Question 52
Let $S = \{1, 2, \dots, 100\}$. Consider a partition of $S$ into $S_1, S_2, \dots, S_n$ for some $n$, i.e. $S_i$ are nonempty, pairwise disjoint and $S = \bigcup_{i=1}^n S_i$. Let $a_i$ be the average of elements of the set $S_i$. Define the score of this partition by
$$
\frac{a_1 + a_2 + \dots + a_n}{n}.
$$
Among all $n$ and partitions of $S$, determine the minimum possible score.
$$
\frac{a_1 + a_2 + \dots + a_n}{n}.
$$
Among all $n$ and partitions of $S$, determine the minimum possible score.