Question 96
Consider $n$ weights, $n \ge 2$, of masses $m_1, m_2, \dots, m_n$, where $m_k$ are positive integers such that $1 \le m_k \le k$ for all $k \in \{1, 2, \dots, n\}$. Prove that we can place the weights on the two pans of a balance such that the pans stay in equilibrium if and only if the number $m_1 + m_2 + \dots + m_n$ is even.