Are there five complex numbers satisfying the following equalities? Can anyone help on the following question? 
Are there five complex numbers $z_{1}$, $z_{2}$ , $z_{3}$ , $z_{4}$
and $z_{5}$ with $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|+\left|z_{5}\right|=1$
such that the smallest among $\left|z_{1}\right|+\left|z_{2}\right|-\left|z_{1}+z_{2}\right|$,
$\left|z_{1}\right|+\left|z_{3}\right|-\left|z_{1}+z_{3}\right|$,
$\left|z_{1}\right|+\left|z_{4}\right|-\left|z_{1}+z_{4}\right|$,
$\left|z_{1}\right|+\left|z_{5}\right|-\left|z_{1}+z_{5}\right|$,
$\left|z_{2}\right|+\left|z_{3}\right|-\left|z_{2}+z_{3}\right|$,
$\left|z_{2}\right|+\left|z_{4}\right|-\left|z_{2}+z_{4}\right|$,
$\left|z_{2}\right|+\left|z_{5}\right|-\left|z_{2}+z_{5}\right|$,
$\left|z_{3}\right|+\left|z_{4}\right|-\left|z_{3}+z_{4}\right|$,
$\left|z_{3}\right|+\left|z_{5}\right|-\left|z_{3}+z_{5}\right|$
and $\left|z_{4}\right|+\left|z_{5}\right|-\left|z_{4}+z_{5}\right|$is
greater than $8/25$?
Thanks!

Suppose you have solutions and express $z_i$ as $r_i e^{\theta_i}$.
(I use $s_i = \sin( \theta_i )$ and $c_i = \sin( \theta_i )$ to make notations shorter)
Then$$\begin{align*}
|z_i| + |z_j| - |z_i + z_j| & = |r_i e^{\theta_i}| + |r_j e^{\theta_j}| - |r_i e^{\theta_i} + r_j e^{\theta_j}| \\
    & = r_i + r_j - |r_i(c_i +is_i) + r_j(c_j +is_j) | \\
    & = r_i + r_j - |(r_ic_i+r_jc_j) + i(r_is_i+r_js_j)| \\
    & = r_i + r_j - \sqrt{(r_ic_i+r_jc_j)^2 + (r_is_i+r_js_j)^2} \\
    & = r_i + r_j - \sqrt{ ( r_i^2c_i^2 + 2r_ir_jc_ic_j + r_j^2c_j^2 ) + ( r_i^2s_i^2 + 2r_ir_js_is_j + r_j^2s_j^2 ) } \\
    & = r_i + r_j - \sqrt{ r_i^2( c_i^2 + s_i ^2 ) + r_j^2(c_j^2 + s_j^2) + 2r_ir_j(c_ic_j+s_is_j) } \\
|z_i| + |z_j| - |z_i + z_j| & = r_i + r_j - \sqrt{r_i^2 + r_j^2+2r_ir_j\cos(\theta_i-\theta_j) }
\end{align*} $$
Then
$$\begin{align*}
|z_i| + |z_j| - |z_i + z_j| > \frac{8}{25} & \Leftrightarrow r_i + r_j - \sqrt{r_i^2 + r_j^2+2r_ir_j\cos(\theta_i-\theta_j) } > \frac{8}{25} \\
 & \Leftrightarrow r_i + r_j - \frac{8}{25} > \sqrt{r_i^2 + r_j^2+2r_ir_j\cos(\theta_i-\theta_j) } \\
    & \Leftrightarrow ( r_i + r_j - \frac{8}{25} ) ^ 2 > r_i^2 + r_j^2+2r_ir_j\cos(\theta_i-\theta_j) \\
    & \Leftrightarrow r_i^2 + r_j^2 + (\frac{8}{25})^2 + 2r_ir_j - 2\frac{8}{25} r_i - 2\frac{8}{25} r_j > r_i^2 + r_j^2+2r_ir_j\cos(\theta_i-\theta_j) \\
    & \Leftrightarrow (\frac{8}{25})^2 + 2r_ir_j - 2\frac{8}{25} r_i - 2\frac{8}{25} r_j > 2r_ir_j\cos(\theta_i-\theta_j) \\
    & \Leftrightarrow (\frac{8}{25})^2 + 2r_ir_j( 1 - \cos(\theta_i-\theta_j) ) - 2\frac{8}{25} r_i - 2\frac{8}{25} r_j > 0 \\
\end{align*} $$
I might update later but that's all I have for now :/
But I would try to express $r_i$ as a function of $r_{\sigma(i)}$ with $\sigma$ a permutation. And by doing that, you would probably get an ugly way of calculating any $r_j$ from all the $\theta_k$ where $j,k \in \{\sigma(i)^n, n\in \mathbb{N}\}$.
