74th NMO Selection Tests for JBMO

To show that $\frac{1}{2}$ is the required minimum, notice that:

$1\le m_A\le n$, for every nonempty subset $A\subset M$, (1); any two one-element subsets are connected with a sequence of subsets, (2).

Indeed, for $k<p$, consider the sequence $\{k\},\{k,k+1\},\{k+1\},\{k+1,k+2\},\dots ,\{p-1,p\},\{p\}$. The absolute value of the difference between the arithmetic means of any two consecutive subsets from this sequence is $\frac{1}{2}$.

Let $A$ and $B$ be two different subsets and $m_A,m_B$ be the arithmetic means of their elements, respectively. Denote by $n_A,n_B$ the closest integers to $m_A$ and $m_B$, respectively. From (1) we have $n_A,n_B\in [1,n]$ and it is enough to consider the sequence $A,s_{n_A,n_B},B$, where $s_{n_A,n_B}$ is the sequence described for the subsets $\{n_A\}$ and $\{n_B\}$.