Question 17
Let $A = \{z \in \mathbb{C} \mid |z| = 1\}$.
a) Prove that $(|z+1| - \sqrt{2})(|z-1| - \sqrt{2}) \le 0$, for all $z \in A$.
b) Prove that, for any $z_1, z_2, \dots, z_{12} \in A$, one can choose the signs "\pm" such that
$$
\sum_{k=1}^{12} |z_k \pm 1| \lt 17.
$$
a) Prove that $(|z+1| - \sqrt{2})(|z-1| - \sqrt{2}) \le 0$, for all $z \in A$.
b) Prove that, for any $z_1, z_2, \dots, z_{12} \in A$, one can choose the signs "\pm" such that
$$
\sum_{k=1}^{12} |z_k \pm 1| \lt 17.
$$