Question 129
Let $n \ge 2$ be a fixed positive integer. The set $M$ consists of $2n^2 - 3n + 2$ positive rational numbers. Prove that there exists an $n$-element subset $A$ of $M$ with the following property: for every $k \in \mathbb{N}$, $2 \le k \le n$, the sum of arbitrary $k$ (not necessarily distinct) elements of $A$ is not an element of $A$.