Question 133
Given an integer $n \ge 3$, prove that there exists a set $S$ of $n$ distinct positive integers such that for any two distinct nonempty subsets $A$ and $B$ of $S$, the numbers
$$
\frac{\sum_{x \in A} x}{|A|}, \quad \frac{\sum_{x \in B} x}{|B|}
$$
are two coprime composite integers, where $\sum_{x \in X} x$ denotes the sum of all elements of a finite set $X$, and $|X|$ denotes the cardinality of $X$.
$$
\frac{\sum_{x \in A} x}{|A|}, \quad \frac{\sum_{x \in B} x}{|B|}
$$
are two coprime composite integers, where $\sum_{x \in X} x$ denotes the sum of all elements of a finite set $X$, and $|X|$ denotes the cardinality of $X$.