275. Maximum Number. Let a set of distinct complex numbers $z_{i}, i=1,2, \ldots, n$, be given, satisfying the inequality

$$
\min _{i \neq j}\left|z_{i}-z_{j}\right| \geqslant \max _{i}\left|z_{i}\right|
$$[^16]

Find the maximum possible $n$ and for this $n$ all sets satisfying the condition of the problem.
275. Let $\left|z_{m}\right|=\max _{i}\left|z_{i}\right|$. Then on the complex plane, all points $z_{i}$ will be located inside the circle $R$ of radius $\left|z_{m}\right|$ centered at the point $z=0$. Clearly, 6 points $z_{i}$, located on the circumference of $R$ and forming a regular hexagon, together with the point $z_{1}=0$ give $n=7$. If, however, $n \geqslant 7$ and some point $z \neq 0$ is not on the circumference of $R$, then by applying the cosine theorem, we will get that $\left|z_{i}-z_{j}\right|0$ and $0 \leqslant \delta<\frac{\pi}{3}$.

[P. Kornya, A. M. M., 76, 91 (January 1969).]