Prove $(\alpha_1 + ........ + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ....... + \alpha_n^2)$ For any real numbers $\alpha_1, \alpha_2, . . . . ., \alpha_n$,
$$(\alpha_1 + ...... + \alpha_n)^2 ≤ n \cdot (\alpha_1^2 + ..... + \alpha_n^2)$$
And when is the inequality strict?

It's just
$$\sum_{1\leq i<j\leq n}(\alpha_i-\alpha_j)^2\geq0.$$
Done!
