For fixed $z_i$s inside the unit disc, can we always choose $a_i$s such that $\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3$? Let $z_1,z_2,\ldots,z_n$ be complex number such that $|z_i|<1$ for all $i=1,2,\ldots,n$. Show that we can choose $a_i \in\{-1,1\}$, $i=1,2,\ldots,n$ such that
$$\left|\sum_{i=1}^n a_iz_i\right|<\sqrt3.$$

I was not able to think it through properly, but here's a sketch:
Use induction as suggested by Berci, but with a little twist. The main idea is that for two numbers $z_i$ and $z_j$ such that $|z_i| < 1$ and $|z_j| < 1$ we can obtain $|z_i\pm z_j| < 1$ as long as some angle (out of four) between them (the difference of arguments) is smaller or equal than $\frac{\pi}{3}$. However, as long as we have 3 or more numbers, we will be able to find such a pair.
Quick illustration of the lemma: $z_i$ is somewhere on the blue line, red cross is the $z_j$ and the violet is their sum. The point is that as long as the red cross belongs to darker green, the violet line will stay in light green region.
$\hspace{70pt}$
I don't know if I will find enough time to work out all the details, so should this idea suit you, feel free to use it.
Cheers!
