Question 4
Let $a_1,a_2,\ldots,a_n$ be positive numbers and $A$ their arithmetic mean. Prove that
$$A^{n-1}\ge\frac1n\sum_{i=1}^n a_1\cdots\widehat{a_i}\cdots a_n,$$
where $\widehat{a_i}$ denotes that the term $a_i$ is omitted.
$$A^{n-1}\ge\frac1n\sum_{i=1}^n a_1\cdots\widehat{a_i}\cdots a_n,$$
where $\widehat{a_i}$ denotes that the term $a_i$ is omitted.