$\textbf{Part 1: }$ Does not exist a partition with the requested condition for $n\le2$.

$\textbf{1.a: }$ For $n=1$: obvious.

$\textbf{1.b: }$ For $n=2$:

Consider the set $B=\{1,2,...,7,8\}$.

$1+2+...+7+8=36$.

Consider a partition of $A$ in $2$ sets $A_1,A_2$.

Denote $B_1$=$A_1$$\cap B$$;\;B_2$

=$A_2$$\cap B$