and $0 \leq x \leq 1$ , if $\left| {{z_1}} \right| \leq \alpha \left| {{z_1} - {z_2}} \right|$ , then $\left| {{z_1} - x{z_2}} \right| \leq \lambda \left| {{z_1} - {z_2}} \right|$ .