Question 86
For a given natural number $k$, let $n_{k}$ be the smallest natural number such that there exists a finite set $A$ of integers with the following properties:
- for every $a \in A$ there exist $x, y \in A$ (not necessarily distinct) such that
$$
n_{k} \mid a-x-y
$$
- there is no subset $B$ of the set $A$ such that $|B| \leqslant k$ and $n_{k} \mid \sum_{b \in B} b$.
Prove that for all $k, k \geqslant 3$, we have
$$
n_{k}\lt\left(\frac{13}{8}\right)^{k+2}
$$
- for every $a \in A$ there exist $x, y \in A$ (not necessarily distinct) such that
$$
n_{k} \mid a-x-y
$$
- there is no subset $B$ of the set $A$ such that $|B| \leqslant k$ and $n_{k} \mid \sum_{b \in B} b$.
Prove that for all $k, k \geqslant 3$, we have
$$
n_{k}\lt\left(\frac{13}{8}\right)^{k+2}
$$