28. For which $n \geqslant 2$ is the inequality

$$
x_{1}^{2}+x_{2}^{2}+\ldots+x_{n}^{2} \geqslant x_{n}\left(x_{1}+x_{2}+\ldots+x_{n-1}\right)
$$

106
valid for all real values of the variables $x_{i}$?
83.28. Answer. The inequality is true only for $n=2,3,4,5$. Hint. Rewrite the inequality as follows:

$$
\left(x_{1}-\frac{x_{n}}{2}\right)^{2}+\left(x_{2}-\frac{x_{n}}{2}\right)^{2}+\ldots+\left(x_{n-1}-\frac{x_{n}}{2}\right)^{2}+(5-n) \frac{x_{n}^{2}}{4} \geqslant 0
$$