Super hard complex numbers problem: there do not exist $n>1$ complex numbers $z_1, z_2, \ldots, z_n$, no two equal, such that for all $1 \le k \le n$

First reduce to the case where all of the $z_i$ are non-zero.

Define polynomials $p,q,r$ by $p(z)=\prod_{i=1}^n(z-z_i)$ and $q(z)=2z\frac{dp}{dz}$ and $r(z)=\prod_{i=1}^n(z+z_i).$ Then your equations are equivalent to $q(z_k)=r(z_k)$ for $1\leq k\leq n.$ This implies $q+(-1)^np=r:$ both sides match at $n$ points and at zero. But then the leading coefficients only match if $2n+(-1)^n=1,$ which forces $n\leq 1.$