5. The Generalized Pigeonhole Principle states that for any function f from a finite set X to a finite set Y and for any positive integer k, if n(X) > k · n(Y), then there is some y in Y such that y is the image of at least k + 1 distinct elements of X.
Call the six people A, B, C, D, E and F. Let X be the set of people other than person A (so X={B,C,D,E,F} and Y be the set {friend of A, enemy of A}. Since n(X) = 5 and n(Y) = 2 we can apply the generalized pigeonhole principle with k = 2 which shows that either at least three people must be friends with person A or at least three people must be enemies with person A.
Suppose (without loss of generality) that people B, C and D are friends with person A. Use the possible relationships between people B, C and D to answer the rest of the question.