Question 84
Consider $n$ weights, $n \geq 2$, with masses $m_{1}, m_{2}, \ldots, m_{n}$, where $m_{k} \in \mathbb{N}$, $1 \leq m_{k} \leq k$ for every $k \in \{1,2, \ldots, n\}$. Prove that we can place the weights on the pans of a balance so that it stays in equilibrium if and only if $m_{1}+m_{2}+\ldots+m_{n}$ is an even number.