Question 74
Weights of $1\ \mathrm{g}$, $2\ \mathrm{g}$, $\ldots$, $200\ \mathrm{g}$ are placed on the two pans of a balance such that on each pan there are $100$ weights and the balance is in equilibrium. Prove that one can swap $50$ weights from one pan with $50$ weights from the other pan such that the balance remains in equilibrium.