Question 3
Let $a_{1}, a_{2}, \ldots, a_{n}$ be positive real numbers, and let $S_{k}$ be the sum of products of $a_{1}, a_{2}, \ldots, a_{n}$ taken $k$ at a time.
Show that
$$
S_{k} S_{n-k} \geq \binom{ n}{k}^{2} a_{1} a_{2} \ldots a_{n}, \quad \text{ for } \quad k=1,2, \ldots, n-1
$$
Show that
$$
S_{k} S_{n-k} \geq \binom{ n}{k}^{2} a_{1} a_{2} \ldots a_{n}, \quad \text{ for } \quad k=1,2, \ldots, n-1
$$