Problem 3. Determine the complex numbers $z$ for which the relation holds

$$
|z|+|z-5 i|=|z-2 i|+|z-3 i|
$$
Solution. We have

$$
|z-2 i|=\left|\frac{2}{5}(z-5 i)+\frac{3}{5} z\right| \leq \frac{2}{5}|z-5 i|+\frac{3}{5}|z|
$$

Similarly,

$$
|z-3 i|=\left|\frac{3}{5}(z-5 i)+\frac{2}{5} z\right| \leq \frac{3}{5}|z-5 i|+\frac{2}{5}|z|
$$

from which we get

$$
|z|+|z-5 i| \geq|z-2 i|+|z-3 i|
$$

Equality holds when there exists $\lambda \geq 0$ such that $z-5 i=\lambda z$, from which we deduce that $z=a i$, with $a \in(-\infty, 0] \cup[5,+\infty)$. $(3 \text{p})$