$$1.)\ \ (n-2)\sum\_{k=1}^{n}|z\_k|^2+|\sum\_{k=1}^{n}z\_k|^2=\sum\_{1\leq k < s \leq n }|z\_k+z\_s|^2$$

$$2.)\ \ n\sum\_{k=1}^{n}|z\_k|^2-|\sum\_{k=1}^{n}z\_k|^2=\sum\_{1 \leq k < s \leq n}|z\_k-z\_s|^2$$
These first two seem closely related.
$$3.) \ \ \ 2(|z\_1|^n+|z\_2|^n)\leq|z\_1+z\_2|^n+|z\_1-z\_2|^n\leq 2^{n-1}(|z\_1|^n+|z\_2|^n),n \geq 2$$

I've tried many things from trig, exponential substitutions\representations of complex numbers, all to no avail, because of a lack of an idea on solving these. Help is very much appreciated.