optimization of non-centrality parameter

Question

minimize $\sum_{i=1}^{k}\alpha_{i}^{2}$ subject to $\sum_{i=1}^{k}\alpha_{i}=0$ and $\max_{1\leq i<j\leq k}|\alpha_{i}-\alpha_{j}|\geq \delta$. I think the solution is $\alpha_{i} = -\frac{\delta}{2}$, $\alpha_{j} = \frac{\delta}{2}$ and $\alpha_{l} = 0$ for all $l\neq i, j$. I need to prove. Thanks