Question 43
Consider inequality
$$
\left(x_{1}+\ldots+x_{n}\right)^{2} \geq 4\left(x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{n} x_{1}\right) .
$$
a. Determine for which $n \geq 3$ is true for each possible choice of positive real numbers $x_{1}, \ldots, x_{n}$.
b. Determine for which $n \geq 3$ is true for each possible choice of real numbers $x_{1}, \ldots, x_{n}$.
$$
\left(x_{1}+\ldots+x_{n}\right)^{2} \geq 4\left(x_{1} x_{2}+x_{2} x_{3}+\ldots+x_{n} x_{1}\right) .
$$
a. Determine for which $n \geq 3$ is true for each possible choice of positive real numbers $x_{1}, \ldots, x_{n}$.
b. Determine for which $n \geq 3$ is true for each possible choice of real numbers $x_{1}, \ldots, x_{n}$.