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
1. **Restate the problem**: We need to determine if \( n \) must be even given that a 2-fair carpool system is possible but a 1-fair carpool system is not possible. 

2. **Define the terms**:
   - \( \chi(p_i) \) is the capacity of the car driven by \( p_i \).
   - A 2-fair carpool system means each MOPper drives exactly 2 times, and all cars are full on each day.
   - A 1-fair carpool system means each MOPper drives exactly 1 time, and all cars are full on each day.

3. **Example construction**: Consider \( n = 23 \) with the following car capacities:
   - \( \chi_1, \chi_2, \chi_3 = 2 \)
   - \( \chi_4, \chi_5 = 3 \)
   - \( \chi_6, \chi_7, \chi_8 = 7 \)
   - \( \chi_9, \chi_{10}, \chi_{11} = 8 \)
   - \( \chi_{12} = 17 \)
   - \( \chi_{13} = 18 \)
   - \( \chi_i = 23 \) for all \( i = 14, \ldots, 23 \)

4. **Construct a 2-fair carpool system**:
   - Day 1: \( p_1, p_2, p_3, p_{12} \) (capacities sum to 23)
   - Day 2: \( p_4, p_5, p_{12} \) (capacities sum to 23)
   - Day 3: \( p_1, p_4, p_{13} \) (capacities sum to 23)
   - Day 4: \( p_2, p_5, p_{13} \) (capacities sum to 23)
   - Day 5: \( p_3, p_6, p_7, p_8 \) (capacities sum to 23)
   - Day 6: \( p_6, p_9, p_{10} \) (capacities sum to 23)
   - Day 7: \( p_7, p_9, p_{11} \) (capacities sum to 23)
   - Day 8: \( p_8, p_{10}, p_{11} \) (capacities sum to 23)
   - Days 9-28: \( p_i \) where \( i = 14, 14, 15, 15, \ldots, 23, 23 \) (each drives twice)

5. **Show that a 1-fair carpool system is not possible**:
   - When \( p_{12} \) drives, the remaining cars must fit 6 people. This requires either \( p_1, p_2, p_3 \) all driving or \( p_4, p_5 \) all driving.
   - When \( p_{13} \) drives, the remaining cars must fit 5 people. This requires one of \( p_1, p_2, p_3 \) driving and one of \( p_4, p_5 \) driving.
   - This setup forces someone in \( \{p_1, p_2, p_3, p_4, p_5\} \) to drive twice, contradicting the 1-fair requirement.

6. **Conclusion**: Since we have constructed a scenario where \( n = 23 \) (an odd number) allows a 2-fair carpool system but not a 1-fair carpool system, \( n \) does not have to be even.

The final answer is \( \boxed{\text{NO}} \)