APMO

(i) Let $S$ be a subset of persons satisfying conditions (a), (b) and (c). Let $x\in S$ be one who knows the maximum number of persons in $S$. Assume that $x$ knows $x_1,x_2,\dots ,x_n$. By (b), $x_i$ and $x_j$ are strangers if $i\ne j$. For each $x_i$, let $N_i$ be the set of persons in $S$ who know $x_i$ but not $x$. Note that, for $i\ne j,N_i$ has no person in common with $N_j$, otherwise there would be more than one person knowing $x_i$ and $x_j$, contradicting (c). By (a) we may assume that $N_1$ is not empty. Let $y_1\in N_1$. By (c), for each $k>1$, there is exactly one person $y_k$ in $N_k$ who knows $y_1$. This means that $y_1$ knows $n$ persons, namely $x_1,y_2,\dots ,y_n$. Because $n$ is the maximal number of persons in $S$ a person in $S$ can know, $y_1$ knows exactly $n$ persons in $S$. By precisely the same reasoning we find that each person in $N_i$, $i=1,2,\dots ,n$, knows exactly $n$ persons in $S$. Letting $y_1$ take the role of $x$ in our argument, we see that also each $x_i$ knows exactly $n$ persons. Note that, by (c), every person in $S$ other than $x,x_1,\dots ,x_n$, must be in some $N_j$. Therefore every person in $S$ knows exactly $n$ persons in $S$ and thus has the same number of acquaintances in $S$.

(ii) To maximize the number of subsets, we have to minimize the size of each group. The smallest possible subset is one in which every person knows exactly two persons, and hence there must be exactly five persons in the subset, forming a cycle where two persons stand side by side only if they know each other. Therefore the maximum possible number of subsets is $1990/5=398$.