Let $a_{1}, \ldots, a_{n}$ be given constants, not all equal.

Use the Lagrangian sufficiency theorem, which you should state clearly, without proof, to minimize $\sum_{i=1}^{n} x_{i}^{2}$ subject to the two constraints $\sum_{i=1}^{n} x_{i}=1, \sum_{i=1}^{n} a_{i} x_{i}=0 .$