ELMO SL 2017 C5

There are $n$ MOPpers $p_1,...,p_n$ designing a carpool system to attend their morning class. Each $p_i$ 's car fits $\chi (p_i)$ people ( $\chi : \{p_1,...,p_n\} \to \{1,2,...,n\}$ ). A $c$ -fair carpool system is an assignment of one or more drivers on each of several days, such that each MOPper drives $c$ times, and all cars are full on each day. (More precisely, it is a sequence of sets $(S_1, ...,S_m)$ such that $|\{k: p_i\in S_k\}|=c$ and $\sum_{x\in S_j} \chi(x) = n$ for all $i,j$ . ) Suppose it turns out that a $2$ -fair carpool system is possible but not a $1$ -fair carpool system. Must $n$ be even?

[i]Proposed by Nathan Ramesh and Palmer Mebane