Question 5
For each integer $n \geq 2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$, we have
$$
\frac{a_1^2+\cdots+a_n^2}{n} \geq\left(\frac{a_1+\cdots+a_n}{n}\right)^2+C_n \cdot\left(a_1-a_n\right)^2
$$
$$
\frac{a_1^2+\cdots+a_n^2}{n} \geq\left(\frac{a_1+\cdots+a_n}{n}\right)^2+C_n \cdot\left(a_1-a_n\right)^2
$$