RS-SMO 2018 Problem 6

For each positive integer $k$ , let $n_k$ be the smallest positive integer such that there exists a finite set $A$ of integers satisfy the following properties:

• For every $a\in A$ , there exists $x,y\in A$ (not necessary distinct) that $$n_k\mid a-x-y$$ [/*]
• There's no subset $B$ of $A$ that $|B|\leq k$ and $$n_k\mid \sum_{b\in B}{b}.$$

Show that for all positive integers $k\geq 3$ , we've $$n_k<\Big( \frac{13}{8}\Big)^{k+2}.$$