Johan Gunardi wrote: Determine all integers $ n > 1$ for which the inequality \[ x_1^2 + x_2^2 + \ldots + x_n^2\ge(x_1 + x_2 + \ldots + x_{n - 1})x_n\] holds for all real $ x_1,x_2,\ldots,x_n$. Id est, it must be $ \left(\sum_{i = 1}^{n - 1}x_i\right)^2\leq4\sum_{i = 1}^{n - 1}x_{i}^2,$ which after assumption $ x_1 = x_2 = ... = x_{n - 1} = 1$ gives $ n\in\{2,3,4,5\}.$