$$\begin{array}{ll} \text{extremize} & x_1 \cdot x_2 \cdots x_n\\ \text{subject to} & (1-x_1) \cdot (1-x_2) \cdots (1-x_n) = a\\ & x_1 + \cdots + x_n = 0\\ & -1 \leq x_i \leq 1\end{array}$$
Without $-1 \leq x_i \leq 1$ one might use the method of Lagrange multipliers, but with it I have no idea what to do. Any help will be appreciated!