Question 174
A party is attended by $n$ people. Assume that there are at most $n$ pairs of friendships among them, and any two people shake hands at the party if and only if they have a common friend at the party.
Suppose $m$ is a positive integer satisfying $m \ge 3$ and $n \le m^3$. Prove that there exists a person $A$ such that the number of people $A$ has shaken hands with at this party does not exceed $(m-1)$ times the number of $A$'s friends.
Suppose $m$ is a positive integer satisfying $m \ge 3$ and $n \le m^3$. Prove that there exists a person $A$ such that the number of people $A$ has shaken hands with at this party does not exceed $(m-1)$ times the number of $A$'s friends.