a) Answer: $1$ If all of them are liars, each person must have exactly one or three friends, so the total degree is odd, a contradiction. On the other hand, we can construct an example where there are 1011 pairs of liars (each pair only know each other), along with a three-people clique of two liars and a knight where they all know one another. b) Answer: $1215$ Let a knook (a knight combined with a rook) denote a pair of a knight and a liar who are friends. Let $k$ be the total number of knooks, and let $N$ and $L$ be the set of knights and liars, respectively. We can see that $2|N| = k \leq 3|L|$, so $|N| \leq \frac{3}{5}(2025) = 1215$. On the other hand, we can construct an example where people are divided into groups of five: three knights and two liars; six knooks among people in the same group are the only friendship.