The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$. (Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.) (a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?