$max\Bigl\{\sum_{i=1}^{n}x_{i}^{2}:\sum_{i=1}^{n}x_{i}=1,x_{i}>\lambda,i=1,...n\Bigl\}= \\ max\Bigl\{\sum_{i=1}^{n-1}x_{i}^{2}+(1-\sum_{i=1}^{n-1}x_{i})^{2} : \sum_{i=1}^{n-1}x_{i} \leq 1-\lambda,x_{i}\geq\lambda, i=1,...,n-1 \Bigl\}= \\ (n-1)\lambda^{2}+(1-(n-1)\lambda)^{2}$
Any idea how to solve this? I tried Lagrange multipliers in both braces. The first one returns critical point at 1/n while the second at $\frac{1-\lambda}{n-1}$.
edit. $(1-\frac{2}{n})\frac{1}{n-1}<\lambda<\frac{1}{n}$
$\sum_{i=1}^{n}x_{i}^{2}=\sum_{i=1}^{n-1}x_{i}^{2}+x_{n}^{2}=\sum_{i=1}^{n-1}x_{i}^{2}+(1-\sum_{i=1}^{n-1}x_{i})^{2}$
