Logic & Reasoning Question At a track meet, every group of $n$ participants shares exactly one common friend.
Suppose runner $P$ has the largest number of friends. Determine how many friends $P$ has.

Assume for this question:


$n \in N$ such that $n ≥ 3$
Friends are mutual, EX: if $X$ is friends with $Y$, then $Y$ is friends with $X$
No one is friends with themselves
a group of runners has a "common friend" $a$ iff each runner in the group is friends with $a$


Prove the answer.

Update for Clarification


$n$ is the size of the group, not the total number of attendees.
There are an arbitrary number of groups
The answer should be expressed in terms of $n$
If $X$ is friends with $Y$ then both $X$ and $Y$ are at the trackmeet
Assume there are at least $n + 1$ attendees.


Thank you very much for your effort so far @fleablood
P can't like as many as n people.

Label n of P's friends as $a_1,... a_n$.

$(P, a_1, ...., a_{n-1})$ have P as the mutual friend.  So $(a_1,..., a_{n-1})$ have no mutual friend.  So $(a_1, ..., a_n)$ have $a_n$ as a mutual friend.  So $(P, a_2, ...., a_n)$ have both P and $a_n$ as mutual friends.  Impossible.

So P has n-1 friends or fewer.

But as any group of n people have 1 mutual friend, that mutual friend has at least n-1 friends.  So P has at least n-1 friends.

So P has n-1 friends.
P can't like as many as n people.

Label n of P's friends as $a_1,... a_n$.

$(P, a_1, ...., a_{n-1})$ have P as the mutual friend.  So $(a_1,..., a_{n-1})$ have no mutual friend.  So $(a_1, ..., a_n)$ have $a_n$ as a mutual friend.  So $(P, a_2, ...., a_n)$ have both P and $a_n$ as mutual friends.  Impossible.

So P has n-1 friends or fewer.

But as any group of n people have 1 mutual friend, that mutual friend has at least n-1 friends.  So P has at least n-1 friends.

So P has n-1 friends.