Question 4
At an IMOTC party, all people have pairwise distinct ages. Some pairs of people are friends and friendship is mutual. Call a person *junior* if they are younger than all their friends, and *senior* if they are older than all their friends. A person with no friends is both *junior* and *senior*. A sequence of pairwise distinct people $A_1, \dots, A_m$ is called *photogenic* if:
* $A_1$ is junior,
* $A_m$ is senior, and
* $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \le i \le m-1$.
Let $k$ be a positive integer such that for every *photogenic* sequence $A_1, \dots, A_m$, $m$ is not divisible by $k$. Prove that the people at the party can be partitioned into $k$ groups so that no two people in the same group are friends.
* $A_1$ is junior,
* $A_m$ is senior, and
* $A_i$ and $A_{i+1}$ are friends, and $A_{i+1}$ is older than $A_i$ for all $1 \le i \le m-1$.
Let $k$ be a positive integer such that for every *photogenic* sequence $A_1, \dots, A_m$, $m$ is not divisible by $k$. Prove that the people at the party can be partitioned into $k$ groups so that no two people in the same group are friends.