Question 16
Let $a_1, a_2, \dots, a_n$ be positive numbers and $A$ their arithmetic mean. Prove that
$$
A^{n-1} \geq \frac{1}{n} \sum_{i=1}^{n} a_1 \cdots \hat{a}_i \cdots a_n, \quad \text{where } \hat{a}_i \text{ denotes the term } a_i \text{ is omitted.}
$$
$$
A^{n-1} \geq \frac{1}{n} \sum_{i=1}^{n} a_1 \cdots \hat{a}_i \cdots a_n, \quad \text{where } \hat{a}_i \text{ denotes the term } a_i \text{ is omitted.}
$$